The regularity of some vector-valued variational inequalities with gradient constraints

We prove the optimal regularity for some class of vector-valued variational inequalities with gradient constraints. We also give a new proof for the optimal regularity of some scalar variational inequalities with gradient constraints. In addition, we prove that some class of variational inequalities with gradient constraints are equivalent to an obstacle problem, both in the scalar and vector-valued case.


Introduction
Let U ⊂ R n be an open bounded set. Suppose K ⊂ R n is a balanced (symmetric with respect to the origin) compact convex set whose interior contains 0. Also suppose that η ∈ R N is a fixed nonzero vector. Consider the following problem of minimizing for an N × n matrix A, and γ K is the norm associated to K defined by (1.4) γ K (x) := inf{λ > 0 | x ∈ λK}.
As K 1 is a closed convex set and I is coercive, bounded and weakly sequentially lower semicontinuous, this problem has a unique solution u. We will show that under some extra assumptions on K u ∈ C 1,1 loc (U ; R N ). This problem is a generalization to the vector-valued case of the elastic-plastic torsion problem, which is the problem of minimizing J η (v) :=ˆU |Dv| 2 − ηv dx for some η > 0, over {v ∈ H 1 0 (U ) | |Dv| ≤ 1 a.e.}. The regularity of the elastic-plastic torsion problem has been studied by Brezis and Stampacchia [2], and Caffarelli and Rivière [3]. There has been several extensions of their results to more general scalar problems with gradient constraints. See for example Jensen [8], Gerhardt [6], Evans [4], Wiegner [14], Ishii and Koike [7]. To 1 the best of author's knowledge, the only work on the regularity of vector-valued problems with gradient constraints is Rozhkovskaya [12].
Our approach is to show that our problem is reducible to a scalar problem with gradient constraint. Then we show that the obtained scalar problem is equivalent to a double obstacle problem with only Lipschitz obstacles. At this point we generalize the proof of Caffarelli and Rivière [3], to obtain the optimal regularity. We should note that Lieberman [9] proves the regularity of a more general double obstacle problem by different methods.
In the process described above, we also show that our vector-valued problem with gradient constraint is equivalent to a vector-valued obstacle problem. This result, which is the first result of its kind as far as the author knows, is a generalization to the vector-valued case of the equivalence between the elastic-plastic torsion problem and an obstacle problem, proved by Brezis and Sibony [1]. Later Treu and Vornicescu [13] proved that the equivalence holds for a larger class of scalar variational inequalities with gradient constraints. We will further generalize their result. Suppose f : R Nm → R and g : R → R are convex functions. Here N m is the number of partial derivatives up to order m of a scalar function on R n . Consider the problem of minimizing where u 0 ∈ W m,p (U ). We will show that under appropriate assumptions, the minimizer of J over W K is the same as its minimizer over for some suitable functions u − , u + . The difference of our result with that of Treu and Vornicescu [13] is that we allow m > 1, f, g to be only convex, and K to have empty interior. Some of our results has been proved using different means by Mariconda and Treu [10].

The Equivalence in the Scalar Case
Suppose K ⊂ R n is a compact convex set whose interior contains the origin. Let J, W K , and W u − ,u + be as above. We assume that on W m,p (U ), J is finite, bounded below and sequentially weakly lower semicontinuous. These assumptions are satisfied if, for example, we impose some growth conditions on f, g and some mild regularity on ∂U . Therefore by our assumption, J attains its minimum on any nonempty closed convex subset of W m,p (U ).
Furthermore, we assume that u 0 is Lipschitz, and Thus in particular, W K is nonempty.
and its polar is the convex set We recall that for all x, y ∈ R n , we have Its proof can be found in Rockafellar [11]. Also, when K is balanced, K • is balanced too, and γ K , γ K • are both norms on R n . Now, let us find u ± ∈ W K such that for all u ∈ W K we have u − ≤ u ≤ u + . Let u ± be respectively the unique minimizers of J ± (v) =´U ∓v(x) dx over W K . We show that they have the desired property. We need the following lemma. Lemma 1. Suppose u is a compactly supported function in W 1,p (R n ) with Du ∈ K a.e.. Then for all x, y.
Proof. Consider the mollifications where η ǫ is a nonnegative smooth function with support in B ǫ (0), and´B ǫ (0) η ǫ dx = 1. Then we know that u ǫ converges to u a.e., and Du ǫ = η ǫ ⋆ Du. Hence where we used Jensen's inequality in the first inequality. Thus for a.e. x, y.
We can redefine u on the measure zero set where this relation fails, in a similar way that we extend Lipschitz functions to the closure of their domains. The extension will satisfy this relation everywhere.
Proof. To see this, let u ∈ W K . Then for some R > 0. Now we can extend v by zero to all of R n , and the extension will satisfy the same gradient bound. Therefore by arguments similar to the previous lemma, we can see that the extension of v, and hence v itself, is Lipschitz with Lipschitz constant 2R. Using the fact that v is zero on the boundary, this also implies that v L ∞ ≤ 2RD, where D is the diameter of U . The result for u follows easily, noting that u 0 is Lipschitz. Now as Du L ∞ < C for some constant C independent of u, we have Du L p < C since U is bounded. Noting that all u ∈ W K have the same boundary value, we get by Poincare inequality u W 1,p < C. Repeating this argument we get u W m,p < C.
Now we can see that J ± are bounded on W K . As J ± are linear, they are weakly continuous. Furthermore W K is convex, closed and bounded in W 1,p (U ). Hence W K is compact with respect to sequential weak convergence. These imply that J ± have minimizers over W K . The uniqueness and the fact that u − ≤ u + a.e. on U , follows from a similar argument to the proof of the next lemma.
Suppose to the contrary that, for example, the set E : The derivative of w is x ∈ E for a.e. x.
Therefore we have Dw(x) ∈ K a.e.. Thus which is a contradiction.
The following characterization of u ± will be used later.
and its derivative has γ K norm less than one. But d K • (x, ∂U ) is a Lipschitz function that vanishes on the boundary of U . It also satisfies As proved by Treu and Vornicescu [13], this last property implies that the γ K norm of the derivative of d K • (x, ∂U ) is less than or equal to 1 a.e.. Now similarly to the proof of Lemma 1, we can show that The following theorem is the generalization of the result of Treu and Vornicescu [13]. We removed the assumptions on the derivatives of g, and allowed m > 1, and also we allowed K to have empty interior. Theorem 2. Suppose K is a compact convex set containing 0, and u 0 is the restriction to U of a compactly supported function in W m,p (R n ) with gradient a.e. in K. Also, suppose f, g are convex and at least one of them is strictly convex. Then the minimizer of Proof. Note that the convexity assumptions on f, g imply that the minimizer of J over any nonempty convex closed set is unique. Also the assumption on u 0 implies First assume that 0 is in the interior of K, and g is C 1 with strictly increasing derivative.
Similarly to Treu and Vornicescu [13], using u 0 we can extend u ± and u to all of R n in a way that the gradient of u ± is still in K. Fix a nonzero vector h ∈ R n , and define By changing the variable from x to x + h in the last integral, we get Adding this to the first integral and using the convexity of f , we havê We divide this inequality by λ > 0 and take the limit as λ → 0. Then, as g is C 1 and u is bounded, by Dominated Convergence Theorem we get But on E + , u(x)−u(x+h)+γ K • (h) < 0. Also g ′ is strictly increasing and therefore g ′ (u(x + h)) − g ′ (u(x)) > 0. Hence E + must have measure zero. This means that for a.e.
Which implies γ K (Du(x)) ≤ 1, and this is equivalent to u ∈ W K . Now suppose that we only have 0 ∈ K. Let Then {K i } is a decreasing family of compact convex sets containing K with 0 ∈ int K i . Therefore {W Ki } is also a decreasing family containing W K . Let u ± i be the corresponding obstacles to W Ki . Then we have } is a decreasing family too and contains W u − ,u + .
Let u i be the minimizer of J over W Ki . We have Du 0 ∈ K ⊂ K i . Therefore we can apply the previous argument and we have J( Therefore there is a subsequence of u i 's, where we denote it by u i k , which converges weakly in u 0 + W m,p 0 (U ) to u. By weak lower semicontinuity of J we get J(u) ≤ lim inf J(u i k ) ≤ J(v) for all v ∈ W u − ,u + . Thus to finish the proof we only need to show that u ∈ W K . To see this note that the sequence u i k is eventually in each W Ki k and as these are closed convex sets they are weakly closed, hence u ∈ W Ki k for all k. This means d(Du, K) ≤ 1 i k a.e.. Thus d(Du, K) = 0 a.e., and by closedness of K we get the desired result.
Next suppose that g is only convex. Consider the mollifications g ǫ := η ǫ ⋆ g. First let us show that g ǫ is convex too. We have

Now let
Then since g ǫ (v)+ ǫv 2 is a smooth strictly convex function, it has strictly increasing derivative. Let u i be the minimizer of J i over W K . Then by the above we have J i (u i ) ≤ J i (v) for all v ∈ W u − ,u + . As the u i 's are in W K , and W K is bounded in W m,p , we can say that there is a subsequence of u i , which we continue to denote it by u i , that converges weakly to u ∈ W K .
Since g ǫ uniformly converges to g on compact sets, and for v ∈ W u − ,u + we have v L ∞ < C for some constant C independent of v, we have for ǫ small enough and independent of v for i large enough. Hence J(u i ) ≤ J(v) + 2δ. Then by weak lower semicontinuity of J we have J(u) ≤ lim inf J(u i ) ≤ J(v) + 2δ. Since δ is arbitrary we get that u is the minimizer of J over W u − ,u + as required.
Remark 1. We can also prove a version of this theorem when 0 / ∈ K, by translating K. But we need to have a bound on the distance of K and the origin.

The Equivalence in The Vector-Valued Case
Suppose K ⊂ R n is a balanced compact convex set whose interior contains 0. Also suppose that η ∈ R N is a fixed nonzero vector. Consider the following problems of minimizing e.}, and over for an n × n matrix A, and γ K , d K are respectively the norm associated to K and the metric of that norm. We show that these problems are equivalent. As both K 1 , K 2 are closed convex sets and I is coercive, bounded and weakly sequentially lower semicontinuous, both problems have unique solution.
Lemma 4. We have Proof. To see this let v ∈ K 1 . Similarly to the proof of Lemma 1 we obtain for a.e. x, y. Using this relation we can redefine v on a set of measure zero the same way that we extend Lipschitz functions. Therefore we can assume that v is continuous. Now as v is 0 on ∂U , we can choose x to be the closest point on ∂U to y with respect to d K , and get the desired result.  Proof. To see this note that T u ∈ W 1,s 0 (U ; R N ) and as T preserves the norm, for a.e. x we have Furthermore as T is orthogonal we have Hence (since T η = η and T is orthogonal) Theorem 3. We have Proof. By the above lemma and uniqueness of the minimizer, we must have T u = u for all orthogonal linear maps T that fix η. This implies that u(x) = u(x)η for some scalar function u. Now we have Hence for a.e. x Also we have It is easy to see that u is the minimizer of J 1 over K 3 . Because for any w ∈ K 3 we have wη ∈ K 2 , therefore Theorem 4. The minimizer of I over K 2 is the same as its minimizer over K 1 .

8
Proof. By the above theorem u(x) = u(x)η, where u is the minimizer of J 1 over K 3 . But we know that the minimizer of J 1 over K 3 is the same as its minimizer over (3.14) Therefore for all z ∈ R n , we have a.e.
Hence u ∈ K 1 . Since K 1 ⊆ K 2 , u is also the minimizer of I over K 1 .

The Optimal Regularity
Let Suppose K ⊂ R n is a balanced compact convex set whose interior contains 0. Let u be the minimizer of J η over W K := {v ∈ c + W 1,s 0 (U ) | γ K (Dv) ≤ k a.e.}, where c, k are constants and γ K is the gauge function of K. We showed that u is also the minimizer of J η over ∂U ) a.e.}, where K • is the polar of K, and d K • is the metric associated to the norm γ K • .
By the above assumptions, there is A > 0 such that γ K • (x) ≤ A|x| for all x. We also need some sort of bound on the second derivative of γ K • , hence we assume that Lemma 6. The above inequality holds when γ K • is the p-norm for p ≥ 2. (In this case, K is the unit disk in the p p−1 -norm.) Proof. Let γ p (x) = ( |x i | p ) 1/p then for γ p (x) = 0 we have By Holder's inequality we get Thus if γ p (z) = 1, we have When γ p (x) > h, γ p is nonzero on the segment L := {x + τ z | −h ≤ τ ≤ h}; and so it is twice differentiable there. Therefore we can apply the mean value theorem to the restriction of γ p and its first derivative to the segment L. Hence we get where 0 < s, t < h and −t < r < s. Now as γ p is convex, its second derivative is nonnegative definite. Hence In the last inequality we used the triangle inequality for γ p .
The following is our main regularity result. Note that by Theorem 3, we also get the regularity for the vector-valued case.
Theorem 5. Suppose u is the minimizer of J η over W K . Then u ∈ W 2,∞ loc (U ), and where C(n) is a constant depending only on the dimension n.
Proof. Let us assume that U has smooth boundary, we will remove this restriction at the end. We know that Let φ ǫ = η ǫ ⋆ φ + δ ǫ and ψ ǫ = η ǫ ⋆ ψ where η ǫ is the standard mollifier and 4kAǫ < δ ǫ < 5kAǫ is chosen such that ∂{φ ǫ < ψ ǫ } is C ∞ (which is possible by Sard's Theorem). Note that We can easily show that γ K (Dφ ǫ ) ≤ k and γ K (Dψ ǫ ) ≤ k. Because of Jensen's inequality and convexity of γ K , we have Since φ ǫ , ψ ǫ are smooth, u ǫ ∈ W 2,p (U ǫ ) for any 1 < p < ∞. Therefore N ǫ is open and Λ i 's are closed. Also we define the free boundaries F i := ∂Λ i ∩ U ǫ . Note that ∂N ǫ consists of F i 's and part of ∂U ǫ . Our strategy for the proof is to show that u ǫ satisfies the bound (4.6) on U ǫ . Then we can let ǫ → 0. Since φ ǫ → φ , ψ ǫ → ψ uniformly, we have u ǫ → u uniformly. Also as for small enough ǫ, u ǫ 's are bounded in W 2,∞ (V ) for V ⊂⊂ U , a subsequence of them is weakly star convergent, and the limit is u. Therefore u ∈ W 2,∞ loc (U ) and |D 2 u| L ∞ ≤ lim inf |D 2 u ǫ | L ∞ gives the desired bound. Now suppose ∂U is not smooth. We approximate U by a shrinking sequence U i of larger domains with smooth boundaries. Let u i be the minimizer of J η on U i , then u i → u uniformly. To see this note that we can consider u as a function on U i , thus J η (u i ) ≤ J η (u). An argument similar to the above implies that a subsequence of u i 's converges weakly star to a function u ⋆ , and u ⋆ satisfies the desired bound. But u ⋆ ∈ W K . Also the lower semicontinuity of J η implies that J η (u ⋆ ) ≤ J η (u). Since the other inequality is satisfied too, we have J η (u ⋆ ) = J η (u). The uniqueness of the minimizer implies that u ⋆ = u. Hence u satsifies the bound (4.6) too. Now let us start proving the bound (4.6) for u ǫ . Lemma 7. We have Proof. Since on ∂U ǫ we have u ǫ = φ ǫ = ψ ǫ we get D z u ǫ = D z φ ǫ = D z ψ ǫ for any direction z tangent to ∂U ǫ , and as u ǫ is between the obstacles inside U ǫ we have D ν φ ǫ ≤ D ν u ǫ ≤ D ν ψ ǫ where ν is the normal direction to ∂U ǫ . Therefore Du ǫ is a convex combination of Dφ ǫ , Dψ ǫ and we get the bound on ∂U ǫ by convexity of γ K . The bound holds on Λ i 's (and hence on F i 's) obviously as u ǫ equals one of the obstacles there.
To obtain the bound for N ǫ note that for any vector z with γ K • (z) = 1 we have on ∂N ǫ , and as D z u ǫ is harmonic in N ǫ we get |D z u ǫ | ≤ k in N ǫ by maximum principle. The result follows from γ K (Du ǫ ) = sup The local behavior of the free boundaries is the same as the case of one obstacle problem as obstacles do not touch inside U ǫ . We need the following lemma from Friedman [5].