ON SELECTION OF SOLUTIONS TO VECTORIAL HAMILTON-JACOBI SYSTEM

ON SELECTION OF SOLUTIONS TO VECTORIALHAMILTON-JACOBI SYSTEMBAISHENG YANAbstract.In this paper,we discuss some principles on the selection ofthe solutions of a special vectorial Hamilton-Jacobi system defined by(1.1)below.Wefirst classify the equivalent solutions based on a well-known construction and then discuss some selection principles usingintuitive descriptions of coarseness or maximality of solutions.We alsodiscuss a selection principle based on comparison of the projections of asolution with certain smooth functions and introduce a notion of partialviscosity solutions for the vectorial Hamilton-Jacobi equation,whichgeneralizes the well-known notion of viscosity solutions in the scalarcase.Since we mainly work in the framework of Sobolev solutions,weshow that,as in the scalar case,for certain domains the partial viscosityprinciple may rule out many interesting Lipschitz solutions.1.IntroductionIn this paper,we discuss some principles on the selection of the solutions of a special vectorial Hamilton-Jacobi system defined by(1.1)Du(x)∈K a.e.x∈Ω,where K is a given set of m×n real matrices,Ωis a bounded domain in R n,u is a mapping fromΩto R m and Du is the gradient matrix of u: Du(x)=(∂u i/∂x j),i=1,2,...,m,j=1,2,...,n.We study the solutions of system(1.1)in the usual Sobolev space W1,p(Ω;R m)for some1≤p≤∞with an affine Dirichlet boundary condition(1.2)u|∂Ω=ξx+b,whereξis a given m×n matrix and b is a given vector in R m.As usual,the(Ω;R m),the closure boundary condition(1.2)means that u−ξx−b∈W1,pin W1,p(Ω;R m)of the set of smooth mappings compactly supported inΩ. It is clear that ifξ∈K the affine map u=ξx+b is always a solu-tion to problem(1.1)-(1.2)and,in view of regularity,this trivial solution is considered as being more favorable than other solutions(if exist).Of Key words and phrases.Selection of solutions,coarse,L p-maximal,and partial viscosity solutions.12BAISHENG YANcourse,a real interest of studying problem(1.1)-(1.2)lies in that even when ξ/∈K the problem may have solutions.The reason we study system(1.1) in the Sobolev spaces rather than some smooth function spaces,such as C1(Ω;R m),is that in application to the phase transition problem for a martensitic material the set K models the set of homogeneous strains min-imizing the material’s elastic potential energy,the so-called energy well, which often has several disjoint connected components;usually,a smooth solution u of(1.1)will force Du to lie only on one of the components of K and thus prohibit the formation of physically important structures in the phase transition(see[2,11]).In the Sobolev space setting,the nontrivial solutions of(1.1)-(1.2),if exist,are not unique and may have differentfine structures(microstructures).However,physically,these solutions may all exhibit the same effective(macroscopically observable)quantities of the ma-terial.This paper is devoted to a mathematical issue regarding the selection of solutions of problem(1.1)-(1.2)in a Sobolev space.The existence of certain weak solutions to the problem(1.1)-(1.2)has been established in a series of papers of Dacorogna and Marcellini[6,7]and M¨u ller andˇSver´a k[9,10],using different methods.They showed that if K is compact and has a sufficiently rich structure then the Dirichlet problem (1.1)-(1.2)possesses infinitely many Lipschitz solutions for certain boundary dataξx+b withξ/∈K(note that the constant b is irrelevant).Their methods seem not applicable to the systems(1.1)defined by unbounded sets K.Nevertheless,in the present paper,we shall assume that,for certain given boundary dataξ,problem(1.1)-(1.2)has solutions in W1,p(Ω;R m). Wefirst classify the equivalent solutions using a principle motivated by a well-known construction of self-similarity(see the proof of Proposition2.1) and then discuss some selection principles based on an intuitive descrip-tion of coarseness or maximality of a solution also motivated by the same construction.All these principles rely essentially on the comparison of all equivalent solutions.For example,an L p-maximal solution defined later is a solution for which a certain L p-norm is not smaller than the norm of other equivalent solutions.Since the class of equivalent solutions is usually difficult to study,these selection principles can hardly be effective in appli-cations;however,they provide some new understanding of the structures of Sobolev solutions to the problem(1.1)-(1.2).To discuss yet another selection principle,we note that if u∈W1,p(Ω;R m) is a solution to the problem(1.1)-(1.2)then,for any unit vector h∈R m, the functionϕ(x)=h·u(x)∈W1,p(Ω)is a solution of the problem(1.3)Dϕ(x)∈Γ a.e.x∈Ω,ϕ|∂Ω=(h tξ)x,ON SELECTION OF SOLUTIONS3 whereΓ=K h={h tη|η∈K}is a subset of M1×n∼=R n.The equation Dϕ(x)∈Γcan be considered as a special scalar Hamilton-Jacobi equation H(Dϕ(x))=0defined by certain Hamiltonians H(p),i.e.,functions vanish-ing exactly onΓ.In the setting of Sobolev solutions,the choice of different Hamiltonians does not affect the solutions of problem(1.3).However,later on,we shall use a particular Hamiltonian to impose some constraints on these solutions.In the scalar case,an important and elegant method to generalize weak solutions of the scalar Hamilton-Jacobi equations to the class of continuous functions is the well-known viscosity solution method introduced by Crandall and Lions[5](see also[4,8]),which is based on the comparison of a solution with certain smooth functions and is compatible with certain regularizing procedures(the vanishing viscosity method).We generalize the notion of the viscosity solutions to the special vectorial Hamilton-Jacobi equations as given by(1.1).Since there is no direct way to compare the vector valued functions,our comparison will be restricted to the projections of a solution in certain directions.This leads us to studying the equations(1.3).Although these equations may be far from completely char-acterizing the original vectorial Hamilton-Jacobi equation,they certainly impose some reasonable partial restrictions on the solutions.In this regard, we define a partial viscosity solution(in direction h)of(1.1)-(1.2)to be those solutions u for whichϕ(x)=h·u(x)is a viscosity solution relevant to prob-lem(1.3).This principle does not involve comparison with other equivalent solutions.Finally,since we mainly work in the framework of Sobolev solutions,even in a scalar case,the existence of W1,p-viscosity solutions seems to depend strongly on the geometry of the domain,as has been recently shown in the paper[3].For instance,they showed that in the scalar case(m=1),for certain domainsΩand boundary dataξx+b,there may exist infinitely many Lipschitz solutions to problem(1.1)-(1.2)but no Lipschitz-viscosity solutions at all.In the vectorial case,we shall see that the partial viscosity principle also rules out many interesting Lipschitz solutions for certain domains(for example,all smooth domains).As an example,we show that for smooth domains the M¨u ller-ˇSver´a k solutions of the two-well problem in[9]are nota partial viscosity solution(in certain directions).2.Classification of Sobolev solutionsIn the rest of the paper,we assumeΩ⊂R n is a bounded domain with Lipschitz boundary∂Ω,although most of the results hold for any open4BAISHENG YANbounded setsΩwith|∂Ω|=0.We use M to denote the space M m×n of all m×n real matrices with norm|ξ|=(ξtξ)1/2.Definition2.1.Let1≤p≤∞,K⊆M be given.We defineβp(K)to be the set of allξ∈M such that the problem(2.1)Du(x)∈K a.e.x∈Ω;u|∂Ω=ξxhas a solution u in W1,p(Ω;R m).Moreover,forξ∈βp(K),let S p(Ω)be the set of solutions u of problem(2.1)in W1,p(Ω;R m).Clearly,K⊆βp(K)andβp(K)⊆βq(K)if p>q.Since any Sobolev map with essentially bounded gradient can be considered as a Lipschitz map,it follows thatβp(K)=βq(K)for all1≤q<p≤∞for a bounded set K. Later,we shall give an example of an unbounded set K such thatβp(K) is strictly contained inβq(K)for p>q.For general sets K,we have the following result.Proposition 2.1.The setβp(K)is independent of the smooth bounded domainΩ.Moreover,if S p(Ω)contains a nontrivial solution u=ξx then it must contain infinitely many solutions.Proof.The proof of this result is not difficult,but involves an important construction that motivates much of the consideration of the present paper. We discuss this construction in the proof.To prove the independence ofβp(K)onΩ,suppose for a bounded Lip-schitz domainΩproblem(2.1)has a solution u∈W1,p(Ω;R m).Let U be another bounded Lipschitz domain in R n.(Note that U can beΩitself.) We show that the Dirichlet problem(2.2)Dv(y)∈K a.e.y∈U;v|∂U=ξyalso has a solution v in W1,p(U;R m);this proves the independence result. To construct a solution to problem(2.2),we define open setsΩa,ǫfor a∈R n,ǫ>0by(2.3)Ωa,ǫ≡a+ǫΩ={a+ǫx|x∈Ω}.Then,for any y∈U,there exists r y>0such thatΩy,r⊂U for all r∈(0,r y).Notice that the family{Ωy,r|y∈U,0<r<r y}forms a Vitali cover for U;thus by the Vitali covering lemma,there exist disjoint setsΩk,and a set N of measure (k=1,2,...),whereΩk=y k+r kΩ,0<r k<r ykzero such that U=∪∞k=1Ωk∪N.Define a map v:U→R m by (2.4)v(y)= ξy k+r k u(y−y k r k),y∈Ωk,k=1,2,...;ξy,otherwise in U.ON SELECTION OF SOLUTIONS5It is easy to verify that v−ξy∈W1,p(U;R m)and that Dv(y)∈K for a.e. y∈U.Therefore v is a solution to the problem(2.2).To prove the second part of the proposition,we choose U=Ω.Let u=ξx be a nontrivial solution of(2.1).The construction above shows that v is a solution of(2.1)and v−ξy L p(Ω)≤(sup y∈Ωr y) u−ξx L p(Ω).Hence,by choosing r y<ǫfor all y∈Ω,we can construct a sequence of solutions{vǫ} in S p(Ω)such that vǫ(y)→ξy in L p(Ω)asǫ→0.This proves that problem (2.1)has infinitely many solutions.The proof of the proposition is now completed. Note that the map v defined by(2.4)above satisfies(2.5)1|U| Uψ(Dv(x))dx=1|Ω|Ωψ(Du(x))dxfor all continuous functionsψsatisfying|ψ(η)|≤C(|η|p+1)(for p=∞we need|ψ(η)|<∞).In the following,we denote by C p(M)the linear space of all such continuous functionsψ.Definition2.2.We say two solutions u,v in S p(Ω)are equivalent and write u∼v if for allψ∈C p(M)(2.6) Ωψ(Dv(x))dx= Ωψ(Du(x))dx.This relation is indeed an equivalence relation on S p(Ω).Any solution v (with U=Ω)as constructed in the proof above is equivalent to u.We denote the quotient set of the equivalence classes by S p(Ω).Therefore the solution set S p(Ω)is decomposed as the union of the disjoint equivalence classes:S p(Ω)=∪{[u]|[u]∈S p(Ω)}.Each[u]in S p(Ω)defines a linear functionalµ[u]on C p(M)throughµ[u],ψ =1|Ω| Ωψ(Du(x))dx,which,by the Riesz representation theorem,also induces a probability Radon measureµ[u]on M supported on the closure¯K.For applications in non-linear elasticity,ifψ(Du(x))is a quantity depending on the deformation gradient Du(x),solutions in an equivalence class all give the same effective or macroscopically observable quantity µ[u],ψ ,although microscopically these solutions are distinct and have differentfine structures.For general energy minimizing sequences,such a description of macroscopically observ-able quantities has been successfully given in terms of Young measures(see [2]).6BAISHENG YANLemma2.2.For each p∈[1,∞],the functionΨ([u])= Du−ξ L p(Ω)is a well-defined function on S p(Ω).Proof.If1≤p<∞we chooseψ(η)=|η−ξ|p in(2.6).While,if p=∞we chooseψ(η)=|η−ξ|q for any q<∞,raise the equality(2.6)to the power 1/q and then let q→∞.Therefore Dv−ξ p= Du−ξ p if v∼u for all 1≤p≤∞;soΨ([u])is well-defined on S p(Ω). Note thatΨ([u])=0if and only ifξ∈K and[u]=[ξx];in this case,the class[ξx]contains only the trivial solutionξx.Therefore,to some extent,the functionΨ([u])provides a scale to measure the L p-deviation of the gradient Du from the constant matrixξ.To keep the L p-deviation as small as possible, the solution class[u]is preferable over the class[v]ifΨ([u])<Ψ([v]).In this sense,it would be interesting to study the problemΨ([u]).¯m=inf[u]∈S p(Ω)Since we do not know much about the structure of S p(Ω),it seems impossible to analyze this minimum problem.In the following,we shallfix a solution class[u]so thatΨ([u])is close to the number¯m with some precision and study the selection principles for solutions in the same class[u].Finally,we give an example of set K for whichβp(K)may depend on p≥1dramatically.Example.Let n≥3and K=C1={ξ∈M n×n||ξ|n=n n/2detξ}be the set of conformal matrices.Supposeξ∈βp(C1).Then there exists a map u∈W1,p(Ω;R n)such that(2.7)|Du(x)|n=n n/2det Du(x) a.e.x∈Ω;u|∂Ω=ξx.If p≥n,we can integrate the equation above overΩand use the boundary condition,the Jensen inequality,and a property of the determinant function to obtain|ξ|n≤n n/2detξ.Hence,by Hadamard’s inequality,ξ∈C1.This showsβn(C1)=C1.Let J be an n×n matrix with J t J=I,det J=−1and let u0(x)= Jx/|x|2.Then an easy calculation shows that u0∈W1,p(B;R n)for all p< n/2,where B is the unit ball in R n,and that u=u0solves(2.7)withΩ=B,ξ=J.This shows J∈βp(C1)for all p<n/2;henceβn(C1)=βp(C1)for all p<n/2.For further information onβp(C1)when n/2≤p<n,see [12,13].3.Selection of solutions in the same classFor any given solution u of problem(2.1),the map v constructed as in the proof of Proposition2.1is an equivalent solution and hasfiner structuresON SELECTION OF SOLUTIONS7 and perhaps more singularities.We don’t know whether or not all solutions v∈[u]are obtained this way.But certainly this construction renders us some reasonable and intuitive standards on selecting solutions in the same solution class.Intuitively,all our selection principles should conform to the principle that solutions with lessfine structures or singularities be more preferable.3.1.Coarseness of solutions.In the construction(2.4),a“patch”of v, v|Ωa,ǫ,totally determines u inΩand has the same intrinsic structure as u (e.g.,boundary data,singularities);such solutions v are considered as having morefine structures or singularities than u and thus being unfavorable. Definition3.1.Let u,v∈S p(Ω).We say u is coarsened from v and write u≻v provided that u∼v and there exists a“patch”Ωa,ǫ⊂⊂Ωsuch that u(y)=ǫ−1(v(a+ǫy)−ξa)for almost every y∈Ω.We say u is self-similar if u≻u.We say u is a coarse solution if no solutions are coarsened from u. Note that the relation“≻”is transitive:u≻v,v≻w=⇒u≻w. Indeed,if forΩa,ǫ,Ωb,δ⊂⊂Ωwe have u(y)=ǫ−1(v(a+ǫy)−ξa)and v(y)=δ−1(w(b+δy)−ξb)for y∈Ω,then for the patchΩc,γ⊂⊂Ω,where c=a+ǫb,γ=ǫδ,we have(3.1)u(y)=γ−1(w(c+γy)−ξc) a.e.y∈Ω.Therefore,if u≻v and v≻u then u,v are both self-similar.Proposition3.1.Let u∈S p(Ω)andΩa,ǫ⊂⊂Ω.Suppose u(y)=ǫ−1(u(a+ǫy)−ξa)for almost every y∈Ω.If u is differentiable at the point aǫ=a1−ǫ,then u≡ξx inΩ.Therefore,the only smooth self-similar solution in S p(Ω) is the trivial solution u=ξx.Proof.Let f(x)=u(x)−ξx.Then,using u(y)=ǫ−1(u(a+ǫy)−ξa)for almost every y∈Ωand iterating(3.1),it is easy to verify that for k=1,2,...f(y)=ǫ−k f(aǫ+ǫk(y−aǫ)).From this and the fact that f is differentiable at aǫandǫ<1,we easily obtain by letting k→∞that f(y)=Df(aǫ)(y−aǫ)for almost every y∈Ω. Since f∈W1,p(Ω;R m)we must have Df(aǫ)=0and thus f≡0;this shows u≡ξx and completes the proof. Remark.Define“ ”by letting w v if and only if w≻v or w=v. The previous result shows that,if u∈S p(Ω)is a nontrivial solution,the relation“ ”is a partial ordering on the set[u]∩C1(Ω;R m).However,as we mentioned in the introduction,in applications,it may very well be possible that the set[u]∩C1(Ω;R m)is empty if u∈S p(Ω)is nontrivial.8BAISHENG YANThe following result shows that every nontrivial solution generates an equivalent(non smooth)self-similar solution.Proposition3.2.If u∈S p(Ω)is a nontrivial solution,then there exists a self-similar solution v∈[u].Proof.LetΩ1=Ωa,ǫ⊂⊂Ωfor some a∈R n,ǫ>0.As in the proof of Proposition2.1,using u∈S p(Ω)we construct a solution u1∈S p(G1),where G1=Ω\Ω1.Let f=u1−ξx∈W1,p(G0;R m).DefineΩk=a+ǫΩk−1and G k=Ωk−1\Ωk inductively for k=2,3,...and note that G k is the imageof G1under the map L k(x)=aǫ+ǫk(x−aǫ),where,as before,aǫ=a1−ǫ.We now define a map v onΩby letting v(x)=ξx+ǫk f(y)if x=L k(y)for some k=1,2,...and y∈G1.Then it can be verified that v∈S p(Ω),v∼u and v≻v. Remark.In some cases,we can construct a self-similar solution v as in the proof above such that v∈C1(Ω\{aǫ};R m).Therefore,the smoothness condition in Proposition3.1can not be dropped.3.2.Structure of a non-coarse solution.Suppose u∈S p(Ω)is not a coarse solution.Then,from the definition,the familyF={Ωa,ǫ⊂⊂Ω u∈S p(Ωa,ǫ), Ωa,ǫψ(Du)=ǫn Ωψ(Du),∀ψ∈C p(M)}is non-empty.Define a relation“≤”on F by letting U≤V if and only if U⊂⊂V or U=V.Then it is easily seen that(F,≤)is a partially ordered set(poset).Therefore,the HausdorffMaximal Principle asserts that there exists a maximal totally ordered subset M⊂F.Letσ=inf{|U| U∈M}. Lemma3.3.If there exists U∈M such that|U|=σ,then u∈S p(U)is a coarse solution on U.Proof.If u∈S p(U)is not a coarse solution,by definition,we have U1= U a,ǫ⊂⊂U such that u∈S p(U1)and U1ψ(Du)=ǫn Uψ(Du)for allψ∈C p(M).As U=Ωb,δ∈F,we easily see that U1∈F.We now show U1∈M and thus obtain the desired contradiction.Since U∈M and M is totally ordered,for any V∈M and V=U,we have either U⊂⊂V or V⊂⊂U; but the latter cannot happen since otherwise|V|<|U|=σ,contradicting the definition ofσ.Therefore,we have U⊂⊂V for all V∈M,V=U. Therefore U1≤V for all V∈M.This shows that the set M′={U1,M}is totally ordered by“≤”and thus by the maximality of M we have U1∈M and hence the contradiction.ON SELECTION OF SOLUTIONS 9Lemma 3.4.If |U |>σfor all U ∈M ,then σ=0.In this case,there exists a sequence {U k }⊂M such that U k +1⊂⊂U k for all k =1,2,...and ∩k U k ={a }for some a ∈U 1.Proof.Let {U k }⊂M be a sequence such that U k +1⊂⊂U k for all k =1,2,...and lim k →∞|U k |=σ.Let U ∞=∩k U k .It is easy to see U ∞=∩k U k is a compact set in U 1and |U ∞|=σ.Since each U k is of the form a k +ǫk Ω,it follows that U ∞=a +ǫΩfor some a ∈R n and ǫ≥0.The result will follow if ǫ=0and thus σ=0.Suppose for the contrary ǫ>0.Since U k ∈F we can also prove that Ωa,ǫ,the interior of U ∞,belongs to F .We claim Ωa,ǫ⊂⊂U for all U ∈M ,and therefore the set {Ωa,ǫ,M}is totally ordered by “≤”;this and the maximality of M would imply Ωa,ǫ∈M ,which contradicts with the assumption and the fact that |Ωa,ǫ|=σ.To prove our claim,we take an arbitrary U ∈M .By assumption,|U |>σ=lim k →∞|U k |,and hence |U |>|U k |for all sufficiently large k.Since M is totally ordered,this implies U k ⊂⊂U and hence Ωa,ǫ⊂⊂U k ⊂⊂U,proving the claim. Summarizing the above results,we have the following structural theorem for non-coarse solutions.In the sequel,we write Ωk ց{a }if Ωk +1⊂⊂Ωk and ∩k Ωk ={a }.Theorem 3.5.Let u ∈S p (Ω)be a non-coarse solution.Then,either (i )for some Ωa,ǫ⊂⊂Ω,u ∈S p (Ωa,ǫ)is a coarse solution on Ωa,ǫand Ωa,ǫψ(Du )=ǫn Ωψ(Du )for all ψ∈C p (M ),or(ii )there exist Ωk =Ωa k ,ǫk ⊂⊂Ωsuch that Ωk ց{a },u ∈S p (Ωk )and Ωkψ(Du )=ǫn k Ωψ(Du )for all ψ∈C p (M ).3.3.L p -maximal solutions.We now discuss another selection principle based on the L p -norm of u −ξx p .We say a solution u ∈S p (Ω)is L p -maximal provided that w −ξx L p (Ω)≥ v −ξx L p (Ω)for all v ∈[u ].If ξ∈K then the trivial solution ξx is L p -maximal since it is the only solution in [ξx ].In general,for a nontrivial solution u ,since [u ]is an infinite set,the existence of an L p -maximal solution in [u ]is not guaranteed.Let Φ(v )= v −ξx L p (Ω)and consider a Φ-maximizing sequence {u j }in [u ]:(3.2)lim j →∞Φ(u j )=ˆm ≡sup {Φ(v ) v ∈[u ]}.Since Du j −ξ p = Du −ξ p <∞is independent of j ,it follows that u j converges weakly (or weakly *if p =∞)to a map ¯u in W 1,p (Ω;R m )as j →∞and ¯u |∂Ω=ξx .From the Sobolev embedding theorem,u j →¯u in L p (Ω)and hence Φ(¯u )=ˆm.However,¯u may not be a solution.10BAISHENG YANFrom the weak lower semicontinuity of the norm,we have D¯u−ξ p≤ Du−ξ p.The following gives a sufficient condition for the weak limit¯u to be an L p-maximal solution.Proposition3.6.Let1<p<∞.Then the limit map¯u above belongs to[u] if and only if D¯u−ξ p= Du−ξ p.In this case,¯u is also an L p-maximal solution.Proof.If¯u∈[u],then D¯u−ξ p= Du−ξ p follows easily from Lemma 2.2.To prove the converse,suppose D¯u−ξ p= Du−ξ p.Since u j∈[u], we have Du j−ξ p= Du−ξ p= D¯u−ξ p for all j and hence Du j→D¯u strongly in L p(Ω),which impliesΩψ(D¯u(x))dx=lim j→∞ Ωψ(Du j(x))dx= Ωψ(Du(x))dxfor allψ∈C p(M).Using this identity withψ(η)=d K(η),the distance function to K,we obtain D¯u(x)∈K for a.e.x∈Ω.Since¯u∈W1,p(Ω;R m) and¯u|∂Ω=ξx,it follows that¯u∈S p(Ω)and¯u∼u,and hence¯u∈[u]. Moreover,by definition,¯u is also an L p-maximal solution.Remark.If it happens that[u]∈S p(Ω)satisfiesΨ([u])=min Sp(Ω)Ψ,thenthe result above shows that any limit map¯u obtained as above is either an L p-maximal solution in[u]or satisfies D¯u−ξ p< Dv−ξ p for all v∈S p(Ω).We now study the structure of the L p-maximal solutions.Lemma3.7.Let u∈S p(Ω)be L p-maximal andΩa,ǫ⊂⊂Ω.If u∈S p(Ωa,ǫ) and Ωa,ǫψ(Du)=ǫn Ωψ(Du)for allψ∈C p(M),then u−ξx L p(Ωa,ǫ)≤ǫ1+n p u−ξx L p(Ω).Proof.Define v(x)=ǫ−1(u(a+ǫx)−ξa)for x∈Ω.Then it is easy to seethat v∼u and u−ξx L p(Ωa,ǫ)=ǫ1+n p v−ξx L p(Ω).Hence the resultfollows from the L p-maximality of u. Combining this lemma with Theorem3.5,we have the following. Theorem3.8.Let u∈S p(Ω)be L p-maximal.Then,either (i)for someΩa,ǫ⊆Ω,u∈S p(Ωa,ǫ)is coarse onΩa,ǫ, Ωa,ǫψ(Du)=ǫn Ωψ(Du)for allψ∈C p(M)and u−ξx L p(Ωa,ǫ)≤ǫ1+n p u−ξx L p(Ω), or(ii)there existΩk=Ωak,ǫk⊂⊂Ωsuch thatΩkց{a}, Ωkψ(Du)=ǫn k Ωψ(Du)for allψ∈C p(M)and u−ξx L p(Ωk)≤ǫ1+n p k u−ξx L p(Ω).ON SELECTION OF SOLUTIONS 113.4.Directionally maximal solutions.In the following,we consider the special case when p =∞.In this case,a solution u ∈S ∞(Ω)can be viewed as a Lipschitz continuous map on ¯Ω.For any given Lipschitz map f :¯Ω→R m ,a unit vector h ∈R m is called a maximal direction of f on ¯Ωprovided f ∞=h ·f (¯x )for some ¯x ∈¯Ω.If f =0,then there exists at least one ¯x ∈¯Ωsuch that f ∞=|f (¯x )|>0,and hence the vector h =f (¯x )/|f (¯x )|is a maximal direction of f.Definition 3.2.Let u ∈S ∞(Ω),h ∈S m −1and u h (x )=h ·(u (x )−ξx ).We say u is h -directionally maximal provided that u h ∞≥ v h ∞for all v ∈[u ].Proposition 3.9.If u ∈S ∞(Ω)is L ∞-maximal and h ∈S m −1is a maximal direction then u is h -directionally maximal.Moreover,u ∈S ∞(Ω)is L ∞-maximal if it is h -directionally maximal for every h ∈S m −1.Proof.If u ∈S ∞(Ω)is L ∞-maximal and h is a maximal direction of u −ξx ,then(3.3) u h ∞= u −ξx ∞≥ v −ξx ∞≥ v h ′ ∞for any v ∈[u ]and h ′∈S m −1.Therefore,by definition,u is h -directionally maximal.On the other hand,if u h ∞≥ v h ∞for all h ∈S m −1and v ∈[u ],choosing h to be a maximal direction of v −ξx,we haveu −ξx ∞≥ u h ∞≥ v h ∞= v −ξx ∞,and hence u is L ∞-maximal.The proof is completed.4.Partial viscosity solutionsIn this section,we assume K ⊂M is a compact set and study the Lipschitz solutions u ∈S ∞(Ω)of problem (2.1).For any unit vector h ∈R m ,the function ϕ(x )=h ·u (x )is a Lipschitz solution of the problem(4.1)Dϕ(x )∈Γ a.e.x ∈Ω,ϕ|∂Ω=(h t ξ)x,where Γ=K h ={h t η|η∈K }is a subset of M 1×n ∼=R n .4.1.Partial viscosity solutions.We first review the definition of viscosity solutions introduced by Crandall and Lions [5](see also [4,8])for a special class of Hamilton-Jacobi equations relevant to the problem (4.1)above.Definition 4.1.Let H :R n →R and φ∈C (¯Ω)be given.Then,(a )φis called a viscosity super-solution of H (Dφ)=0if,for each w ∈C ∞(Ω)and for each ¯x ∈Ωat which φ−w has a local minimum,we have H (Dw (¯x ))≥0;12BAISHENG YAN(b)φis called a viscosity sub-solution of H(Dφ)=0if,for each w∈C∞(Ω)and for each¯x∈Ωat whichφ−w has a local maximum,we have H(Dw(¯x))≤0;(c)φis called a viscosity solution of H(Dφ)=0ifφis both viscosity super-solution and viscosity sub-solution of H(Dφ)=0.Remark.A viscosity solutionφto the problemH(Dφ(x))=0,φ|∂Ω=g(g given)can be studied as a certain limit of the solutionsφǫof the following regular-izing problem asǫ→0+(4.2)−ǫ∆φǫ+H(Dφǫ)=0,φǫ|∂Ω=g.The problem(4.2)makes no sense for the vector valued functionsφ:Ω→R m when m≥2.be the distance function to K h.We can write(4.1)asLet d Kh(Dϕ(x))=0 a.e.x∈Ω,ϕ|∂Ω=(h tξ)x.(4.3)d Kh,we may use other Of course,instead of using the distance function d KhHamiltonians to write the equation(4.1).For instance,if f:R n→R is any continuous function such that the zero set f−1(0)=K h then the equation Dϕ(x)∈K h can be written as f(Dϕ(x))=0.The class of Lipschitz solutions will not be affected by the choice of different Hamiltonians f. However,a different Hamiltonian f will give a different class of viscosity solutions to the problem(4.1).Definition4.2.We say u∈S∞(Ω)is a partial viscosity solution in direction h if the functionϕ(x)=h·u(x)is a viscosity solution of(4.3).Lemma 4.1.A Lipschitz continuous functionϕ:¯Ω→R withϕ|∂Ω= (h tξ)x is a viscosity solution of(4.3)if and only if D+ϕ(x)⊆K h for all x∈Ω,where(4.4)D+ϕ(x)≡ p∈R n lim sup y→x,y∈Ωϕ(y)−ϕ(x)−p·(y−x)≤0 .|y−x|≥0,any Lipschitz functionϕon¯Ωwithϕ|∂Ω=(h tξ)x Proof.Since d Khis automatically a viscosity super-solution of(4.3).Therefore,any such functionϕis a viscosity solution if and only if it is a viscosity sub-solution; the latter condition is equivalent to D+ϕ(x)⊆K h for all x∈Ω(see[4, 8]). Proposition4.2.Let u∈S∞(Ω)be a partial viscosity solution in direction h.Then either h tξ∈K h orϕ(x)=h·u(x)<(h tξ)x for all x∈Ω.ON SELECTION OF SOLUTIONS13 Proof.Let u h(x)=ϕ(x)−(h tξ)x and consider max¯Ωu h=ˆm.Then either (i)there exists a¯x∈Ωsuch that u h(¯x)=ˆm,or(ii)ˆm>u h(x)for all x∈Ω. In case(i),we have u h(y)≤u h(¯x)for all y∈¯Ωand henceϕ(y)−ϕ(¯x)−h tξ(y−¯x)≤0.This implies h tξ∈D+ϕ(¯x)and thus by the lemma above h tξ∈K h.In case(ii),we haveˆm=0,u h(x)<0and henceϕ(x)<(h tξ)x for all x∈Ω.The proof is completed.4.2.Geometric restrictions.We follow the paper[3]to study the rela-tionships between the boundary values and the domainΩfor a Lipschitz partial viscosity solution of problem(2.1).First of all,we need some defini-tions[1].Given a locally compact set E⊂R n and a point¯x∈E,a vectorθ∈R n is called a generalized tangent to E⊂R n at¯x provided that there exist sequences h k→0+and x k→¯x,x k∈E such that(x k−¯x)/h k→θas k→∞.The set of generalized normals of E at¯x,denoted by N E(¯x),is defined to be the set ofν∈R n such thatν·θ≤0for all generalized tangentsθof E at¯x.Obviously,N E(¯x)is a cone and may only consist of the single element0.The following theorem shows that the existence of Lipschitz partial vis-cosity solutions of(2.1)depends heavily on the set K,the boundary dataξ, h and the domainΩ.Theorem 4.3.If u∈S∞(Ω)is a Lipschitz partial viscosity solution in direction h,then for any x∈∂Ωand anyνx∈N R n\Ω(x)\{0},there exists a numberλ≥0such that h tξ+λνx∈K h.Proof.The result has been proved in[3]under the condition that h tξ∈int(conv(K h))(see Theorem3.4in[3]).However,in their proof,this condi-tion is not needed.Therefore,the theorem follows from the result of that paper.4.3.M¨u ller-ˇSver´a k’s solutions for the two-well problem.In this sub-section,we show that the Lipschitz solution constructed in M¨u ller and ˇSver´a k[9]for the two dimensional two-well problem([2,11])cannot bea partial viscosity solution in certain directions.In the following,let K=SO(2)∪SO(2)H be a two-well set,where SO(2) is the set of rotations and H=diag(λ,µ)with0<λ<λ−1<µ.In this case,the two-well K has rank-one connections;that is,for any A∈K there exist exactly two matrices B∈K such that rank(A−B)=1(see[2,11]). As in[9,11],it is convenient to identify the space M2×2with R2×R2using。