Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave and Conve
The entropy formula for the Ricci flow and its geometric applications
a rX iv:mat h /211159v1[ma t h.DG]11Nov22The entropy formula for the Ricci flow and its geometric applications Grisha Perelman ∗November 20,2007Introduction 1.The Ricci flow equation,introduced by Richard Hamilton [H 1],is the evolution equation d ∗St.Petersburg branch of Steklov Mathematical Institute,Fontanka 27,St.Petersburg191011,Russia.Email:perelman@pdmi.ras.ru or perelman@ ;I was partially supported by personal savings accumulated during my visits to the Courant Institute in the Fall of 1992,to the SUNY at Stony Brook in the Spring of 1993,and to the UC at Berkeley as a Miller Fellow in 1993-95.I’d like to thank everyone who worked to make those opportunities available to me.1in dimension four converge,modulo scaling,to metrics of constant positivecurvature.Without assumptions on curvature the long time behavior of the metricevolving by Ricciflow may be more complicated.In particular,as t ap-proaches somefinite time T,the curvatures may become arbitrarily large in some region while staying bounded in its complement.In such a case,it isuseful to look at the blow up of the solution for t close to T at a point where curvature is large(the time is scaled with the same factor as the metric ten-sor).Hamilton[H9]proved a convergence theorem,which implies that asubsequence of such scalings smoothly converges(modulo diffeomorphisms) to a complete solution to the Ricciflow whenever the curvatures of the scaledmetrics are uniformly bounded(on some time interval),and their injectivity radii at the origin are bounded away from zero;moreover,if the size of thescaled time interval goes to infinity,then the limit solution is ancient,thatis defined on a time interval of the form(−∞,T).In general it may be hard to analyze an arbitrary ancient solution.However,Ivey[I]and Hamilton[H4]proved that in dimension three,at the points where scalar curvatureis large,the negative part of the curvature tensor is small compared to the scalar curvature,and therefore the blow-up limits have necessarily nonneg-ative sectional curvature.On the other hand,Hamilton[H3]discovered a remarkable property of solutions with nonnegative curvature operator in ar-bitrary dimension,called a differential Harnack inequality,which allows,inparticular,to compare the curvatures of the solution at different points and different times.These results lead Hamilton to certain conjectures on thestructure of the blow-up limits in dimension three,see[H4,§26];the presentwork confirms them.The most natural way of forming a singularity infinite time is by pinchingan(almost)round cylindrical neck.In this case it is natural to make a surgery by cutting open the neck and gluing small caps to each of the boundaries,andthen to continue running the Ricciflow.The exact procedure was describedby Hamilton[H5]in the case of four-manifolds,satisfying certain curvature assumptions.He also expressed the hope that a similar procedure wouldwork in the three dimensional case,without any a priory assumptions,and that afterfinite number of surgeries,the Ricciflow would exist for all timet→∞,and be nonsingular,in the sense that the normalized curvatures ˜Rm(x,t)=tRm(x,t)would stay bounded.The topology of such nonsingular solutions was described by Hamilton[H6]to the extent sufficient to makesure that no counterexample to the Thurston geometrization conjecture can2occur among them.Thus,the implementation of Hamilton program would imply the geometrization conjecture for closed three-manifolds.In this paper we carry out some details of Hamilton program.The more technically complicated arguments,related to the surgery,will be discussed elsewhere.We have not been able to confirm Hamilton’s hope that the so-lution that exists for all time t→∞necessarily has bounded normalized curvature;still we are able to show that the region where this does not hold is locally collapsed with curvature bounded below;by our earlier(partly unpublished)work this is enough for topological conclusions.Our present work has also some applications to the Hamilton-Tian con-jecture concerning K¨a hler-Ricciflow on K¨a hler manifolds with positivefirst Chern class;these will be discussed in a separate paper.2.The Ricciflow has also been discussed in quantumfield theory,as an ap-proximation to the renormalization group(RG)flow for the two-dimensional nonlinearσ-model,see[Gaw,§3]and references therein.While my back-ground in quantum physics is insufficient to discuss this on a technical level, I would like to speculate on the Wilsonian picture of the RGflow.In this picture,t corresponds to the scale parameter;the larger is t,the larger is the distance scale and the smaller is the energy scale;to compute something on a lower energy scale one has to average the contributions of the degrees of freedom,corresponding to the higher energy scale.In other words,decreasing of t should correspond to looking at our Space through a microscope with higher resolution,where Space is now described not by some(riemannian or any other)metric,but by an hierarchy of riemannian metrics,connected by the Ricciflow equation.Note that we have a paradox here:the regions that appear to be far from each other at larger distance scale may become close at smaller distance scale;moreover,if we allow Ricci flow through singularities,the regions that are in different connected compo-nents at larger distance scale may become neighboring when viewed through microscope.Anyway,this connection between the Ricciflow and the RGflow sug-gests that Ricciflow must be gradient-like;the present work confirms this expectation.3.The paper is organized as follows.In§1we explain why Ricciflow can be regarded as a gradientflow.In§2,3we prove that Ricciflow,considered as a dynamical system on the space of riemannian metrics modulo diffeomor-phisms and scaling,has no nontrivial periodic orbits.The easy(and known)3case of metrics with negative minimum of scalar curvature is treated in§2; the other case is dealt with in§3,using our main monotonicity formula(3.4) and the Gaussian logarithmic Sobolev inequality,due to L.Gross.In§4we apply our monotonicity formula to prove that for a smooth solution on a finite time interval,the injectivity radius at each point is controlled by the curvatures at nearby points.This result removes the major stumbling block in Hamilton’s approach to geometrization.In§5we give an interpretation of our monotonicity formula in terms of the entropy for certain canonical ensemble.In§6we try to interpret the formal expressions,arising in the study of the Ricciflow,as the natural geometric quantities for a certain Riemannian manifold of potentially infinite dimension.The Bishop-Gromov relative volume comparison theorem for this particular manifold can in turn be interpreted as another monotonicity formula for the Ricciflow.This for-mula is rigorously proved in§7;it may be more useful than thefirst one in local considerations.In§8it is applied to obtain the injectivity radius control under somewhat different assumptions than in§4.In§9we consider one more way to localize the original monotonicity formula,this time using the differential Harnack inequality for the solutions of the conjugate heat equation,in the spirit of Li-Yau and Hamilton.The technique of§9and the logarithmic Sobolev inequality are then used in§10to show that Ricciflow can not quickly turn an almost euclidean region into a very curved one,no matter what happens far away.The results of sections1through10require no dimensional or curvature restrictions,and are not immediately related to Hamilton program for geometrization of three manifolds.The work on details of this program starts in§11,where we describe the ancient solutions with nonnegative curvature that may occur as blow-up limits offinite time singularities(they must satisfy a certain noncollaps-ing assumption,which,in the interpretation of§5,corresponds to having bounded entropy).Then in§12we describe the regions of high curvature under the assumption of almost nonnegative curvature,which is guaranteed to hold by the Hamilton and Ivey result,mentioned above.We also prove, under the same assumption,some results on the control of the curvatures forward and backward in time in terms of the curvature and volume at a given time in a given ball.Finally,in§13we give a brief sketch of the proof of geometrization conjecture.The subsections marked by*contain historical remarks and references. See also[Cao-C]for a relatively recent survey on the Ricciflow.41Ricciflow as a gradientflow1.1.Consider the functional F= M(R+|∇f|2)e−f dV for a riemannian metric g ij and a function f on a closed manifold M.Itsfirst variation can be expressed as follows:δF(v ij,h)= M e−f[−△v+∇i∇j v ij−R ij v ij−v ij∇i f∇j f+2<∇f,∇h>+(R+|∇f|2)(v/2−h)]= M e−f[−v ij(R ij+∇i∇j f)+(v/2−h)(2△f−|∇f|2+R)], whereδg ij=v ij,δf=h,v=g ij v ij.Notice that v/2−h vanishes identically iffthe measure dm=e−f dV is keptfixed.Therefore,the symmetric tensor −(R ij+∇i∇j f)is the L2gradient of the functional F m= M(R+|∇f|2)dm, where now f denotes log(dV/dm).Thus given a measure m,we may consider the gradientflow(g ij)t=−2(R ij+∇i∇j f)for F m.For general m thisflow may not exist even for short time;however,when it exists,it is just the Ricciflow,modified by a diffeomorphism.The remarkable fact here is that different choices of m lead to the sameflow,up to a diffeomorphism;that is, the choice of m is analogous to the choice of gauge.1.2Proposition.Suppose that the gradientflow for F m exists for t∈[0,T]. Then at t=0we have F m≤nNow we computeF t≥2n( (R+△f)e−f dV)2=2t1and t2,are called Ricci solitons.(Thus,if one considers Ricciflow as a dy-namical system on the space of riemannian metrics modulo diffeomorphism and scaling,then breathers and solitons correspond to periodic orbits and fixed points respectively).At each time the Ricci soliton metric satisfies an equation of the form R ij+cg ij+∇i b j+∇j b i=0,where c is a number and b i is a one-form;in particular,when b i=1log V=1dtV(2−n)/nλ −RdV≥n2V2/n[ |R ij+∇i∇j f−1( (R+△f)2e−f dV−( (R+△f)e−f dV)2)]≥0,nwhere f is the minimizer for F.72.4.The arguments above also show that there are no nontrivial(that is with non-constant Ricci curvature)steady or expanding Ricci solitons(on closed M).Indeed,the equality case in the chain of inequalities above requires that R+△f be constant on M;on the other hand,the Euler-Lagrange equation for the minimizer f is2△f−|∇f|2+R=const.Thus,△f−|∇f|2=const=0, because (△f−|∇f|2)e−f dV=0.Therefore,f is constant by the maximum principle.2.5*.A similar,but simpler proof of the results in this section,follows im-mediately from[H6,§2],where Hamilton checks that the minimum of RV22e−f dV,(3.1)restricted to f satisfying(4πτ)−nM,τt=−1(3.3)2τThe evolution equation for f can also be written as follows:2∗u=0,where u=(4πτ)−ng ij|2(4πτ)−n2τalong the Ricciflow.It is not hard to show that in the definition ofµthere always exists a smooth minimizer f(on a closed M).It is also clear that limτ→∞µ(g ij,τ)=+∞whenever thefirst eigenvalue of−4△+R is positive. Thus,our statement that there is no shrinking breathers other than gradient solitons,is implied by the followingClaim For an arbitrary metric g ij on a closed manifold M,the function µ(g ij,τ)is negative for smallτ>0and tends to zero asτtends to zero.Proof of the Claim.(sketch)Assume that¯τ>0is so small that Ricci flow starting from g ij exists on[0,¯τ].Let u=(4πτ)−n2τ−1g ij,fτ,12τ−1g ij”converge”to the euclidean metric,and if we couldextract a converging subsequence from fτ,we would get a function f on R n, such that R n(2π)−n2|∇f|2+f−n](2π)−n2−t)=µ(g ij(0),12)satisfiesR ij+∇i∇j f−g ij=0.Of course,this argument requires the existence of minimizer,and justification of the integration by parts;this is easy if M is closed,but can also be done with more efforts on some complete M,for instance when M is the Gaussian soliton.93.3*The no breathers theorem in dimension three was proved by Ivey[I]; in fact,he also ruled out nontrivial Ricci solitons;his proof uses the almost nonnegative curvature estimate,mentioned in the introduction.Logarithmic Sobolev inequalities is a vast area of research;see[G]for a survey and bibliography up to the year1992;the influence of the curvature was discussed by Bakry-Emery[B-Em].In the context of geometric evolution equations,the logarithmic Sobolev inequality occurs in Ecker[E1].4No local collapsing theorem IIn this section we present an application of the monotonicity formula(3.4) to the analysis of singularities of the Ricciflow.4.1.Let g ij(t)be a smooth solution to the Ricciflow(g ij)t=−2R ij on[0,T). We say that g ij(t)is locally collapsing at T,if there is a sequence of times t k→T and a sequence of metric balls B k=B(p k,r k)at times t k,such that r2k/t k is bounded,|Rm|(g ij(t k))≤r−2k in B k and r−n k V ol(B k)→0.Theorem.If M is closed and T<∞,then g ij(t)is not locally collapsing at T.Proof.Assume that there is a sequence of collapsing balls B k=B(p k,r k) at times t k→T.Then we claim thatµ(g ij(t k),r2k)→−∞.Indeed one(x,p k)r−1k)+c k,whereφis a function of one can take f k(x)=−logφ(dist tkvariable,equal1on[0,1/2],decreasing on[1/2,1],and very close to0on [1,∞),and c k is a constant;clearly c k→−∞as r−n k V ol(B k)→0.Therefore, applying the monotonicity formula(3.4),we getµ(g ij(0),t k+r2k)→−∞. However this is impossible,since t k+r2k is bounded.4.2.Definition We say that a metric g ij isκ-noncollapsed on the scaleρ,if every metric ball B of radius r<ρ,which satisfies|Rm|(x)≤r−2for every x∈B,has volume at leastκr n.It is clear that a limit ofκ-noncollapsed metrics on the scaleρis also κ-noncollapsed on the scaleρ;it is also clear thatα2g ij isκ-noncollapsed on the scaleαρwhenever g ij isκ-noncollapsed on the scaleρ.The theorem above essentially says that given a metric g ij on a closed manifold M and T<∞,one canfindκ=κ(g ij,T)>0,such that the solution g ij(t)to the Ricciflow starting at g ij isκ-noncollapsed on the scale T1/2for all t∈[0,T), provided it exists on this interval.Therefore,using the convergence theorem of Hamilton,we obtain the following10Corollary.Let g ij (t ),t ∈[0,T )be a solution to the Ricci flow on a closed manifold M,T <∞.Assume that for some sequences t k →T,p k ∈M and some constant C we have Q k =|Rm |(p k ,t k )→∞and |Rm |(x,t )≤CQ k ,whenever t <t k .Then (a subsequence of)the scalings of g ij (t k )at p k with factors Q k converges to a complete ancient solution to the Ricci flow,which is κ-noncollapsed on all scales for some κ>0.5A statistical analogyIn this section we show that the functional W ,introduced in section 3,is in a sense analogous to minus entropy.5.1Recall that the partition function for the canonical ensemble at tem-perature β−1is given by Z = exp (−βE )dω(E ),where ω(E )is a ”density of states”measure,which does not depend on β.Then one computes the average energy <E >=−∂(∂β)2log Z.Now fix a closed manifold M with a probability measure m ,and suppose that our system is described by a metric g ij (τ),which depends on the temper-ature τaccording to equation (g ij )τ=2(R ij +∇i ∇j f ),where dm =udV,u =(4πτ)−n 2)dm.(We do not discuss here what assumptions on g ij guarantee that the corre-sponding ”density of states”measure can be found)Then we compute<E >=−τ2 M(R +|∇f |2−n 2τg ij |2dmAlternatively,we could prescribe the evolution equations by replacing the t -derivatives by minus τ-derivatives in (3.3),and get the same formulas for Z,<E >,S,σ,with dm replaced by udV.Clearly,σis nonnegative;it vanishes only on a gradient shrinking soliton.<E >is nonnegative as well,whenever the flow exists for all sufficiently small τ>0(by proposition 1.2).Furthermore,if (a)u tends to a δ-function as τ→0,or (b)u is a limit of a sequence of functions u i ,such that each u i11tends to aδ-function asτ→τi>0,andτi→0,then S is also nonnegative. In case(a)all the quantities<E>,S,σtend to zero asτ→0,while in case (b),which may be interesting if g ij(τ)goes singular atτ=0,the entropy S may tend to a positive limit.If theflow is defined for all sufficiently largeτ(that is,we have an ancient solution to the Ricciflow,in Hamilton’s terminology),we may be interested in the behavior of the entropy S asτ→∞.A natural question is whether we have a gradient shrinking soliton whenever S stays bounded.5.2Remark.Heuristically,this statistical analogy is related to the de-scription of the renormalization groupflow,mentioned in the introduction: in the latter one obtains various quantities by averaging over higher energy states,whereas in the former those states are suppressed by the exponential factor.5.3*An entropy formula for the Ricciflow in dimension two was found by Chow[C];there seems to be no relation between his formula and ours.The interplay of statistical physics and(pseudo)-riemannian geometry occurs in the subject of Black Hole Thermodynamics,developed by Hawking et al.Unfortunately,this subject is beyond my understanding at the moment.6Riemannian formalism in potentially infi-nite dimensionsWhen one is talking of the canonical ensemble,one is usually considering an embedding of the system of interest into a much larger standard system of fixed temperature(thermostat).In this section we attempt to describe such an embedding using the formalism of Rimannian geometry.6.1Consider the manifold˜M=M×S N×R+with the following metric:˜g ij=g ij,˜gαβ=τgαβ,˜g00=N2N .It turns out that the components of the curvaturetensor of this metric coincide(modulo N−1)with the components of the matrix Harnack expression(and its traces),discovered by Hamilton[H3]. One can also compute that all the components of the Ricci tensor are equal12to zero(mod N−1).The heat equation and the conjugate heat equation on M can be interpreted via Laplace equation on˜M for functions and volume forms respectively:u satisfies the heat equation on M iff˜u(the extension of u to˜M constant along the S Nfibres)satisfies˜△˜u=0mod N−1;similarly,u satisfies the conjugate heat equation on M iff˜u∗=τ−N−12e−f dV).To achieve this,first apply to˜g a(small)diffeomor-phism,mapping each point(x i,yα,τ)into(x i,yα,τ(1−2fN)˜gαβ,˜g m00=˜g00−2fτ−fN)˜gαβ,g m00=˜g m00−|∇f|2=12−[τ(2△f−|∇f|2+R)+f−n]),g m i0=g mα0=g m iα=0Note that the hypersurfaceτ=const in the metric g m has the volume form τN/2e−f times the canonical form on M and S N,and the scalar curvatureof this hypersurface is12+τ(2△f−|∇f|2+R)+f)mod N−1.Thus theentropy S multiplied by the inverse temperatureβis essentially minus the total scalar curvature of this hypersurface.6.3Now we return to the metric˜g and try to use its Ricci-flatness by interpreting the Bishop-Gromov relative volume comparison theorem.Con-sider a metric ball in(˜M,˜g)centered at some point p whereτ=0.Then clearly the shortest geodesic between p and an arbitrary point q is always orthogonal to the S Nfibre.The length of such curveγ(τ)can be computedas τ(q)2τ+R+|˙γM(τ)|2dτ= √τ(R+|˙γM(τ)|2)dτ+O(N−3Thus a shortest geodesic should minimize L(γ)= τ(q)0√2Nτ(q) centered at p is O(N−1)-close to the hypersurfaceτ=τ(q),and its volume can be computed as V(S N) M( 2N L(x)+O(N−2))N dx,so the ratio of this volume to 2timesτ(q)−nMτ(R(γ(τ))+|˙γ(τ)|2)dτ(of course,R(γ(τ))and|˙γ(τ)|2are computed using g ij(τ))Let X(τ)=˙γ(τ),and let Y(τ)be any vectorfield alongγ(τ).Then the first variation formula can be derived as follows:δY(L)=14τ2τ1√τ(<Y,∇R >+2<∇X Y,X >)dτ= τ2τ1√dτ<Y,X >−2<Y,∇X X >−4Ric(Y,X ))dτ=2√τ<Y,∇R −2∇X X −4Ric(X,·)−12∇R +1τX (τ)has a limit as τ→0.From now on we fix p and τ1=0and denote by L (q,¯τ)the L -length of the L -shortest curve γ(τ),0≤τ≤¯τ,connecting p and q.In the computations below we pretend that shortest L -geodesics between p and q are unique for all pairs (q,¯τ);if this is not the case,the inequalities that we obtain are still valid when understood in the barrier sense,or in the sense of distributions.The first variation formula (7.1)implies that ∇L (q,¯τ)=2√¯τ(R +|X |2)−<X,∇L >=2√¯τ(R +|X |2)To evaluate R +|X |2we compute (using (7.2))dτR +2<∇R,X >−2Ric(X,X )−1τ(R +|X |2),(7.3)where H (X )is the Hamilton’s expression for the trace Harnack inequality (with t =−τ).Hence,¯τ32L (q,¯τ),(7.4)15where K =K (γ,¯τ)denotes the integral ¯τ0τ3¯τR −1¯τK (7.5)|∇L |2=−4¯τR +2¯τL −4¯τK (7.6)Finally we need to estimate the second variation of L.We computeδ2Y (L )=¯τ0√τ(Y ·Y ·R +2<∇X ∇Y Y,X >+2<R (Y,X ),Y,X >+2|∇X Y |2)dτNowd¯τ+¯τ0√2τY (7.8)We computed τ<Y,Y >,16so |Y (τ)|2=ττ(∇Y ∇Y R +2<R (Y,X ),Y,X >+2∇X Ric(Y,Y )−4∇Y Ric(Y,X )+2|Ric(Y,·)|2−22τ¯τ)dτTo put this in a more convenient form,observe thatdτRic(Y,Y )−2|Ric(Y,·)|2,so Hess L (Y,Y )≤1¯τ−2√τH (X,Y )dτ,(7.9)whereH (X,Y )=−∇Y ∇Y R −2<R (Y,X )Y,X >−4(∇X Ric(Y,Y )−∇Y Ric(Y,X ))−2Ric τ(Y,Y )+2|Ric(Y,·)|2−1τR +n τ−1dτ|Y |2=2Ric(Y,Y )+2<∇X Y,Y >=2Ric(Y,Y )+2<∇Y X,Y >=2Ric(Y,Y )+1¯τHess L (Y,Y )≤1√2H (X,˜Y )dτ,(7.11)where ˜Y is obtained by solving ODE (7.8)with initial data ˜Y (¯τ)=Y (¯τ).Moreover,the equality in (7.11)holds only if ˜Y is L -Jacobi and hence d √¯τ.17Now we can deduce an estimate for the jacobian J of the L-exponential map,given by L exp X(¯τ)=γ(¯τ),whereγ(τ)is the L-geodesic,starting at p and having X as the limit of√dτlog J(τ)≤n2¯τ−3√¯τg.Let l(q,τ)=1τL(q,τ)be thereduced distance.Then along an L-geodesicγ(τ)we have(by(7.4))d2¯τl+12¯τ−32exp(−l(τ))J(τ)is nonincreasing inτalongγ, and monotonicity is strict unless we are on a gradient shrinking soliton. Integrating over M,we get monotonicity of the reduced volume function ˜V(τ)= Mτ−n2¯τ≥0,(7.13) which follows immediately from(7.5),(7.6)and(7.10).Note also a useful inequality2△l−|∇l|2+R+l−nτL(q,τ),then from(7.5), (7.10)we obtain¯L¯τ+△¯L≤2n(7.15) Therefore,the minimum of¯L(·,¯τ)−2n¯τis nonincreasing,so in particular, the minimum of l(·,¯τ)does not exceed n2(τ0−τ),whenever theflow exists forτ∈[0,τ0].)7.2If the metrics g ij(τ)have nonnegative curvature operator,then Hamil-ton’s differential Harnack inequalities hold,and one can say more about the behavior of l.Indeed,in this case,if the solution is defined forτ∈[0,τ0],then H(X,Y)≥−Ric(Y,Y)(1τ0−τ)≥−R(1τ0−τ)|Y|2and18H(X)≥−R(1τ0−τ).Therefore,wheneverτis bounded away fromτ0(say,τ≤(1−c)τ0,c>0),we get(using(7.6),(7.11))|∇l|2+R≤Cldτlog|Y|2≤1n.We claim that˜V k(ǫk r2k)<3ǫn2ǫ−12k;on the otherhand,the contribution of the longer vectors does not exceed exp(−12k)by the jacobian comparison theorem.However,˜V k(t k)(that is,at t=0)stays bounded away from zero.Indeed,since min l k(·,t k−12,we can pick a point q k,where it is attained,and obtain a universal upper bound on l k(·,t k)by considering only curvesγwithγ(t k−12T].Sincethe monotonicity of the reduced volume requires˜V k(t k)≤˜V k(ǫk r2k),this is a contradiction.A similar argument shows that the statement of the corollary in4.2can be strengthened by adding another property of the ancient solution,obtained as a blow-up ly,we may claim that if,say,this solution is defined for t∈(−∞,0),then for any point p and any t0>0,the reduced volume function˜V(τ),constructed using p andτ(t)=t0−t,is bounded below byκ.7.4*The computations in this section are just natural modifications of those in the classical variational theory of geodesics that can be found in any textbook on Riemannian geometry;an even closer reference is[L-Y],where they use”length”,associated to a linear parabolic equation,which is pretty much the same as in our case.198No local collapsing theorem II8.1Let usfirst formalize the notion of local collapsing,that was used in7.3.Definition.A solution to the Ricciflow(g ij)t=−2R ij is said to be κ-collapsed at(x0,t0)on the scale r>0if|Rm|(x,t)≤r−2for all(x,t) satisfying dist t(x,x0)<r and t0−r2≤t≤t0,and the volume of the metric ball B(x0,r2)at time t0is less thanκr n.8.2Theorem.For any A>0there existsκ=κ(A)>0with the fol-lowing property.If g ij(t)is a smooth solution to the Ricciflow(g ij)t=−2R ij,0≤t≤r20,which has|Rm|(x,t)≤r−20for all(x,t),satisfying dist0(x,x0)<r0,and the volume of the metric ball B(x0,r0)at time zero is at least A−1r n0,then g ij(t)can not beκ-collapsed on the scales less than r0at a point(x,r20)with dist r20(x,x0)≤Ar0.Proof.By scaling we may assume r0=1;we may also assume dist1(x,x0)= A.Let us apply the constructions of7.1choosing p=x,τ(t)=1−t.Arguing as in7.3,we see that if our solution is collapsed at x on the scale r≤1,then the reduced volume˜V(r2)must be very small;on the other hand,˜V(1)can not be small unless min l(x,12(x,x0)≤13Kr0+r−10)(the inequality must be understood in the barrier sense,when necessary)(b)(cf.[H4,§17])Suppose Ric(x,t0)≤(n−1)K when dist t(x,x0)<r0, or dist t(x,x1)<r0.Thend3Kr0+r−10)at t=t0 Proof of Lemma.(a)Clearly,d t(x)= γ−Ric(X,X),whereγis the shortest geodesic between x and x0and X is its unit tangent vector,On the other hand,△d≤ n−1k=1s′′Y k(γ),where Y k are vectorfields alongγ,vanishing at20x0and forming an orthonormal basis at x when complemented by X,ands′′Yk (γ)denotes the second variation along Y k of the length ofγ.Take Y k to beparallel between x and x1,and linear between x1and x0,where d(x1,t0)=r0. Then△d≤n−1k=1s′′Y k(γ)= d(x,t0)r0−Ric(X,X)ds+ r00(s2r20)ds= γ−Ric(X,X)+ r00(Ric(X,X)(1−s2r20)ds≤d t+(n−1)(220),andrapidly increasing to infinity on(110),in such a way that2(φ′)2/φ−φ′′≥(2A+100n)φ′−C(A)φ,(8.1) for some constant C(A)<∞.Note that¯L+2n+1≥1for t≥12)is achieved for some y satisfying d(y,110.Now we compute2h=(¯L+2n+1)(−φ′′+(d t−△d−2A)φ′)−2<∇φ∇¯L>+(¯L t−△¯L)φ(8.2)∇h=(¯L+2n+1)∇φ+φ∇¯L(8.3) At a minimum point of h we have∇h=0,so(8.2)becomes2h=(¯L+2n+1)(−φ′′+(d t−△d−2A)φ′+2(φ′)2/φ)+(¯L t−△¯L)φ(8.4)Now since d(y,t)≥120),we can apply our lemma(a)to get d t−△d≥−100(n−1)on the set where φ′=0.Thus,using(8.1)and(7.15),we get2h≥−(¯L+2n+1)C(A)φ−2nφ≥−(2n+C(A))hThis implies that min h can not decrease too fast,and we get the required estimate.219Differential Harnack inequality for solutions of the conjugate heat equation9.1Proposition.Let g ij(t)be a solution to the Ricciflow(g ij)t=−2R ij,0≤t≤T,and let u=(4π(T−t))−ng ij|2(9.1)2(T−t)Proof.Routine computation.Clearly,this proposition immediately implies the monotonicity formula (3.4);its advantage over(3.4)shows up when one has to work locally.9.2Corollary.Under the same assumptions,on a closed manifold M,or whenever the application of the maximum principle can be justified,min v/u is nondecreasing in t.9.3Corollary.Under the same assumptions,if u tends to aδ-function as t→T,then v≤0for all t<T.Proof.If h satisfies the ordinary heat equation h t=△h with respect to the evolving metric g ij(t),then we have ddt hv≥0.Thus we only need to check that for everywhere positive h the limit of hv as t→T is nonpositive.But it is easy to see,that this limit is in fact zero.9.4Corollary.Under assumptions of the previous corollary,for any smooth curveγ(t)in M holds−d2(R(γ(t),t)+|˙γ(t)|2)−1and2(T−t)v≤0we get f t+12|∇f|2−f dt f(γ(t),t)=−f t−<∇f,˙γ(t)>≤−f t+12|˙γ|2.Summing these two inequalities, we get(9.2).9.5Corollary.If under assumptions of the previous corollary,p is the point where the limitδ-function is concentrated,then f(q,t)≤l(q,T−t),where l is the reduced distance,defined in7.1,using p andτ(t)=T−t.22。
非线性临界Kirchhoff型问题的正基态解
2021,41A (3):666-685数学物理学报http: // a ct a 非线性临界Kirchhoff 型问题的正基态解成艺群滕凯民**收稿日期：2020-04-17;修订日期：2020-10-26E-mail: *****************; t *********************基金项目：国家自然科学基金(11501403)、山西省留学回国择优项目(2018)和山西省自然科学基金面上项目(201901D111085)Supported by the NSFC(11501403), the Scientific Activities of Selected Returned Overseas Professionals in Shanxi Province (2018) and the NSF of Shanxi Province(201901D111085)*通讯作者(太原理工大学数学学院 太原030024)摘要：该文研究如下Kirchhoff 型方程(a + b △u + V (x )u = —2 u + s \u \4 u,x e R 3,u e H 1 (R 3),其中a > 0, b> 0,4 <p< 6, V (x ) e L l |c (R 3)是一个给定的非负函数且满足lim V (x ):= 抵•对V (x )给定适当的假设条件，当s 充分小时，证明了基态解的存在性.关键词：Kirchhoff 型方程；临界非线性；基态解.MR(2010)主题分类：35B09; 35J20 中图分类号：O175.2 文献标识码：A 文章编号：1003-3998(2021)03-666-201引言本文研究如下Kirchhoff 型问题(a + b△u + V (x)u = |u |p-2u + s |u |4u, x u > 0, u e H 1 (R 3)(1.1)e R 3,正基态解的存在性，其中a > 0, b> 0,4 <p< 6, s> 0•此外，V (x )是一个非负函数且满足他)：V (x ) e 厶仁(R 3), lim V (x ) = V ^, V (x ) > V 0 > 0 a .e . x e R 3.问题(1.1)与下面方程p 兽-(牛+2L /dudx /(u )(1.2)No.3成艺群等：非线性临界Kirchhoff 型问题的正基态解667对应的稳态相关.1983年，作为经典D'Alembert 波动方程的延伸，Kirchhoff^在研究拉 伸弦的横向振动，特别是考虑到横向振动引起的弦的长度变化时首次提出方程(1.2),其中u 表示变量，且当b 与弦的内在属性相关时，b 是外力，a 是初始应变量，如Young's 模.近年来，采用非线性分析的工具和变分方法，许多学者对下列非线性Kirchhoff 型方程G H X (R 3)|V u |2d x)△u + V (x)u = /(x, u), x G R 3,(1.3)进行了大量的研究，建立了基态解，束缚态解和半经典态解等的存在性和多重性.例如：当 V (x ) := C 时，Xu 和Chen [2]证明了方程(1.3)具有径向基态解，其中/(x, u )满足临界 Berestycki-Lions 型条件.当C = 0时，问题(1.3)可简化为如下方程|V u |2d x)△u = /(x, u ), x G R 3,u G H X (R 3),(1.4)Liu 和Guo [3〕研究了临界增长中具有一般非线性的问题(1.4)的基态解的存在性•当(1.4)式 中 /(x,u ) = g (x )|u |2*-2u + Xh(x)|u|q -^u 时，Li [4]通过 Nehari 法和变分法得到了问题(1.4) 的正基态解的存在性.有关问题(1.4)基态解的更多结果，参看文献[5-7].关于基态解的研究方面，最近，Li 和YeB ]假设V (x )满足以下假设.(i) V (x ) G C (R 3, R )弱可微且满足(V V (x ),x ) G L TO (R 3) U L 3 (R 3)和V (x ) — 2(V V (x ), x ) > 0 a .e . x G R 3;(ii) 对任意的x G R 3,V (x ) < liminf V (y ) := % <十^且其在Lebesgue 正测度的子集 中是严格的；(iii) 存在C > 0使得0= inf u E H 1 (R 3\{0})J r 3 |V u |2 十 V (x )|u |2JR3 |u |2> 0,采用单调性技巧，Pohozaev-Nehari 流形和全局紧性引理证明了当/(x,u ) = |u |p -1u , 2 < p < 5时问题(1.3)正基态解的存在性.Wu [9]利用Pohozaev 流形证明了问题(1.3)存在正 基态解•当/(x,u )在无穷远是次临界且在原点附近是超线性的，V (x )满足与上面⑴和 (ii)相似的一些条件时，Guo [10〕用变分方法证明了问题(1.3)存在正基态解•当/(x,u )= K (x )|u |4u + g (x,u )且V (x )满足渐近周期条件时，作者们在文献[11]中证明了问题(1.3)存 在正基态解•当/(x,u )在无穷远是次临界，在原点是超线性的且满足Berestycki-Lions 条件 和位势V (x )满足与上面(i)-(iii)类似的一些条件时，Liu 和Guo [12]利用Jeanjean 建立的 抽象临界点定理和一个新全局紧性弓I 理证明了问题(1.3)至少存在一个基态解.随后，Tang 和Chen [13〕对位势V (x ) G C (R 3, [0, Q)提出一些更强的条件，他们证明了问题(1.3)存在一 个Nehari-Pohozaev 型的基态解.后来，Chen 和Tang [14〕又证明了问题(1.3)存在一个基 态解，其中/(x,u )满足一般的Berestycki-Lions 假设和位势V (x ) G C (R 3, [0, g ))满足类似 文献[13]的条件.YeW 证明了问题(1.3)存在正基态解，其中/(x,u ) = a (x )f (u ) + u 5且 V (x )在无穷远处满足指数阶衰减•当位势V (x )有一个井位势，Sun 和Wu [16l 得到了一类668数学物理学报Vol.41A 如下Kirchhoff型问题基态解的存在性—(a/|V u|2d x+b)△u+AV(x)u=f(x,u),x e R N,'丿R N)u e H1(r n).关于问题(1.3)基态解的更多结果，参看文献[17-26].受上述文献的启发，本文事■虑具有小临界扰动项的问题(1.1)基态解的存在性.与上述文献相比，我们只需求V(x)e r|c(R3)或V(x)可能在局部区域比％大.这是本文主要结果的新奇之处，其方法是基于约束极小化方法•主要的困难在于非局部项J R b|V u|2d x^u的出现，由于R3的无界性以及带有临界扰动项的非线性而缺乏紧性.此外，由于V(x)非径向对称，故不能将问题限制在径向对称Sobolev空间用(便)中，其中H^R3)j L q(R3)(2<s<6)是紧的.为克服这些困难，必须进行更仔细的分析.特别地，对于序列{u…}c H1(R3)且u…在H1(R3)弱收敛于u,将仔细分析J r3|V u”|2d x和J r3|V u|2d x的不同来恢复紧性.若£=0,则问题(1.1)有一个正基态解，显然，当£趋于0时，对于方程(1.1)来说，我们期望这种结果不会改变.本文将试图证明该现象.本文的主要结果如下.定理1.1假设V(x)满足(旳)和V(x)<a.e.x e R3,(1.5)那么存在£0>0,对任意的£e(0,£o),问题(1.1)有一个正基态解.当位势V(x)不满足(1.5)式时，通过考虑下列极限问题的正基态解—(a+b J|V u|2d x)+V^u=|u|p-2u,x e R3,(1.6)u e H1(R3)证明问题(1.1)的基态解的存在性.事实上根据文献[18,27],在相差平移的情形下，方程(1.6)存在唯一的正的径向对称解，记为w.现陈述如下结果.定理1.2假设V(x)满足(V0),如果存在z e R3满足V(x)|w z|2d x</%w2d x,(1.7)R3其中W z(x):=w(x—z),且w是问题(1.6)的正径向基态解.那么对任意小的£，问题(1.1)存在正的基态解.注意到当V(x)三％时，那么对任意z e R3,(1.7)式成立.此篇论文的结构如下.在第2节，将给出一些符号且回忆了一些学过的知识.在第3节，将给出定理1.1和定理1.2的证明.2准备工作不失一般性，假设％=1.在下文中，将使用以下符号.•H1(R3)是Sobolev空间，其内积和范数如下||训2:=/(a|V u|2+u2)d x.JR3No.3成艺群等：非线性临界Kirchhoff 型问题的正基态解669同时，在这里我们将引入一种等价范数||u||V ：= / (a |V u |2 + V (x )u 2)d x.J r 3予实.上/ (a |V u |2 + u 2)d x < C (a |V u |2 + V (x )u 2)d xJ r 3 — J r 3显然成立.反之,根据(兀)可得，存在R> 0使得当|x | > R 时有V (x ) < 2%,从而有/ (a |V u |2 + V (x )u 2)d x R 3=a |V u |2d x + V (x )u 2d x +V (x )u 2d x J r 3 J\x \>R J\x\<R <a |V u |2d x + 2 / V ^u 2d x 7r 3 J\x \>R+ / ((V (x ))2d x )u 6dx) 3J \x \<R _3 丿< c / (a |V u |2 + u 2)d x.7r 3• D 1，2(R 3)是通常的 Sobolev 空间，其标准范数为 ||u||D ：=(J r 3 |V u |2d x )2.• (O )表示Lebesgue 空间，其中1 < q < g , OC R 3是一个可测集•当O 是R 3的 适当可测子集时，L (O )上的范数为| • |£q (o ).当O = R 3时，其范数为| • |,• c,C i ,C,C i ,…表示一些正常数.问题(1.1)对应的能量泛函是厶:H i (R 3) t R 定义为厶(u )=2(a |V u |2 + V (x )u 2)d x |u |p d x 一； [ |u |6d x.6 J r 3显然, 如下厶G C i (H i (R 3),R )且厶的临界点是问题(1.1)的弱解•问题(1.1)对应的极限方程G H i (R 3),△u + V ^u = |u |p-2u + s |u |4u,x G R 3,(2.1)且其对应的泛函为 —:H i (R 3) t R ,定义如下1 / (a |V u |2 + u 2)d x + -( / |V u |2d x) — - [ |u |p d x —三/ |u |6d x.2 J R34 ' 丿R 3 丿 p J R 3 6 丿R 3与厶具有相同性质的泛函定义如下I (u ) = 1 / (a |V u |2 + V (x )u 2)d x + -( / |V u |2d x) — - [ |u |p d x,2 J r34 ' J r 3 丿 p J r 3I x (u ) = 1 / (a |V u |2 + u 2)d x + [ |V u |2d x) — - [ |u |p d x.2 丿r34 ' J r 3 ) P J r 3定义泛函I ,厶，：和厶心上的Nehari 流形如下N = {u G H 0(R 3)\{0} : I z (u )[u ] = 0}, N = {u G H 0(R 3)\{0} : I ((u )[u ] = 0},N g = {u G H 1(R 3)\{0} : I Q (u )[u ] = 0}, = {u G H 1(R 3)\{0} : I ]；^(u )[u ] = 0}.670数学物理学报Vol.41A 定义m ：= inf Ig 〕 ：= inf /^x . (2.2)N so N e , g 注意到，当£> 0时有m e < m .事实上，设w 满足/g (w ) = m 且存在r e > 0使得r e w e 那么m e < I e ；g (r e w ) < I g (r e w ) < I g (w ) = m. (2.3)弓|理2.1 (i)存在唯一 t u > 0使得t u u eN ,有i (t u u ) = max I (tu ),且u 一 t u 是从H 1(R 3)\{0}到R +的连续映射.若在N g , M 和上分别考虑I g , I 和厶,g ,类似的结论成立.(ii) 对任意的u e M,g ，存在正常数C 〉0使得||训> C> 0,其中C 与£无关.(iii) 对任意的£ e (0,£o ),设u 是约束在M,g 上的厶,g 的极小元，那么存在正常数 C 1 > 0使得|u |p > C 1 > 0,其中C 1与£无关.证 根据标准的讨论，可得⑴成立.(ii)对任意的u e M,g ，由Sobolev 嵌入定理得0 = ||训2 + b |V u |4 — |u |p — £|u |6 > ||u||2 — C 1||u 卩一C 2£||u||6,即||u||2 <C 1||u||p + C 2£||u||6，其中C 1,C 2 > 0与£和U 无关•因此，对任意的£>0,有||训 > C > 0, V u eM ,g(2.4)成立，其中C 是与£和u 无关的正常数.(iii)设u 是厶,g 约束在M,g 上的极小元，可得I £,g (u ) = 4ll u ^2 + 〃4p |u |p + 12£|u |6 = m Q 由上式和(2.3)式可得||u |2 < 4m e + o (1) < 4m. (2.5)从(2.4)式可得||训有正下界，且从(2.5)式可得||训有正上界.因此，利用Sobolev 嵌入定 理，可得C 2|u |p = ||训2 + b ll Vu ll 2 ― £|u |6 > ll u|2 ― c £II 训6 > C 2 ― c 1£ > ~2 > C 1? V £ e (0, £o ),其中C, C 1 > 0与£和u 无关•证毕. I命题2.1下列估计成立a /b 宀3 叫 < 3(2£S +s 6+a 切+桔住s 3+-----------------\ 2S 6 + a S 3 )S a,b , V £ > 0, (2.6)其中s 是最佳Sobolev 常数.证 注意到(2.8)式中的S a ；b 是下列方程解的基态水平lim u (x ) = 0,|x|—>g a + b|V u △u = £|u |4u,x e R 3,(2.7)No.3成艺群等：非线性临界Kirchhoff 型问题的正基态解671即-----------------\ 2S 6 + a S 3 )和=3许+s 6+期)+包汀+2^/r 3 |u |6d x因此，由上式可知a [ |V (tu )|2d x |V (tu )|2d x)—訂 |tu |6d x 2 J r 3 4 ' J r 3 丿 6 J r 3=at 2 / |V u |2d x + 寻4( / |V u |2d x 『3J r 3 12 W r 3 丿a [- 2 -(J r 3 |V u |2d x )2 + \l -2(J r 3 |V u |2d x )4 +4a8 J r 3 |V u |2d x J r 3 |u |6d x =5 |Vu|2dx -----------------------------y ------------------------------------------------------------------------3丿R 3=min (a / |V u |*2d x + -( / |V u |2d x) — - / |u |6d x : u G D 1,2(R 3),I 2 J 4 ' 丿R 3 J 6 丿R 3/ a |V u |2d x + -( [ |V u |2d x) = - / |u |6d x{.7r 3 ' J r 3 丿 7r 3 丿事实上，下列问题R 3(2.8)的正解具有如下形式—△u = |u |4u, u G D 1，2(R 3)x G R 3,(2.9)(3$)1肿 _ (J + |x — x 0|2)2 :事实上，最佳Sobolev 常数S 可通过下式定义血 |V u |2dx _J > 0, x o G R 3.注意到,S = inf ------厂应 .ueD 1,2(R 3)\{0} |u |2 u£D 1.2(R 3)\{0}, u 6 = ^/R 3对任意的u G D X '2(R 3)\{0},存在唯一的t > 0使得tu 满足a [ |V (tu )|2d x + b( [ |V (tu )|2d x) = 8 / |tu |6d x, 丿R 3 '丿R 3 丿 丿R 3inf |V u |2d x.—a [ |V u |2d x = 0,丿R 3(2.10)即通过计算得t 2-(J r 3 |V u |2d x )2 + J -2(J r 3 |V u |2d x )4 十 4«£ J r 3 |V u |2d x J r 3 |u |6d x2^/r 3 |u |6d x ■ -(J r 3 |V u |2d x )2 + j -2(J r 3 |V u |2d x )4 +4处 J r 3 |V u |2d x j R 3 |u |6d x -28 J r 3 |u |6d xJ r 3 |V u |2d x - ” • c 2-I J R 八…I —| +-MJ r 3 |u |6d x ) 3 丿J r 3 |V u |2d x L/ J r 3 |V u |2d x a 68 (J r 3 |u |6d x ) 3VC/r 3 |u |6d x ) 3“ 'I2672数学物理学报Vol.41A\( J r 3 |V "|2dx )2 + -MJ r3 |u |6d x ) 3 丿> S (b S 2 + v b 2S4 + 4a£S ) + 召a / b 宀3/ b 2 十 ab / b 宀33(2£ V 4£2 £ 12(2£+ W J r 3 |V u |2d x 1^2£(J r 3 |u |6d x ) 3护(J r 3 |V u |2d xMJ r 3 |u |6d x )■ s ________________________12 (b S 2 + 7b 2S 4 + 4a£S )J + 4a£ I 2⑵11)另一方面，关于Sobolev 常数S ，设U o 是满足|U o |6 = 1的达到S 的函数，那么存在唯一的 t u o > 0使得t u o U o 满足(2.10)式*因此，通过计算，有|V U o |2d x +12t U 。
PbI2
Remnant PbI2, an unforeseen necessity in high-efficiency hybrid perovskite-based solar cells?a)Duyen H. Cao, Constantinos C. Stoumpos, Christos D. Malliakas, Michael J. Katz, Omar K. Farha, Joseph T. Hupp, and Mercouri G. KanatzidisCitation: APL Materials 2, 091101 (2014); doi: 10.1063/1.4895038View online: /10.1063/1.4895038View Table of Contents: /content/aip/journal/aplmater/2/9?ver=pdfcovPublished by the AIP PublishingArticles you may be interested inParameters influencing the deposition of methylammonium lead halide iodide in hole conductor free perovskite-based solar cellsAPL Mat. 2, 081502 (2014); 10.1063/1.4885548Air stability of TiO2/PbS colloidal nanoparticle solar cells and its impact on power efficiencyAppl. Phys. Lett. 99, 063512 (2011); 10.1063/1.3617469High efficiency mesoporous titanium oxide PbS quantum dot solar cells at low temperatureAppl. Phys. 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Phys. 101, 114503 (2007); 10.1063/1.2737977APL MATERIALS2,091101(2014)Remnant PbI2,an unforeseen necessity in high-efficiency hybrid perovskite-based solar cells?aDuyen H.Cao,1Constantinos C.Stoumpos,1Christos D.Malliakas,1Michael J.Katz,1Omar K.Farha,1,2Joseph T.Hupp,1,band Mercouri G.Kanatzidis1,b1Department of Chemistry,and Argonne-Northwestern Solar Energy Research(ANSER)Center,Northwestern University,2145Sheridan Road,Evanston,Illinois60208,USA2Department of Chemistry,Faculty of Science,King Abdulaziz University,Jeddah,Saudi Arabia(Received2May2014;accepted26August2014;published online18September2014)Perovskite-containing solar cells were fabricated in a two-step procedure in whichPbI2is deposited via spin-coating and subsequently converted to the CH3NH3PbI3perovskite by dipping in a solution of CH3NH3I.By varying the dipping time from5s to2h,we observe that the device performance shows an unexpectedly remark-able trend.At dipping times below15min the current density and voltage of thedevice are enhanced from10.1mA/cm2and933mV(5s)to15.1mA/cm2and1036mV(15min).However,upon further conversion,the current density decreases to9.7mA/cm2and846mV after2h.Based on X-ray diffraction data,we determinedthat remnant PbI2is always present in these devices.Work function and dark currentmeasurements showed that the remnant PbI2has a beneficial effect and acts as ablocking layer between the TiO2semiconductor and the perovskite itself reducingthe probability of back electron transfer(charge recombination).Furthermore,wefind that increased dipping time leads to an increase in the size of perovskite crys-tals at the perovskite-hole-transporting material interface.Overall,approximately15min dipping time(∼2%unconverted PbI2)is necessary for achieving optimaldevice efficiency.©2014Author(s).All article content,except where otherwisenoted,is licensed under a Creative Commons Attribution3.0Unported License.[/10.1063/1.4895038]With the global growth in energy demand and with compelling climate-related environmental concerns,alternatives to the use of non-renewable and noxious fossil fuels are needed.1One such alternative energy resource,and arguably the only legitimate long-term solution,is solar energy. Photovoltaic devices which are capable of converting the photonflux to electricity are one such device.2Over the last2years,halide hybrid perovskite-based solar cells with high efficiency have engendered enormous interest in the photovoltaic community.3,4Among the perovskite choices, methylammonium lead iodide(MAPbI3)has become the archetypal light absorber.Recently,how-ever,Sn-based perovskites have been successfully implemented in functional solar cells.5,6MAPbI3 is an attractive light absorber due to its extraordinary absorption coefficient of1.5×104cm−1 at550nm;7it would take roughly1μm of material to absorb99%of theflux at550nm.Further-more,with a band gap of1.55eV(800nm),assuming an external quantum efficiency of90%,a maximum current density of ca.23mA/cm2is attainable with MAPbI3.Recent reports have commented on the variability in device performance as a function of perovskite layer fabrication.8In our laboratory,we too have observed that seemingly identicalfilmsa Invited for the Perovskite Solar Cells special topic.b Authors to whom correspondence should be addressed.Electronic addresses:j-hupp@ and m-kanatzidis@2,091101-12166-532X/2014/2(9)/091101/7©Author(s)2014FIG.1.X-ray diffraction patterns of CH3NH3PbI3films with increasing dipping time(%composition of PbI2was determined by Rietveld analysis(see Sec.S3of the supplementary material for the Rietveld analysis details).have markedly different device performance.For example,when ourfilms of PbI2are exposed to MAI for several seconds(ca.60s),then a light brown coloredfilm is obtained rather than the black color commonly observed for bulk MAPbI3(see Sec.S2of the supplementary material for the optical band gap of bulk MAPbI3).23This brown color suggests only partial conversion to MAPbI3and yields solar cells exhibiting a J sc of13.4mA/cm2and a V oc of960mV;these values are significantly below the21.3mA/cm2and1000mV obtained by others.4Under the hypothesis that fully converted films will achieve optimal light harvesting efficiency,we increased the conversion time from seconds to2h.Unexpectedly,the2-h dipping device did not show an improved photovoltaic response(J sc =9.7mA/cm2,V oc=846mV)even though conversion to MAPbI3appeared to be complete.With the only obvious difference between these two devices being the dipping time,we hypothesized that the degree of conversion of PbI2to the MAPbI3perovskite is an important parameter in obtaining optimal device performance.We thus set out to understand the correlation between the method of fabrication of the MAPbI3layer,the precise chemical compositions,and both the physical and photo-physical properties of thefilm.We report here that remnant PbI2is crucial in forming a barrier layer to electron interception/recombination leading to optimized J sc and V oc in these hybrid perovskite-based solar cells.We constructed perovskite-containing devices using a two-step deposition method according to a reported procedure with some modifications.4(see Sec.S1of the supplementary material for the experimental details).23MAPbI3-containing photo-anodes were made by varying the dipping time of the PbI2-coated photo-anode in MAI solution.In order to minimize the effects from unforeseen variables,care was taken to ensure that allfilms were prepared in an identical manner.The composi-tions offinal MAPbI3-containingfilms were monitored by X-ray diffraction(XRD).Independently of the dipping times,only theβ-phase of the MAPbI3is formed(Figure1).9However,in addition to theβ-phase,allfilms also showed the presence of unconverted PbI2(Figure1,marked with*) which can be most easily observed via the(001)and(003)reflections at2θ=12.56◦and38.54◦respectively.As the dipping time is increased,the intensities of PbI2reflections decrease with a concomitant increase in the MAPbI3intensities.In addition to the decrease in peak intensities of PbI2,the peak width increases as the dipping time increases indicating that the size of the PbI2 crystallites is decreasing,as expected,and the converse is observed for the MAPbI3reflections.This observation suggests that the conversion process begins from the surface of the PbI2crystallites and proceeds toward the center where the crystallite domain size of the MAPbI3phase increases and that of PbI2diminishes.Interestingly,the remnant PbI2phase can be seen in the data of other reports, but has not been identified as a primary source of variability in cell performance.8,10 Considering that the perovskite is the primary light absorber within the device,we wantedto further investigate how the optical absorption of thefilm changes with increasing dipping timeFIG.2.Absorption spectra of CH3NH3PbI3films as a function of unconverted PbI2phase fraction.FIG.3.(a)J-V curves and(b)EQE of CH3NH3PbI3-based devices as a function of unconverted PbI2phase fraction.(Figure2).11,12The pure PbI2film shows a band gap of2.40eV,consistent with the yellow color of PbI2.As the PbI2film is gradually converted to the perovskite,the band gap is progressively shifted toward1.60eV.The deviation of MAPbI3’s band gap(1.60eV)from that of the bulk MAPbI3 material(1.55eV)could be explained by quantum confinement effects related with the sizes of TiO2and MAPbI3crystallites and their interfacial interaction.13,14Interestingly,we also noticed the presence of a second absorption in the light absorber layer,in which the gap gradually red shifts from1.90eV to1.50eV as the PbI2concentration is decreased from9.5%to0.3%(Figure2—blue arrow).Having established the chemical compositions and optical properties of the light absorberfilms, we proceeded to examine the photo-physical responses of the corresponding functional devices in order to determine how the remnant PbI2affects device performance.The pure PbI2based device remarkably achieved a0.4%efficiency with a J sc of2.1mA/cm2and a V oc of564mV (Figure3(a)).Upon progressive conversion of the PbI2layer to MAPbI3,we observe two different regions(Figure4,Table I).In thefirst region,the expected behavior is observed;as more PbI2is converted to MAPbI3,the trend is toward higher photovoltaic efficiency,due both to J sc and V oc, until1.7%PbI2is reached.The increase in J sc is attributable,at least in part,to increasing absorption of light by the perovskite.We speculate that progressive elimination of PbI2,present as a layer between TiO2and the perovskite,also leads to higher net yields for electron injection into TiO2and therefore,higher J values.For a sufficiently thick PbI2spacer layer,electron injection would occur instepwise fashion,i.e.,perovskite→PbI2→TiO2.Finally,the photovoltage increase is attributable toFIG.4.Summary of J-V data vs.PbI2concentration of CH3NH3PbI3-based devices(Region1:0to15min dipping time, Region2:15min to2h dipping time).TABLE I.Photovoltaic performance of CH3NH3PbI3-based devices as a function of unconverted PbI2fraction.Dipping time PbI2concentration a J sc(mA/cm2)V oc(V)Fill factor(%)Efficiency(%) 0s100% 2.10.564320.45s9.5%10.10.93352 4.960s7.2%13.40.96052 6.72min 5.3%14.00.964557.45min 3.7%14.70.995578.315min 1.7%15.1 1.036629.730min0.8%13.60.968648.51h0.4%12.40.938657.62h0.3%9.70.84668 5.5a Determined from the Rietveld analysis of X-ray diffraction data.the positive shift in TiO2’s quasi-Fermi level as the population of photo-injected electrons is higher with increased concentration of MAPbI3.The second region yields a notably different trend;surprisingly,below a concentration of2% PbI2,J sc,V oc,and ultimatelyηdecrease.Considering that the light-harvesting efficiency would increase when the remaining2%PbI2is converted to MAPbI3(albeit to only a small degree),then the remnant PbI2must have some other role.We posit that remnant PbI2serves to inhibit detrimental electron-transfer processes(Figure5).Two such processes are back electron transfer from TiO2to holes in the valence band of the perovskite(charge-recombination)or to the holes in the HOMO of the HTM(charge-interception).This retardation of electron interception/recombination observation is reminiscent of the behavior of atomic layer deposited Al2O3/ZrO2layers that have been employed in dye-sensitized solar cells.15–18It is conceivable that the conversion of PbI2to MAPbI3occurs from the solution interface toward the TiO2/PbI2interface and thus would leave sandwiched between TiO2and MAPbI3a blocking layer of PbI2that inhibits charge-interception/recombination.For this hypothesis to be correct,it is crucial that the conduction-band-edge energy(E cb)of the PbI2be higher than the E cb of the TiO2.19–21 The work function of PbI2was measured by ultraviolet photoelectron spectroscopy(UPS)and was observed to be at6.35eV vs.vacuum level,which is0.9eV lower than the valence-band-edge energy(E vb)of MAPbI3(see Sec.S7of the supplementary material23for the work function of PbI2);the E cb(4.05eV)was calculated by subtracting the work function from the band gap(2.30eV).solar cell.FIG.6.Dark current of CH3NH3PbI3-based devices as a function of unconverted PbI2phase fraction.The E cb of PbI2is0.26eV higher than the E cb of TiO2and thus PbI2satisfies the conditions of a charge-recombination/interception barrier layer.In order to probe the hypothesis that PbI2acts as a charge-interception barrier,dark current measurements,in which electronsflow from TiO2to the HOMO of the HTM,were made.Consistent with our hypothesis,Figure6illustrates that the onset of the dark current occurs at lower potentials as the PbI2concentration decreases.In the absence of other effects,the increasing dark current with increasing fraction of perovskite(and decreasing fraction of PbI2)should result in progressively lower open-circuit photovoltages.Instead,the photocurrent density and the open-circuit photovoltage bothincrease,at least until to PbI2fraction reaches1.7%.As discussed above,thinning of a PbI2-basedFIG.7.Cross-sectional SEM images of CH3NH3PbI3film with different dipping time.sandwich layer should lead to higher net injection yields,but excessive thinning would diminish the effectiveness of PbI2as a barrier layer for back electron transfer reactions.Given the surprising role of remnant PbI2in these devices,we further probed the two-step conversion process by using scanning-electron microscopy(SEM)(Figure7).Two domains of lead-containing materials(PbI2and MAPbI3)are present.Thefirst domain is sited within the mesoporous TiO2network(area1)while the second grows on top of the network(area2).Area2initially contains 200nm crystals.As the dipping time is increased,the crystals show marked changes in size and morphology.The formation of bigger perovskite crystals is likely the result of the thermodynamically driven Ostwald ripening process,i.e.,smaller perovskite crystals dissolves and re-deposits onto larger perovskite crystals.22The rate of charge-interception,as measured via dark current,is proportional to the contact area between the perovskite and the HTM.Thus,the eventual formation of large, high-aspect-ratio crystals,as shown in Figure7,may well lead to increases in contact area and thereby contributes to the dark-current in Figure6.Regardless,we found that the formation of large perovskite crystals greatly decreased our success rate in constructing high-functioning,non-shorting solar cells.In summary,residual PbI2appears to play an important role in boosting overall efficiencies for CH3NH3PbI3-containing photovoltaics.PbI2’s role appears to be that of a TiO2-supported blocking layer,thereby slowing rates of electron(TiO2)/hole(perovskite)recombination,as well as decreasing rates of electron interception by the hole-transporting material.Optimal performance for energy conversion is observed when ca.98%of the initially present PbI2has been converted to the perovskite. Conversion to this extent requires about15min.Pushing beyond98%(and beyond15min of reaction time)diminishes cell performance and diminishes the success rate in constructing non-shorting cells.The latter problem is evidently a consequence of conversion of small and more-or-less uniformly packed perovskite crystallites to larger,poorly packed crystallites of varying shape and size.Finally,the essential,but previously unrecognized,role played by remnant PbI2 provides an additional explanation for why cells prepared dissolving and then depositing pre-formed CH3NH3PbI3generally under-perform those prepared via the intermediacy of PbI2.We thank Prof.Tobin Marks for use of the solar simulator and EQE measurement system. Electron microscopy was done at the Electron Probe Instrumentation Center(EPIC)at Northwestern University.Ultraviolet Photoemission Spectroscopy was done at the Keck Interdisciplinary SurfaceScience facility(Keck-II)at Northwestern University.This research was supported as part of theANSER Center,an Energy Frontier Research Center funded by the U.S Department of Energy, Office of Science,Office of Basic Energy Sciences,under Award No.DE-SC0001059.1R.Monastersky,Nature(London)497(7447),13(2013).2H.J.Snaith,J.Phys.Chem.Lett.4(21),3623(2013).3M.M.Lee,J.Teuscher,T.Miyasaka,T.N.Murakami,and H.J.Snaith,Science338(6107),643(2012).4J.Burschka,N.Pellet,S.J.Moon,R.Humphry-Baker,P.Gao,M.K.Nazeeruddin,and M.Gratzel,Nature(London) 499(7458),316(2013).5F.Hao,C.C.Stoumpos,D.H.Cao,R.P.H.Chang,and M.G.Kanatzidis,Nat.Photonics8(6),489(2014);F.Hao,C.C. Stoumpos,R.P.H.Chang,and M.G.Kanatzidis,J.Am.Chem.Soc.136,8094–8099(2014).6N.K.Noel,S.D.Stranks,A.Abate,C.Wehrenfennig,S.Guarnera,A.Haghighirad,A.Sadhanala,G.E.Eperon,M.B. Johnston,A.M.Petrozza,L.M.Herz,and H.J.Snaith,Energy Environ.Sci.7,3061(2014).7H.S.Kim,C.R.Lee,J.H.Im,K.B.Lee,T.Moehl,A.Marchioro,S.J.Moon,R.Humphry-Baker,J.H.Yum,J.E.Moser, M.Gratzel,and N.G.Park,Sci.Rep.2,591(2012).8D.Y.Liu and T.L.Kelly,Nat.Photonics8(2),133(2014).9C.C.Stoumpos,C.D.Malliakas,and M.G.Kanatzidis,Inorg.Chem.52(15),9019(2013).10J.H.Noh,S.H.Im,J.H.Heo,T.N.Mandal,and S.I.Seok,Nano Lett.13(4),1764(2013).11Diffuse reflectance measurements of MAPbI3films were converted to absorption spectra using the Kubelka-Munk equation,α/S=(1-R)2/2R,where R is the percentage of reflected light,andαand S are the absorption and scattering coefficients, respectively.The band gap values are the energy value at the intersection point of the absorption spectrum’s tangent line and the energy axis.12L.F.Gate,Appl.Opt.13(2),236(1974).13O.V oskoboynikov,C.P.Lee,and I.Tretyak,Phys.Rev.B63(16),165306(2001).14X.X.Xue,W.Ji,Z.Mao,H.J.Mao,Y.Wang,X.Wang,W.D.Ruan,B.Zhao,and J.R.Lombardi,J.Phys.Chem.C 116(15),8792(2012).15E.Palomares,J.N.Clifford,S.A.Haque,T.Lutz,and J.R.Durrant,J.Am.Chem.Soc.125(2),475(2003).16C.Prasittichai,J.R.Avila,O.K.Farha,and J.T.Hupp,J.Am.Chem.Soc.135(44),16328(2013).17A.K.Chandiran,M.K.Nazeeruddin,and M.Gratzel,Adv.Funct.Mater.24(11),1615(2014).18M.J.Katz,M.J.D.Vermeer,O.K.Farha,M.J.Pellin,and J.T.Hupp,Langmuir29(2),806(2013).19M.J.DeVries,M.J.Pellin,and J.T.Hupp,Langmuir26(11),9082(2010).20C.Prasittichai and J.T.Hupp,J.Phys.Chem.Lett.1(10),1611(2010).21F.Fabregat-Santiago,J.Garcia-Canadas,E.Palomares,J.N.Clifford,S.A.Haque,J.R.Durrant,G.Garcia-Belmonte, and J.Bisquert,J.Appl.Phys.96(11),6903(2004).22Alan D.McNaught and Andrew Wilkinson,IUPAC Compendium of Chemical Terminology(Blackwell Scientific Publica-tions,Oxford,1997).23See supplementary material at /10.1063/1.4895038for experimental details,absorption spectrum of bulk CH3NH3PbI3,fraction,size,absorption spectrum,work function of unconverted PbI2,and average photovoltaic perfor-mance.。
《经济法》经济法主体
五、国家经济监督主体
按职能和法律地位划分，主要有： 权力监督主体：人大 行政监督主体： （一）一般行政监督主体： （二）专门行政监督主体：监察部、审计署
六、行为
宏观调控行为 微观规制行为 国有参与行为 涉外管制行为 市场监管行为
第三节 市场主业性与非行业性
根据是否有成员的行业或职业集体性
意义：明确其在中介组织体系中的特殊地 位；明晰与其他中介组织在性质和职能上 的区别；
3、评价性与非评价性
根据所提供的服务是否属于评价性服务
法律意义：明确社会责任和法律义务；确 定共同的行为准则
4、专业服务性与佣金服务性
根据提供服务时是否直接以专业技术服务 为标的
第二节 企业法
一、企业法的概念
企业法是调整企业在设立、存续和终止过 程中所发生的社会关系的法律规范的总称。
二、企业法的调整对象
企业以及企业与投资者之间的经济关系
企业在设立、存续和终止过程中所发生的 经济管理关系 企业内部的经营管理关系
三、企业法律体系
关于公司的规范性文件 关于合伙企业 关于独资企业的 关于国有企业的
第三节 专业服务中介组织
一、概念
专业服务中介组织，是指依法设立，由一 定专业资格的人员组成，并以提供专业技 术性中间服务为经营手段的中介组织。 特征：专业法定性、社会评价性、企业经 营性
种类：合格评定机构、资产评估机构与保 险公估机构、专业委托代理服务机构、公 证与仲裁服务机构
二、组织形式与法律地位
二、经济管理主体的职权
退化线性椭圆方程非常弱解的存在唯一性
龙源期刊网 退化线性椭圆方程非常弱解的存在唯一性作者：晏华辉顾广泽来源：《湖南大学学报·自然科学版》2014年第07期摘要：定义了在所谓的具有一片平的边界的有界光滑区域内退化线性椭圆的非常弱解的概念，然后利用变法方法与退化椭圆方程的极值原理等证明了该问题非常弱解的存在唯一性结果.关键词：存在性；唯一性；非常弱解；退化椭圆方程中图分类号：O175.25 文献标识码：A他们需要得到上面问题非常弱解的存在唯一性结果.[1]QUITTNER P， REICHEL W. Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions [J]. Calc Var Partial Diff Equ，2008，32（4）： 429-452.[2]BIDAUTVERON M F， PONCE A， VERON L. Boundary singularities of positive solutions of some nonlinear elliptic equations [J]. C R Acad Sci Paris Ser I Math， 2007，344（2）： 83-88.[3]HU B. Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition [J]. Differential Integral Equations. 1994，7（2）： 301-313.[4]MCKENNA P J， REICHEL W. A priori bounds for semilinear equations and a new class of critical exponents for Lipschitz domains [J]. J Funct Anal， 2007，244（1）： 220-246.[5]PACARD F. Existence de solutions faibles positive de dans des ouverts bornes de [J]. C R Acad Sci Paris Ser. I Math， 1992，315（7）： 793-798.[6]PACARD F. Existence and convergence of positive weak solutions of in a bounded domains of [J]. Calc Var Partial Diff Equ， 1993， 1（3）： 243-265.[7]QUITTNER P， SOUPLET PH. A priori estimates and existence for elliptic systems via bootstrap in a weighted Lebesgue spaces [J]. Arch Ration Mech Anal， 2004， 174（1）： 49-81.[8]CABRE X， SIRE Y. Nonlinear equations for fractional Laplacians I： regularity，maximum principles， and Hamiltonian estimates [J]. Ann Inst H Poincar＼'{e} Anal NonLin＼'{e}aire， 2014，31（1）： 23-53.。
反应扩散方程
Article history: Received 27 August 2012 Accepted 17 January 2013 Keywords: Predator–prExistence and uniqueness Predator restrict
Qiu Xiao-xiao, Xiao Hai-bin ∗
Department of Mathematics, Ningbo University, Ningbo, Zhejiang, 315211, PR China
article
info
abstract
This paper is devoted to investigation of Holling type II predator–prey systems with prey refuges and predator restricts. Using a transformation technique, we change the system into a generalized Liénard system and give sufficient conditions to ensure the global stability of the positive equilibrium and existence and uniqueness of a stable limit cycle. We also find the property of alternation for phase structure of the system. © 2013 Elsevier Ltd. All rights reserved.
journal homepage: /locate/nonrwa
PDE数值
Mathematics Subject Classification (2000): 93E20, 60G40, 91B28
1 Introduction
This paper focuses on the problem of a company which wants to expand its stochastic production capacity. The investments in capital for expanding the capacity are irreversible in the sense that the company cannot recover the investment by reducing the capacity. In addition, there is a transaction cost for purchasing capital. We refer to the book by Dixit and Pindick (1994) for a review where such problems occur. There are several papers in the literature dealing with irreversible investments models. For instance, Kobila (1993) consider a model with deterministic capacity in an uncertain market and without transaction costs on buying capital. Recently, Chiarolla and Haussmann
2 Formulation of the problem
一类脉冲捕食-食饵系统的多个正周期解问题
一类脉冲捕食-食饵系统的多个正周期解问题方亚运;王奇;郑兆岳【期刊名称】《佳木斯大学学报（自然科学版）》【年(卷),期】2014(000)001【摘要】In this paper , an impulsive predator -prey system with Beddington -DeAngelis functional re-sponse and harvesting terms was considered using the continuation theorem .The sufficient condition forthe exist-ence of four positive periodic solutions was obtain .%利用Mawhin延拓定理研究了一类具有Beddington-DeAngelis功能反应及收获项的时滞脉冲捕食-食饵系统，得到系统存在多个正周期解的充分条件，推广了已有结论。
【总页数】4页(P150-152,155)【作者】方亚运;王奇;郑兆岳【作者单位】安徽大学数学科学学院，安徽合肥230601;安徽大学数学科学学院，安徽合肥 230601;安徽工贸职业技术学院基础部，安徽淮南232007【正文语种】中文【中图分类】O175.7【相关文献】1.时间尺度上带有收获项的一类捕食-食饵系统多个正周期解的存在性 [J], 李周红;杨成莲2.一类具有脉冲作用和HollingⅡ型功能性反应的非自治捕食者-食饵系统正周期解的存在性 [J], 王斌;朱勇3.一类具有脉冲效应和可变时滞的非自治捕食者-食饵系统正周期解的存在性 [J], 王斌;刘萍4.一类捕食者-食饵三次Kolmogorov脉冲系统正周期解的存在性 [J], 王晓阳;朱天晓;高海音;盖永杰5.一类捕食-食饵系统的八个正周期解问题 [J], 陆地成;王奇;张友梅因版权原因，仅展示原文概要，查看原文内容请购买。
一类动力学方程及流体力学方程解的Gevrey类正则性
Boltzmann 方程 . . . . . . . . . . . . . . . . . . . . . . . . 碰撞算子 Q(f, f ) 的基本性质 . . . . . . . . . . . . . . . . . Fokker-Planck 方程、Landau 方程以及 Boltzmann 方程线性 化模型 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Navier-Stokes 方程 . . . . . . . . . . . . . . . . . . . . . . . Gevrey 函数空间 . . . . . . . . . . . . . . . . . . . . . . . .
研究现状及本文主要结果 . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 1.2.2 1.2.3 1.2.4 存在性及唯一性 . . . . . . . . . . . . . . . . . . . . . . . . . 动力学方程的正则性理论: 空间齐次情形 . . . . . . . . . . . 动力学方程的正则性理论: 空间非齐次情形 . . . . . . . . . . Navier-Stokes 方程的正则性理论 . . . . . . . . . . . . . . .
第二章 预备知识 2.1 2.2 2.3 基本记号
Fourier 变换 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 基本函数空间及常用不等式 . . . . . . . . . . . . . . . . . . . . . . 2.3.1 2.3.2 Lp 空间及其性质 . . . . . . . . . . . . . . . . . . . . . . . . Sobolev 空间及其性质 . . . . . . . . . . . . . . . . . . . . .
Stochastic Functional Differential Equations with Markovian Switching
Department of StXatuiesrtiocnsgaMndaoModelling Science University of Strathclyde, Glasgow G1 1XH, U.K.
1. INTRODUCTION
Stochastic modelling has come to play an important role in many branches of science and industry. An area of particular interest has been the automatic control of stochastic systems, with consequent emphasis being placed on the analysis of stability in stochastic models (cf. Arnold 1], Has'minskii 5] and Mao 10]). The stability of stochastic functional di erential equations has been studied by many authors and we here mention Kolmanovskii & Myshkis 7], Kolmanovskii & Nosov 8], Ladde & Lakshmikantham 9], Mao 11] and Mohammed 13] among others. However, there is little work on the stability of stochastic functional di erential equations with Markovian switching although there are several papers on the stability of stochastic di erential equations with Markovian switching e.g. Basak et al. 2], Ghosh et al. 4] and Skorohod 14]. The main aim of this paper is to close this gap.
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Ｊｏｕｒｎａｌ ｏｆＰｈｙｓｉｃａｌ Ｓｃｉｅｎｃｅ ａｎｄ Ａｐｐｌｉｃａｔｉｏｎ ５（２０１ ５）７１—８１ ｄｏｉ：１０．１７２６５／２１５９．５３４８／２０１ ５，０１．０１ １
Ｅｘｉｓｔｅｎｃｅ ｏｆ Ｐｏｓｉｔｉｖｅ Ｓｏｌｕｔｉｏｎｓ ｔｏ Ｓｅｍｉｌｉｎｅａｒ Ｅｌｌｉｐｔｉｃ Ｓｙｓｔｅｍｓ Ｉｎｖｏｌｖｉｎｇ Ｃｏｎｃａｖｅ ａｎｄ Ｃｏｎｖｅｘ Ｎｏｎｌｉｎｅａｒｉｔｉｅｓ
Ｍｏｈａｍｅｄ Ｅｌ ＭＯｋｈｔａｒ ｏｕｌｄ Ｅｌ Ｍｏｋｈｔａｒ Ｄｅｐａｒｔｅｍｅｎｔ ｏｆＭａｔｈｅｍａｔｉｃｓ，Ｏａｓｓｉｍ Ｕｎｉｖｅｒｓｉｔｙ，ＢＯ ６６４４，Ｂｕｒａｉｄａｈ ５１４５２，Ｋｉｎｇｄｏｍ ｏｆＳａｕｄｉ Ａｒａｂｉａ
Ａｂｓｔｒａｃｔ：Ｉｎ ｔｈｉｓ ｐａｐｅｒ，ｗｅ ｓｔｕｄｉｅｄ ｔｈｅ ｃｏｍｂｉｎｅｄ ｅｆｆｅｃｔ ｏｆ ｃｏｎｃａｖｅ ａｎｄ ｃｏｎｖｅｘ ｎｏｎｌｉｎｅａｒｉｔｉｅｓ ｏｎ ｔｈｅ ｎｕｍｂｅｒ ｏｆ ｐｏｓｉｔｉｖｅ ｓｏｌｕｔｉｏｎｓ ｆｏｒ ａ ｓｅｍｉｌｉｎｅａｒ ｅｌｌｉｐｔｉｃ ｓｙｓｔｅｍ．Ｗｅ ｐｒｏｖｅ ｔｈｅ ｅｘｉｓｔｅｎｃｅ ｏｆ ａｔ ｌｅａｓｔ ｆｏｕｒ ｐｏｓｉｔｉｖｅ ｓｏｌｕｔｉｏｎｓ ｆｏｒ ａ ｓｅｍｉｌｉｎｅａｒ ｅｌｌｉｐｔｉｃ ｓｙｓｔｅｍ ｉｎｖｏｌｖｉｎｇ ｃｏｎｃａｖｅ ａｎｄ ｃｏｎｖｅｘ ｎｏｎｌｉｎｅａｒｉｔｉｅｓ ｂｙ ｕｓｉｎｇ ｔｈｅ Ｎｅｈａｒｉ ｍａｎｉｆｏｌｄ ａｎｄ ｔｈｅ ｃｅｎｔｅｒ ｍａｓｓ ｆｕｎｃｔｉｏｎ．
Ｋｅｙ ｗｏｒｄｓ：Ｅｌｌｉｐｔｉｃ ｓｙｓｔｅｍｓ，ｃｏｎｃａｖｅ ａｎｄ ｃｏｎｖｅｘ ｎｏｎｌｉｎｅａｒｉｔｉｅｓ，Ｎｅｈａｒｉ ｍａｎｉｆｏｌｄ，ｍａｘｉｍｕｍ ｐｒｉｎｃｉｐｌｅ
１．Ｉｎｔｒｏｄｕｃｔｉｏｎ Ｔｈｉｓ ｐａｐｅｒ ｄｅａｌｓ ｗｉｔｈ ｔｈｅ ｅｘｉｓｔｅｎｃｅ ａｎｄ ｎｏｎｅｘｉｓｔｅｎｃｅ ｏｆ ｐｏｓｉｔｉｖｅ ｓｏｌｕｔｉｏｎｓ ｔｏ ｔｈｅ ｆｏｌｌｏｗｉｎｇ ｐｒｏｂｌｅｍ： 一△ ＋ ＝（ａ十１） ｖ ＋ （０［ ＋１ Ｕａ＇ｖｐ ｉｎ Ｑ
一△ｖ＋ｖ＝（ ＋１）ｚｆ ｖ卢＋ （ ＋１） ｖｐ’ｉｎ Ｑ 扰＞０．ｖ＞０ ｉｎ Ｑ
＝ｖ＝０ ｏｎ Ｑ （１） ｗｈｅｒｅ，Ｑ ｉｓ ａ ｂｏｕｎｄｅｄ ｒｅｇｕｌａｒ ｄｏｍａｉｎ ｉｎ冗Ⅳ（Ｎ ３１ ｃｏｎｔａｉｎｉｎｇ ０ ｉｎ ｉｔｓ ｉｎｔｅｒｉｏｒ， １＜Ｐ＝ｏｔ＋ ＋１＜２ 一１，２ ＝２Ｎ／（Ｎ一２）
ｉｓ ｔｈｅ Ｓｏｂｏｌｅｖ ｃｒｉｔｉｃａｌ ｅｘｐｏｎｅｎｔ， ０ ｑ： ＋ ＋１＜１ ａｎｄ ｉｓ ａ ｒｅａｌ
ｐａｒａｍｅｔｅｒ． Ｓｅｍｉｌｉｎｅａｒ ｓｃａｌａｒ ｅｌｌｉｐｔｉｃ ｅｑｕａｔｉｏｎｓ ｗｉｔｈ ｃｏｎｃａｖｅ
ａｎｄ ｃｏｎｖｅｘ ｎｏｎｌｉｎｅａｒｉｔｉｅｓ ａｒｅ ｗｉｄｅｌｙ ｓｔｕｄｉｅｄ；ｗｅ ｒｅｆｅｒ ｔｈｅ ｒｅａｄｅｒｓ ｔｏ【２，３，７，１３，１４，２１】ｅｔｃ．．Ｆｏｒ ｔｈｅ ｓｅｍｉｌｉｎｅａｒ ｅｌｌｉｐｔｉｃ ｓｙｓｔｅｍｓ，ｗｅ ｒｅｆｅｒ ｔｏ Ａｈａｍｍｏｕ［１， ５，６，９，１２，１５，１７，１９］．Ｔｈｅ ｍｏｄｅｌ ｔｙｐｅ ｉｓ ｗｒｉｔｔｅｎ ａｓ ｆｏｌｌｏｗｓ：
－Ａｕ＋ ＝ ＋ｐｇｕ ｉｎ Ｑ Ｏ＜ ∈ ） （２）
Ｗｈｅｎ ｈ三ｇ三１，Ｅｑ．（１）ｃａｎ ｂｅ ｒｅｇａｒｄｅｄ ａｓ ａ Ｃｏｒｒｅｓｐｏｎｄｉｎｇ ａｕｔｈｏｒ：Ｍｏｈａｍｅｄ Ｅ１ Ｍｏｋｈｔａｒ ｏｕｌｄ Ｅｌ Ｍｏｋｈｔａｒ，Ｅ－ｍａｉｌ：ｍｅｄ．ｍｏｋｈｔａｒ６６＠ｙａｈｏｏ．ｆｒ，Ｍ．１ａｂｄｉ＠ｑｕ．ｅｄｕ．ｓａ．
ｐｅｒｔｕｒｂａｔｉｏｎ ｐｒｏｂｌｅｍ ｏｆ ｔｈｅ ｆｏｌｌｏｗｉｎｇ ｅｑｕａｔｉｏｎ ＝ Ｑ ６（Ｑ）． Ｔｈｅ ｎｕｍｂｅｒ ｏｆ ｐｏｓｉｔｉｖｅ ｓｏｌｕｔｉｏｎｓ ｏｆ Ｅｑ．（３）ｉｓ ａｆｆｅｃｔｅｄ ｂｙ ｔｈｅ ｓｈａｐｅ ｏｆ ｔｈｅ ｄｏｍａｉｎ Ｓ２．ｗｈｉｃｈ ｈａｓ ｂｅｅｎ ｔｈｅ ｆｏｃｕｓ ｏｆ ａ ｇｒｅａｔ ｄｅａｌ ｏｆ ｒｅｓｅａｒｃｈ ｉｎ ｒｅｃｅｎｔ ｙｅａｒｓ．Ｆｏｒ ｅｘａｍｐｌｅ，ｗｅ ｑｕｏｔｅ ｔｈｅ ｗｏｒｋｓ ｏｆ Ｂｙｅｏｎ【８１， Ｄａｎｃｅｒ『１０］，Ｄａｍａｓｃｅｌｌｉ，Ｌ．ｅｔ ａｌ［１ １］ａｎｄ Ｗｕ［２２］． Ｃｏｎｓｉｄｅｒｉｎｇ ｔｈｅ ｍｕｌｔｉｐｌｉｃｉｔｙ ｏｆ ｐｏｓｉｔｉｖｅ ｓｏｌｕｔｉｏｎｓ ｔｏ ｐｒｏｂｌｅｍ Ｅｑ．（１），ｗｅ ｅｘｔｅｎｄｅｄ ｔｈｉｓ ｍｅｔｈｏｄ ｉｎ ｔｈｉｓ ｐａｐｅｒ． Ｗｅ （Ｑ）＝
ｎＯｒｎ１
ｃｏｎｓｉｄｅｒ ｔｈｅ Ｓｏｂｏｌｅｖ ｓｐａｃｅｓ （Ｑ）×７－／ （Ｑ）ｗｉｔｈ ｒｅｓｐｅｃｔ ｔｏ ｔｈｅ
（ｕ，ｖ）ｌｌ＝（ ＩＩ ＋ＩｌｕＩＩ ） ； ｗｈｅｒｅ， ＩｌｕＩＩ＝Ｉｌｕ『Ｉ ５＝（』ＱＩＶ ｌ ＋ ）¨
ｉｓ ａ ｓｔａｎｄａｒｄ ｎｏｌＴｒｌ ｉｎ ） Ｓｉｎｃｅ ｏｕｒ ａｐｐｒｏａｃｈ ｉｓ ｖａｒｉａｔｉｏｎａｌ，ｗｅ ｄｅｆｉｎｅ ｔｈｅ ｆｕｎｃｔｉｏｎａｌ ｏｎ （Ｑ）ｂｙ （甜，ｖ） （１／２）ｌｌ（ ， ） 一 ｆＱ／．／ａ＋１ Ｖｆｌ＋ｌ ｄｒ一
Ｊｆ
Ｑ ｂ／ａｌ＋ｌ ｖｆｌ＇＋ｌ ｄｒ
Ｊ Ｑ Ｊ Ｓ
ｃｃａ，卢，≥＝ ｉｎ ｆ
。 ｌｕｌ ｄｘ “∈ ｛ｏ＞【ｌ ）
∈ ＋ ＜ 一 Ｏ ＼－＿● ７２ Ｅｘｉｓｔｅｎｃｅ ｏｆ Ｐｏｓｉｔｉｖｅ Ｓｏｌｕｔｉｏｎｓ ｔｏ Ｓｅｍｉｌｉｎｅａｒ Ｅｌｌｉｐｔｉｃ Ｓｙｓｔｅｍｓ Ｉｎｖｏｌｖｉｎｇ Ｃｏｎｃａｖｅ ａｎｄ Ｃｏｎｖｅｘ Ｎｏｎｌｉｎｅａ ｒｉｔＩｅｓ
（ａ，卢） ， 。，０ ，
：：寿 （“，Ｖ）∈ ＼｛（ｏ，）＞ｆ Ｉ． ｌｖ 出１
Ｆｒｏｍ［１ ６］，Ｃ（口，ｐ）ｉｓ ａｃｈｉｅｖｅｄ．
Ｌｅｍｍａ １ Ｌｅｔ ｂｅ ａ ｄｏｍａｉｎ（ｎｏｔ ｎｅｃｅｓｓａｒｉｌｙ ｂｏｕｎｄｅｄ）ａｎｄ １＜Ｐ＝ ＋ ＋１＜２ 一１．，ｗｅ
Ｃ（ ， ）
［ ） ＋ ｆｌ＋ｌ Ｊ” ’Ｃ 卢
ｗｉｔｈ ［（ ） ¨ ’＋（ ） 叶¨ ’］
ｓｉｍｐｌｙ ｗｒｉｔｔｅｎ ａｓ Ｋ（ａ，ｆｉ）． Ｐｒｏｏｆ Ｔｈｅ ｐｒｏｏｆｉｓ ｅｓｓｅｎｔｉａｌｌｙ ｔｈｅ ｓａｍｅ ａｓ ｉｎ Ｒｅｆ．［５】，ｂｕｔ
ｗｉｔｈ ｍｉｎｏｒ ｍｏｄｉｆｉｃａｔｉｏｎｓ．Ｕｎｄｅｒ ｔｈｅ ａｓｓｕｍｐｔｉｏｎ ≠０，ｏｕｒ ｐｒｏｂｌｅｍ Ｅｑ．（１）ｃａｎ ｒｅｇａｒｄｅｄ ａｓ ａ ｐｅｒｔｕｒｂａｔｉｏｎ ｏｆ ｔｈｅ ｓｙｓｔｅｍ ｗｈｅｎ ：０；ｌｅｔ ｚ＝ ，Ｙ）∈ ｘ ａｎｄ ｌｅｔ Ｏ ｂｅ ａ ｂｏｕｎｄｅｄ ｓｍｏｏｔｈ ｄｏｍａｉｎ ｉｎ ￣’ ；ｄｅｎｏｔｅ ｔｈｅⅣ－ｂａ１１ ＢＮ， ｔｈｅ ｉｎｆｉｎｉｔｅ ｓｔｒｉｐ Ｓ ａｎｄ ｔｈｅ ｆｉｎｉｔｅ ｓｔｒｉｐ Ｓｔ，ｆ ａｓ ｆｏｌｌｏｗｓ：
ＢＸ（Ｚｏ，尸）＝ ∈ Ⅳ／ －ｚ０Ｉ＜ ＞；
Ｓ＝＜ ， ）∈７ Ⅳ／ｘ∈０ ；
＝＜ ，Ｊ，）∈ Ⅳ／，＜Ｙ＜，＞． Ｗｕ［２ １】ｃｏｎｓｉｄｅｒｅｄ ａ ｐｅｒｔｕｒｂａｔｉｏｎ ｏｆ ｔｈｅ ｆｉｎｉｔｅ ｓｔｒｉｐ Ｓ—ｔ．ｔ ｔｈａｔ ｉｓ，ａ ｆｉｎｉｔｅ ｓｔｒｉｐ ｗｉｔｈ ａ ｈｏｌｅ：
Ｏ，＝ ｔ ． ｗｈｅｒｅ ｉｓ ａ ｂｏｕｎｄｅｄ ｄｏｍａｉｎ ｉｎ 冗Ⅳｗｉｔｈ （ｔ）ｃｃ Ｓ，ｆｏｒ ｓｏｍｅｔ，＞０．
Ｉｎ ｏｕｒ ｗｏｒｋ，ｗｅ ｒｅｓｅａｒｃｈ ｔｈｅ ｃｒｉｔｉｃａｌ ｐｏｉｎｔｓ ａｓ ｔｈｅ ｍｉｎｉｍｉｚｅｒｓ ｏｆ ｔｈｅ ｅｎｅｒｇｙ ｆｕｎｃｔｉｏｎａｌ ａｓｓｏｃｉａｔｅｄ ｗｉｔｈ ｔｈｅ ｐｒｏｂｌｅｍ Ｅｑ．（１）ｏｎ ｔｈｅ ｃｏｎｓｔｒａｉｎｔ ｄｅｆｉｎｅｄ ｂｙ ｔｈｅ Ｎｅｈａｒｉ ｍａｎｉｆｏｌｄ，ｗｈｉｃｈ ａｒｅ ｓｏｌｕｔｉｏｎｓ ｏｆ ｏｕｒ ｓｙｓｔｅｍ． Ｔｈｅｏｒｅｍ １ Ａｓｓｕｍｅ ｔｈａｔ
Ｎ ３，１＜Ｐ＝ ＋ ＋１＜２ 一１ ａｎｄ ０ ｑ＝ ＋ ＋１＜１．Ｆｏｒ ｅａｃｈ ｔ＞ｔ０，ｔｈｅｒｅ
ｅｘｉｓｔｓ人＞０，ｓｏ ｔｈｅ ｐｒｏｂｌｅｍ Ｅｑ．（１）ｈａｓ ａｔ ｌｅａｓｔ ｆｏｕｒ
ｐｏｓｉｔｉｖｅ ｓｏｌｕｔｉｏｎｓ ｆｏｒ ａ１１ ∈（Ｏ，人）．
Ｔｈｉｓ ｐａｐｅｒ ｉｓ ｏｒｇａｎｉｚｅｄ ａｓ ｆｏｌｌｏｗｓ：ｉｎ ｔｈｅ Ｓｅｃｔｉｏｎ ２， ｗｅ ｇｉｖｅ ｓｏｍｅ ｐｒｅｌｉｍｉｎａｒｉｅｓ；ｗｈｉｌｅ Ｓｅｃｔｉｏｎ ３ ｉｓ ｄｅｖｏｔｅｄ ｔｏ ｔｈｅ ｐｒｏｏｆｏｆｔｈｅｏｒｅｍ １． ２．Ｐｒｅｌｉｍｉｎａｒｉｅｓ ２．，Ｎｅｈａｒｉ Ｍａｎｉｆｏｌｄ Ｃｏｎｓｉｄｅｒｉｎｇ ｔｈｅ Ｎｅｈａｒｉ ｍａｎｉｆｏｌｄ： Ｍ （Ｑ）＝ ＜（ ）∈ （Ｑ）＼＜（０，ｏ）＞／（ （ ），（ ））＝ Ｄｅｆｉｎｅ （ ，Ｖ）＝ （ ，Ｖ），（ ，ｖ）） ＝ＩＩ（ｈｉ，ｖ）ＩＩ 一（ ＋１）ｌ ｇａ＋１ １，卢制ｄｘ— Ｐ（ｑ＋１）Ｉ甜口 ｖｐ ＋ ｄｘ ￡２ Ｔｈｅｎ，ｆｏｒ（ ，ｖ）∈Ａ４ （Ｑ） ＋ （ ｔ（ ），ｕ，ｖ））：２１１（ｕ，ｖ）ｌｌ 一 ０＞ １） ．ｆＱ １／ａ＋ｌ Ｖｆｌ＋ｌ 一 （ｇ＋】） ／ｇａ＇＋１ Ｖｆｌ＇＋ｌ ＝（１～ｐ）ｌｌ（ｕ，ｖ）ｌｌ 一 （ｇ— ）（１＋ｇ）ｆ Ｕｄ＋ｌ 卢 ＋ ｄｘ ＝（１～ｑ）ｌＩ（ ，Ｖ）ｆｆ 一（ 十１）（ —ｇ）ｆ ｂｔａ＋１ｖ ｄｘ． ０ Ｓ２ （４） Ｎｏｗ，ａｓ ｉｎ Ｔａｒａｎｔｅｌｌｏ Ｉ１ ８】，ｗｅ ｓｐｌｉｔ ＭⅣ（ｆ２）ｉｎ ｔｈｒｅｅ ｐａｒｔｓ： Ｍ （Ｑ） ＜（ ）∈Ｍ （Ｑ）：（ ｖ），（ ））＞０＞ 朋 （Ｑ） ＜（ ） （Ｑ）：（ ｖ），（ ））：０＞ Ｍ （Ｑ） ＜（ ）∈Ｍ （Ｑ）：（ ｖ），（ ））＜０） Ｉｔ ｉｓ ｗｅｌｌ ｋｎｏｗｎ ｔｈａｔ ｉｓ ｏｆｃｌａｓｓ Ｃｌ ｉｎ ７－ｔ（Ｑ） ａｎｄ ｔｈｅ ｓｏｌｕｔｉｏｎｓ ｏｆ Ｅｑ．（１）ａｒｅ ｔｈｅ ｃｒｉｔｉｃａｌ ｐｏｉｎｔｓ ｏｆ ｗｈｉｃｈ ｉｓ ｎｏｔ ｂｏｕｎｄｅｄ ｂｅｌｏｗ ｏｎ （Ｑ）．Ｎｏｔｅ ｔｈａｔ ）ｃｏｎｔａｉｎｓ ｅｖｅｒｙ ｎｏｎｔｒｉｖｉａ１ ｓｏｌｕｔｉｏｎ ｏｆ ｔｈｅ ｐｒｏｂｌｅｍ Ｅｑ．（１）．Ｍｏｒｅｏｖｅｒ，ｗｅ ｈａｖｅ ｔｈｅ ｆｏｌｌｏｗｉｎｇ ｒｅｓｕｌｔｓ： Ｌｅｔ ０≥ ［ ］［ ］ （１＿叮） １） １）．（１＋ｑ） （ ， ）Ｃ（ ， ）） ｆＱｆ ＋ （５） ｗｈｅｒｅ， ｉｓ ｔｈｅ Ｌｅｂｅｓｇｕｅ ｍｅａｓｕｒｅ ｏｆ ｄｏｍａｉｎ Ｑ． Ｌｅｍｍａ １ Ｗｅ ｈａｖｅ Ｍ ）＝ ｆｏｒ ａ１１ ∈（Ｏ，／２０）． Ｐｒｏｏｆ Ｌｅｔ ｕｓ ｒｅａｓｏｎ ｂｙ ｃｏｎｔｒａｄｉｃｔｉｏｎ．Ｓｕｐｐｏｓｅ