ON SELECTION OF SOLUTIONS TO VECTORIAL HAMILTON-JACOBI SYSTEM

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1 ON SELECTION OF SOLUTIONS TO VECTORIAL HAMILTON-JACOBI SYSTEM BAISHENG YAN Abstract. In this paper, we discuss some principles on the selection of the solutions of a special vectorial Hamilton-Jacobi system defined by (1.1) below. We first classify the equivalent solutions based on a wellknown construction and then discuss some selection principles using intuitive descriptions of coarseness or maximality of solutions. We also discuss a selection principle based on comparison of the projections of a solution with certain smooth functions and introduce a notion of partial viscosity solutions for the vectorial Hamilton-Jacobi equation, which generalizes the well-known notion of viscosity solutions in the scalar case. Since we mainly work in the framework of Sobolev solutions, we show that, as in the scalar case, for certain domains the partial viscosity principle may rule out many interesting Lipschitz solutions. 1. Introduction In 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, Ω is a bounded domain in R n, u is a mapping from Ω to 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 W 1,p (Ω;R m ) for some 1 p with an affine Dirichlet boundary condition (1.2) u Ω = ξx + b, where ξ is a given m n matrix and b is a given vector in R m. As usual, the boundary condition (1.2) means that u ξx b W 1,p 0 (Ω;R m ), the closure in W 1,p (Ω;R m ) of the set of smooth mappings compactly supported in Ω. It is clear that if ξ K the affine map u = ξx + b is always a solution 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. 1

2 2 BAISHENG YAN course, 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 C 1 (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 minimizing 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 different fine structures (microstructures). However, physically, these solutions may all exhibit the same effective (macroscopically observable) quantities of the material. 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 Müller and Šverák [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 data ξx + b with ξ / 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 data ξ, problem (1.1)-(1.2) has solutions in W 1,p (Ω;R m ). We first classify the equivalent solutions using a principle motivated by a well-known construction of self-similarity (see the proof of Proposition 2.1) and then discuss some selection principles based on an intuitive description 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 applications; 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 W 1,p (Ω;R m ) is a solution to the problem (1.1)-(1.2) then, for any unit vector h R m, the function ϕ(x) = h u(x) W 1,p (Ω) is a solution of the problem (1.3) Dϕ(x) Γ a.e. x Ω, ϕ Ω = (h t ξ)x,

3 ON SELECTION OF SOLUTIONS 3 where Γ = K h = {h t η η K} is a subset of M 1 n = R n. The equation Dϕ(x) Γ can be considered as a special scalar Hamilton-Jacobi equation H(Dϕ(x)) = 0 defined by certain Hamiltonians H(p), i.e., functions vanishing exactly on Γ. 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 characterizing 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 which ϕ(x) = h u(x) is a viscosity solution relevant to problem (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 W 1,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 domains Ω and boundary data ξx+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 Müller-Šverák solutions of the two-well problem in [9] are not a partial viscosity solution (in certain directions). 2. Classification of Sobolev solutions In the rest of the paper, we assume Ω R n is a bounded domain with Lipschitz boundary Ω, although most of the results hold for any open

4 4 BAISHENG YAN bounded sets Ω with Ω = 0. We use M to denote the space M m n of all m n real matrices with norm ξ = (ξ t ξ) 1/2. Definition 2.1. Let 1 p, K M be given. We define β p (K) to be the set of all ξ M such that the problem (2.1) Du(x) K a.e. x Ω; u Ω = ξx has a solution u in W 1,p (Ω;R m ). Moreover, for ξ β p (K), let S p (Ω) be the set of solutions u of problem (2.1) in W 1,p (Ω;R m ). Clearly, K β p (K) and β p (K) β q (K) if p > q. Since any Sobolev map with essentially bounded gradient can be considered as a Lipschitz map, it follows that β p (K) = β q (K) for all 1 q < p for a bounded set K. Later, we shall give an example of an unbounded set K such that β p (K) is strictly contained in β q (K) for p > q. For general sets K, we have the following result. Proposition 2.1. The set β p (K) is independent of the smooth bounded domain Ω. 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 of β p (K) on Ω, suppose for a bounded Lipschitz domain Ω problem (2.1) has a solution u W 1,p (Ω;R m ). Let U be another bounded Lipschitz domain in R n. (Note that U can be Ω itself.) We show that the Dirichlet problem (2.2) Dv(y) K a.e. y U; v U = ξy also has a solution v in W 1,p (U;R m ); this proves the independence result. To construct a solution to problem (2.2), we define open sets Ω a,ǫ for a R n, ǫ > 0 by (2.3) Ω a,ǫ a + ǫω = {a + ǫx x Ω}. Then, for any y U, there exists r y > 0 such that Ω y,r U for all r (0, r y ). Notice that the family {Ω y,r y U, 0 < r < r y } forms a Vitali cover for U; thus by the Vitali covering lemma, there exist disjoint sets Ω k (k = 1, 2,...), where Ω k = y k + r k Ω, 0 < r k < r yk, and a set N of measure zero such that U = k=1 Ω k N. Define a map v : U R m by { ξyk + r k u( y y k r (2.4) v(y) = k ), y Ω k, k = 1, 2,...; ξy, otherwise in U.

5 ON SELECTION OF SOLUTIONS 5 It is easy to verify that v ξy W 1,p 0 (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 Ω r y ) u ξx L p (Ω). Hence, by choosing r y < ǫ for all y Ω, we can construct a sequence of solutions {v ǫ } in S p (Ω) such that v ǫ (y) ξy in L p (Ω) as ǫ 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 1 (2.5) ψ(dv(x))dx = 1 ψ(du(x))dx U Ω U for all continuous functions ψ satisfying 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 functions ψ. Definition 2.2. We say two solutions u, v in S p (Ω) are equivalent and write u v if for all ψ C p (M) (2.6) ψ(dv(x)) dx = ψ(du(x)) dx. Ω 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 functional µ [u] on C p (M) through µ [u], ψ = 1 ψ(du(x)) dx, Ω which, by the Riesz representation theorem, also induces a probability Radon measure µ [u] on M supported on the closure K. For applications in nonlinear elasticity, if ψ(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 different fine structures. For general energy minimizing sequences, such a description of macroscopically observable quantities has been successfully given in terms of Young measures (see [2]). Ω Ω Ω

6 6 BAISHENG YAN Lemma 2.2. For each p [1, ], the function Ψ([u]) = Du ξ L p (Ω) is a well-defined function on S p (Ω). Proof. If 1 p < we choose ψ(η) = η ξ p in (2.6). While, if p = we choose ψ(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 ; so Ψ([u]) is well-defined on S p (Ω). Note that Ψ([u]) = 0 if and only if ξ K and [u] = [ξx]; in this case, the class [ξx] contains only the trivial solution ξx. Therefore, to some extent, the function Ψ([u]) provides a scale to measure the L p -deviation of the gradient Du from the constant matrix ξ. To keep the L p -deviation as small as possible, the solution class [u] is preferable over the class [v] if Ψ([u]) < Ψ([v]). In this sense, it would be interesting to study the problem m = inf Ψ([u]). [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 shall fix a solution class [u] so that Ψ([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 which β p (K) may depend on p 1 dramatically. Example. Let n 3 and K = C 1 = {ξ M n n ξ n = n n/2 det ξ} be the set of conformal matrices. Suppose ξ β p (C 1 ). Then there exists a map u W 1,p (Ω;R n ) such that (2.7) Du(x) n = n n/2 det Du(x) a.e. x Ω; u Ω = ξx. If p n, we can integrate the equation above over Ω and use the boundary condition, the Jensen inequality, and a property of the determinant function to obtain ξ n n n/2 det ξ. Hence, by Hadamard s inequality, ξ C 1. This shows β n (C 1 ) = C 1. Let J be an n n matrix with J t J = I, detj = 1 and let u 0 (x) = Jx/ x 2. Then an easy calculation shows that u 0 W 1,p (B;R n ) for all p < n/2, where B is the unit ball in R n, and that u = u 0 solves (2.7) with Ω = B, ξ = J. This shows J β p (C 1 ) for all p < n/2; hence β n (C 1 ) β p (C 1 ) for all p < n/2. For further information on β p (C 1 ) when n/2 p < n, see [12, 13]. 3. Selection of solutions in the same class For any given solution u of problem (2.1), the map v constructed as in the proof of Proposition 2.1 is an equivalent solution and has finer structures

7 ON SELECTION OF SOLUTIONS 7 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 less fine structures or singularities be more preferable Coarseness of solutions. In the construction (2.4), a patch of v, v Ωa,ǫ, totally determines u in Ω and has the same intrinsic structure as u (e.g., boundary data, singularities); such solutions v are considered as having more fine structures or singularities than u and thus being unfavorable. Definition 3.1. 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 Ω a,ǫ Ω such 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 for Ω a,ǫ, Ω b,δ Ω we have u(y) = ǫ 1 (v(a + ǫy) ξa) and v(y) = δ 1 (w(b + δy) ξb) for y Ω, then for the patch Ω c,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. Proposition 3.1. Let u S p (Ω) and Ω a,ǫ Ω. Suppose u(y) = ǫ 1 (u(a+ ǫy) ξa) for almost every y Ω. If u is differentiable at the point a ǫ = a 1 ǫ, then u ξx in Ω. 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 Ω and iterating (3.1), it is easy to verify that for k = 1, 2,... f(y) = ǫ k f(a ǫ + ǫ k (y a ǫ )). From this and the fact that f is differentiable at a ǫ and ǫ < 1, we easily obtain by letting k that f(y) = Df(a ǫ )(y a ǫ ) for almost every y Ω. Since f W 1,p 0 (Ω;R m ) we must have Df(a ǫ ) = 0 and 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] C 1 (Ω;R m ). However, as we mentioned in the introduction, in applications, it may very well be possible that the set [u] C 1 (Ω;R m ) is empty if u S p (Ω) is nontrivial.

8 8 BAISHENG YAN The following result shows that every nontrivial solution generates an equivalent (non smooth) self-similar solution. Proposition 3.2. If u S p (Ω) is a nontrivial solution, then there exists a self-similar solution v [u]. Proof. Let Ω 1 = Ω a,ǫ Ω for some a R n, ǫ > 0. As in the proof of Proposition 2.1, using u S p (Ω) we construct a solution u 1 S p (G 1 ), where G 1 = Ω \ Ω 1. Let f = u 1 ξx W 1,p 0 (G 0 ;R m ). Define Ω k = a + ǫω k 1 and G k = Ω k 1 \ Ω k inductively for k = 2, 3,... and note that G k is the image of G 1 under the map L k (x) = a ǫ + ǫ k (x a ǫ ), where, as before, a ǫ = a 1 ǫ. We now define a map v on Ω by letting v(x) = ξx + ǫ k f(y) if x = L k (y) for some k = 1, 2,... and y G 1. 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 C 1 (Ω \ {a ǫ };R m ). Therefore, the smoothness condition in Proposition 3.1 can not be dropped Structure of a non-coarse solution. Suppose u S p (Ω) is not a coarse solution. Then, from the definition, the family F = {Ω a,ǫ Ω u S p (Ω a,ǫ ), ψ(du) = ǫ n ψ(du), ψ C p (M)} Ω a,ǫ 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 Hausdorff Maximal Principle asserts that there exists a maximal totally ordered subset M F. Let σ = inf{ U U M}. Lemma 3.3. 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 U 1 = U a,ǫ U such that u S p (U 1 ) and U 1 ψ(du) = ǫ n U ψ(du) for all ψ C p (M). As U = Ω b,δ F, we easily see that U 1 F. We now show U 1 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 of σ. Therefore, we have U V for all V M, V U. Therefore U 1 V for all V M. This shows that the set M = {U 1, M} is totally ordered by and thus by the maximality of M we have U 1 M and hence the contradiction. Ω

9 ON SELECTION OF SOLUTIONS 9 Lemma 3.4. If U > σ for 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 1 and U = σ. Since each U k is of the form a k + ǫ k Ω, it follows that U = a + ǫω for some a R n and ǫ 0. The result will follow if ǫ = 0 and thus σ = 0. Suppose for the contrary ǫ > 0. Since U k F we can also prove that Ω a,ǫ, the interior of U, belongs to F. We claim Ω a,ǫ U for all U M, and therefore the set {Ω a,ǫ, M} is totally ordered by ; this and the maximality of M would imply Ω a,ǫ M, which contradicts with the assumption and the fact that Ω a,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 Ω a,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 Ω k ց {a} if Ω k+1 Ω k and k Ω k = {a}. Theorem 3.5. Let u S p (Ω) be a non-coarse solution. Then, either (i) for some Ω a,ǫ Ω, u S p (Ω a,ǫ ) is a coarse solution on Ω a,ǫ and Ω a,ǫ ψ(du) = ǫ n Ω ψ(du) for all ψ C p(m), or (ii) there exist Ω k = Ω ak,ǫ k Ω such that Ω k ց {a}, u S p (Ω k ) and Ω k ψ(du) = ǫ n k Ω ψ(du) for all ψ C p(m) 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 ū in W 1,p (Ω;R m ) as j and ū Ω = ξx. From the Sobolev embedding theorem, u j ū in L p (Ω) and hence Φ(ū) = ˆm. However, ū may not be a solution.

10 10 BAISHENG YAN From the weak lower semicontinuity of the norm, we have Dū ξ p Du ξ p. The following gives a sufficient condition for the weak limit ū to be an L p -maximal solution. Proposition 3.6. Let 1 < p <. Then the limit map ū above belongs to [u] if and only if Dū ξ p = Du ξ p. In this case, ū is also an L p -maximal solution. Proof. If ū [u], then Dū ξ p = Du ξ p follows easily from Lemma 2.2. To prove the converse, suppose Dū ξ p = Du ξ p. Since u j [u], we have Du j ξ p = Du ξ p = Dū ξ p for all j and hence Du j Dū strongly in L p (Ω), which implies ψ(dū(x))dx = lim ψ(du j (x))dx = ψ(du(x))dx j Ω Ω for all ψ C p (M). Using this identity with ψ(η) = d K (η), the distance function to K, we obtain Dū(x) K for a.e. x Ω. Since ū W 1,p (Ω;R m ) and ū Ω = ξx, it follows that ū S p (Ω) and ū u, and hence ū [u]. Moreover, by definition, ū is also an L p -maximal solution. Remark. If it happens that [u] S p (Ω) satisfies Ψ([u]) = min Sp(Ω) Ψ, then the result above shows that any limit map ū obtained as above is either an L p -maximal solution in [u] or satisfies Dū ξ p < Dv ξ p for all v S p (Ω). We now study the structure of the L p -maximal solutions. Lemma 3.7. Let u S p (Ω) be L p -maximal and Ω a,ǫ Ω. If u S p (Ω a,ǫ ) and Ω a,ǫ ψ(du) = ǫ n Ω ψ(du) for all ψ C p(m), then u ξx L p (Ω a,ǫ) ǫ 1+ 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 see that v u and u ξx L p (Ω a,ǫ) = ǫ 1+ n p v ξx L p (Ω). Hence the result follows from the L p -maximality of u. Combining this lemma with Theorem 3.5, we have the following. Theorem 3.8. Let u S p (Ω) be L p -maximal. Then, either (i) for some Ω a,ǫ Ω, u S p (Ω a,ǫ ) is coarse on Ω a,ǫ, Ω a,ǫ ψ(du) = ǫ n Ω ψ(du) for all ψ C p(m) and u ξx L p (Ω a,ǫ) ǫ 1+ n p u ξx L p (Ω), or (ii) there exist Ω k = Ω ak,ǫ k Ω such that Ω k ց {a}, Ω k ψ(du) = ǫ n k Ω ψ(du) for all ψ C p(m) and u ξx L p (Ω k ) ǫ 1+ n p k u ξx L p (Ω). Ω

11 ON SELECTION OF SOLUTIONS 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 Ω provided f = h f( x) for some x Ω. If f 0, then there exists at least one x Ω such 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 1 and 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 1 is 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 1 and v [u], choosing h to be a maximal direction of v ξx, we have u ξx u h v h = v ξx, and hence u is L -maximal. The proof is completed. 4. Partial viscosity solutions In 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 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) φ is called a viscosity super-solution of H(Dφ) = 0 if, for each w C (Ω) and for each x Ω at which φ w has a local minimum, we have H(Dw( x)) 0;

12 12 BAISHENG YAN (b) φ is called a viscosity sub-solution of H(Dφ) = 0 if, for each w C (Ω) and for each x Ω at which φ w has a local maximum, we have H(Dw( x)) 0; (c) φ is called a viscosity solution of H(Dφ) = 0 if φ is both viscosity super-solution and viscosity sub-solution of H(Dφ) = 0. Remark. A viscosity solution φ to the problem H(Dφ(x)) = 0, φ Ω = g (g given) can be studied as a certain limit of the solutions φ ǫ of the following regularizing problem as ǫ 0 + (4.2) ǫ φ ǫ + H(Dφ ǫ ) = 0, φ ǫ Ω = g. The problem (4.2) makes no sense for the vector valued functions φ: Ω R m when m 2. Let d Kh be the distance function to K h. We can write (4.1) as (4.3) d Kh (Dϕ(x)) = 0 a.e. x Ω, ϕ Ω = (h t ξ)x. Of course, instead of using the distance function d Kh, we may use other Hamiltonians 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). Definition 4.2. We say u S (Ω) is a partial viscosity solution in direction h if the function ϕ(x) = h u(x) is a viscosity solution of (4.3). Lemma 4.1. A Lipschitz continuous function ϕ: Ω R with ϕ Ω = (h t ξ)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 ϕ(y) ϕ(x) p (y x) lim sup 0 }. y x, y Ω y x Proof. Since d Kh 0, any Lipschitz function ϕ on Ω with ϕ Ω = (h t ξ)x is automatically a viscosity super-solution of (4.3). Therefore, any such function ϕ is 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]). Proposition 4.2. Let u S (Ω) be a partial viscosity solution in direction h. Then either h t ξ K h or ϕ(x) = h u(x) < (h t ξ)x for all x Ω.

13 ON SELECTION OF SOLUTIONS 13 Proof. Let u h (x) = ϕ(x) (h t ξ)x and consider max Ω u h = ˆm. Then either (i) there exists a x Ω such 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 Ω and hence ϕ(y) ϕ( x) h t ξ (y x) 0. This implies h t ξ D + ϕ( x) and thus by the lemma above h t ξ K h. In case (ii), we have ˆm = 0, u h (x) < 0 and hence ϕ(x) < (h t ξ)x for all x Ω. The proof is completed Geometric restrictions. We follow the paper [3] to study the relationships between the boundary values and the domain Ω for a Lipschitz partial viscosity solution of problem (2.1). First of all, we need some definitions [1]. Given a locally compact set E R n and a point x E, a vector θ 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 θ as k. The set of generalized normals of E at x, denoted by N E ( x), is defined to be the set of ν R n such that ν θ 0 for all generalized tangents θ of E at x. Obviously, N E ( x) is a cone and may only consist of the single element 0. The following theorem shows that the existence of Lipschitz partial viscosity solutions of (2.1) depends heavily on the set K, the boundary data ξ, h and the domain Ω. Theorem 4.3. If u S (Ω) is a Lipschitz partial viscosity solution in direction h, then for any x Ω and any ν x N R n \Ω(x) \ {0}, there exists a number λ 0 such that h t ξ + λ ν x K h. Proof. The result has been proved in [3] under the condition that h t ξ int(conv(k h )) (see Theorem 3.4 in [3]). However, in their proof, this condition is not needed. Therefore, the theorem follows from the result of that paper Müller-Šverák s solutions for the two-well problem. In this subsection, we show that the Lipschitz solution constructed in Müller and Šverák [9] for the two dimensional two-well problem ([2, 11]) cannot be a 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) with 0 < 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 M 2 2 with R 2 R 2 using

14 14 BAISHENG YAN the map J : ( a1 a 2 a 2 a 1 ) + ( b1 b 2 b 2 b 1 )( ) λ 0 (a,b), 0 µ where a = (a 1, a 2 ), b = (b 1, b 2 ). The set K = SO(2) SO(2)H thus becomes the union of two disjoint circles; more precisely, J(K) = {(a, 0) a = 1} {(0,b) b = 1}. Let K be an open set defined by { (4.5) K = F = J 1 (a,b) λµ det F a <, b < det F 1 }. λµ 1 λµ 1 The following theorem has been proved by Müller and Šverák in [9] (see also [7, 10]). Theorem 4.4 (Müller and Šverák [9]). K β (K). That is, for any ξ K and any bounded domain Ω R 2 there exists a Lipschitz map u: Ω R 2 such that Du(x) K a.e. and u Ω = ξx. For any unit vector h R 2, since K = SO(2) SO(2)H is invariant under rotations, it is easy to see that the set K h = {h t η η K} is independent of h and (4.6) K h = K 0 {(p, q) M 1 2 = R 2 } p 2 + q 2 = 1 or p2 λ 2 + q2 µ 2 = 1, that is, K h is the union of a circle and an ellipse in R 2. Let ( K) h = {h t η η K}. Since K is left-invariant under SO(2), that is, RF K for all F K and R SO(2), we have (4.7) ( K) h = K 0 { (p, q) M 1 2 (r, s) M 1 2 : ( ) } p q r s K for all h, and K 0 M 1 2 is an open set contained in the convex hull conv(k 0 ). Note that neither K 0 nor K 0 is invariant under SO(2). Using equations (4.5) and (4.7), we easily see that (p, q) K 0 if and only if there exists 1 < t < λµ such that (4.8) (4.9) (4.10) qr + ps = t, r 2 + s 2 < f(t) p 2 q 2, µ 2 r 2 + λ 2 s 2 < g(t) µ 2 p 2 λ 2 q 2, where f(t) = κ(t 1) 2 + 2t, g(t) = κ(λµ t) 2 + 2λµt, with κ = (µ λ)2 (λµ 1) 2. For each t, we denote the sets of (r, s) R 2 determined by equations (4.8), (4.9), (4.10) by D 1 (t), D 2 (t), D 3 (t), respectively; these are line, open (perhaps empty) disk and ellipse, all depending on (p, q). From this, we easily have the following.

15 ON SELECTION OF SOLUTIONS 15 Lemma 4.5. (p, q) K 0 if and only if there exists a number t (1, λµ) such that D 1 (t) D 2 (t) D 3 (t). It would be lengthy and complicated to precisely describe the set of all these points (p, q). Instead, we will focus on the set of (p, q) in K 0 with p 0, q 0 for which there exists t (1, λµ) such that D 2 (t) D 3 (t). Note that the functions f, g defined above satisfy λ 2 (µ 2 λ 2 ) < g(t) λ 2 f(t) < µ 2 λ 2 < µ 2 f(t) g(t) < µ 2 (µ 2 λ 2 ) for all 1 < t < λµ. Therefore, we can define µ a(t) = 2 f(t) g(t) g(t) λ µ 2 λ 2, b(t) = 2 f(t) µ 2 λ 2 for 1 < t < λµ. It then follows that (p, q) if and only if there exists 1 < t < λµ such that 0 q < a(t), 0 p < b(t), t < p a 2 (t) q 2 + q b 2 (t) p 2. Introducing 0 α, β < π/2 with q = a(t)sinα, p = b(t)sin β, it follows that t < a(t)b(t)sin(α + β) for (p, q). One also easily sees that t < a(t)b(t) t for 1 < t < λµ; hence let 0 < c(t) < π/2 be such that sinc(t) = a(t)b(t). Then, using γ = α + β, we can write = 1<t<λµ t, where t is the set defined by t = { (p, q) R 2 p = b(t)sin β, q = a(t)sin(γ β), (γ, β) Γt }, Γ t being the set of (γ, β) satisfying 0 β < π 2, β γ < π 2 + β and c(t) < γ < π c(t). The line γ = π/2, 0 < β < π/2 is always in Γ t for all 1 < t < λµ, and hence its image in t, which is a quarter of the ellipse p 2 b 2 (t) + q2 a 2 (t) = 1, lies always in the set t and thus in the open set K 0. After some calculations, we see that at least for t sufficiently close to 1 or λµ the ellipse p2 b 2 (t) + q2 a 2 (t) = 1 intersects the set K 0 and also contains a portion which is outside the region bounded by K 0. Therefore, we arrive at the following result. Proposition 4.6. The set K 0 contains certain points (p, q) with p 2 +q 2 > 1 and p2 + q2 > 1 and also contains certain points (p, q) K λ 2 µ 2 0. Remark. For any θ = (p, q) K 0 with p 2 + q 2 > 1 and p2 + q2 > 1, let λ 2 µ 2 N(θ) be the set of all non-zero points ν M 1 2 such that θ + tν / K 0 for all t 0. We easily see that N(θ) for these θ s. We conclude this paper with some non partial viscosity solution results concerning the Müller-Šverák solutions stated in Theorem 4.4.

16 16 BAISHENG YAN Proposition 4.7. Let h R 2 be a unit vector. Suppose ξ K is such that the point (p, q) = h t ξ K 0. Then the solution u as stated in Theorem 4.4 is not a partial viscosity solution in the direction h. Proof. Without loss of generality, we assume (p, q) = h t ξ K 0 satisfies p 2 + q 2 = 1. By changing ξ K to Rξ K with R SO(2), we can also assume h = (1, 0) t. Let u: Ω R 2 be a Lipschitz solution of Du(x) K a.e. and u Ω = ξx, as given by Theorem 4.4. By definition, u is not a partial viscosity solution in direction h = (1, 0) t if we show that ϕ(x) = h u(x) = u 1 (x) is not a viscosity solution of problem (4.11) d K0 (Dφ(x)) = 0, φ Ω = px 1 + qx 2. Note that, since (p, q) K 0, by the uniqueness of viscosity solution (see [4, 5, 8]), the only viscosity solution of (4.11) is the trivial solution φ = px 1 + qx 2. However, from Du(x) K = SO(2) SO(2)H, u Ω = ξx and ξ / SO(2), we easily obtain ϕ(x) = u 1 (x) px 1 + qx 2. This completes the proof. Assume now the domain Ω satisfies the property that for each non-zero point ν M 1 2 there exists a point y Ω such that ν N R 2 \Ω(y). (This condition is satisfied for instance if Ω is smooth.) Then, from Theorem 4.3 and the remark following Proposition 4.6, we easily have the following result. Proposition 4.8. Let ξ K and let h R 2 be a unit vector. Suppose (p, q) = h t ξ K 0 satisfies p 2 + q 2 > 1 and p2 + q2 > 1. Then the solution λ 2 µ 2 u as stated in Theorem 4.4 is not a partial viscosity solution in direction h. References [1] J. Aubin and H. Frankowska, Set Valued Analysis, Birkhauser, [2] J. M. Ball and R. D. James, Proposed experimental tests of a theory of fine microstructures and the two well problem, Phil. Trans. Roy. Soc. London, A 338 (1992), [3] P. Cardaliaguet, B. Dacorogna, W. Gangbo and N. Georgy, Geometric restrictions for the existence of viscosity solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16 (1999), [4] M. G. Crandall, L. C. Evans and P. L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), [5] M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 277 (1983), [6] B. Dacorogna and P. Marcellini, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases, Acta Math., 178 (1) (1997), [7] B. Dacorogna and P. Marcellini, Cauchy-Dirichlet problem for first order nonlinear systems, J. Funct. Anal., 152 (1998), [8] W. Fleming and M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, 1993.

17 ON SELECTION OF SOLUTIONS 17 [9] S. Müller and V. Šverák, Attainment results for the two-well problem by convex integration, Geometric analysis and the calculus of variations, , Internat. Press; Cambridge, MA, [10] S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity, preprint. [11] V. Šverák, On the problem of two wells, in Microstructure and Phase Transition, (D. Kinderlehrer et al. eds.), , Springer-Verlag; Berlin, Heidelberg, New York, [12] B. Yan, Remarks about W 1,p -stability of the conformal set in higher dimensions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), [13] B. Yan and Z. Zhou, Stability of weakly almost conformal mappings, Proc. Amer. Math. Soc., 126 (1998), Department of Mathematics, Michigan State University, East Lansing, MI address: yan@math.msu.edu