Overdetermined problems for p-Laplace and generalized Monge–Ampére equations

We investigate overdetermined problems for p-Laplace and generalized Monge–Ampére equations. By using the theory of domain derivative, we find duality results and characterization of the overdetermined boundary conditions via minimization of suitable functionals with respect to the domain.


Introduction
Let D be a bounded smooth domain in R N . A point x ∈ D will be denoted with x = (x 1 , . . . , x N ). We also denote u i = ∂u ∂x i , u ij = ∂ 2 u ∂x i ∂x j , etc., the partial derivatives of u. Let us recall the following well-known overdetermined problem. Let c be a constant. If there exists a solution u to the Dirichlet problem u = 1 in D, u = 0 on ∂D (1) such that u satisfies the additional condition |∇u| = c on ∂D (2) then D must be a ball. This result has been proved by Serrin [1] in 1971 using the moving plane method. At the same time, Weinberger [2] yields a different proof of the same result by using a Pohozaev identity and the maximum principle applied to a suitable P-function. The method of Weinberger requires less regularity of the boundary ∂D, but the method CONTACT Yichen Liu yichen.liu07@yahoo.com This article has been republished with minor changes. These changes do not impact the academic content of the article. of Serrin can be easily applied to a large class of non-linear and fully non-linear operators. These two celebrated papers have inspired a great number of mathematicians, and the corresponding literature is nowadays very prominent. We refer to [3][4][5][6][7][8] and references therein. For recent progress on this topic, we refer to the survey [9]. Among several ideas related to this overdetermined problem, we recall the following duality result [7]. Theorem 1.1: Let u ∈ C 2 (D) ∩ C 1 (D) be a solution to Problem (1). The following statements are equivalent: (i) u satisfies condition (2).

holds for all functions v harmonic in D.
Motivated by this result, we shall prove duality theorems for overdetermined problems involving p-Laplace equations as well as generalized Monge-Ampére equations. In the case of generalized Monge-Ampére equations, the overdetermined boundary condition is not the same as (2), but condition (27) below. In the linear case (κ = 1) , condition (27) reduces to the familiar condition |∇u| = c on ∂D. If 1 < κ ≤ N, this condition involves ∇u as well as the second derivatives of u throughout the Newton tensor T κ−1 (u). Furthermore, we consider suitable functionals of the domain D whose minimizers must satisfy overdetermined boundary condition (2) for the p-Laplace problem, and condition (27) for the generalized Monge-Ampére problems. A crucial tool serving us shall be the domain derivative.
The paper is organized as follows. In Section 2, we introduce the notion of domain derivative. Some of our descriptions are formal; for a precise treatment of the domain derivative, we refer to [10]. In particular, we find a sort of linearized equation of the p-Laplace equation p u = f (u) (see Equation (10)), as well as a linearized equation of the generalized Monge-Ampére equation S κ (u) = f (u) (see Equation (19)). These linearized equations are crucial to get our duality results. Sections 3 and 4 contain our main results. Section 3 is made of two subsections. In Section 3.1, we prove a duality result for a p-Laplace boundary value problem (see Theorem 3.1). In Section 3.2, we prove a duality result for a boundary value problem corresponding to a generalized Monge-Ampére equation (see Theorem 3.2). Also, Section 4 is made of two subsections. In Section 4.1, we introduce a special functional associated with our p-Laplace equation in a domain D. We shall prove that the minimum of such functional with respect to D under the condition |D| = constant yields a condition for ∇u on ∂D which is the same as used in Theorem 3.1(i). In Section 4.2, we introduce a special functional associated with a generalized Monge-Ampére equation in a domain D. We shall prove that the minimum of such functional with respect to D under the condition |D| = constant yields a condition for ∇u on ∂D which is the same as used in Theorem 3.2(i).

Domain derivative
The theory of domain derivative is very useful in fields as shape optimization. From a mathematical point of view, it goes back to Hadamard [11] and Schiffer [12]. We recall shortly the definitions and refer to [10] for a careful treatment. If L(u) is a differential operator, we consider the Dirichlet problem: where f is a smooth function such that problem (4) has a unique solution. Let I be the identity map. For a smooth (C 2 is enough) vector field V : R N → R N , and |t| small, define Now, we consider the Dirichlet problem in D t : Clearly, since D t depends on the vector field V, also v depends on V. By [10], v satisfies the boundary condition where ν = (ν 1 , . . . , ν N ) is the unit exterior normal on ∂D.
To obtain the equation for v, we compute If f is differentiable, we have The computation of the left-hand side of (8) depends on the structure of the differential operator L. If L(u) = u we find Consider now the p-Laplacian L(u) = div(|∇u| p−2 ∇u). We have Hence, in this case, the equation corresponding to (8) for v reads as Now we recall the definition of generalized Monge-Ampére operators. Let 1 ≤ κ ≤ N, and let S κ (u) be the κth elementary symmetric function of the eigenvalues of the Hessian matrix H = D 2 u = [u ij ] (that is, the sum of all principal minors of order κ of H). Clearly, we have S 1 (u) = u (Laplace operator) and S N (u) = det[D 2 u] (Monge-Ampére operator). Given a positive smooth function f (t), we consider the problem Suppose the domain D ⊂ R N is bounded and smooth. In addition, for κ fixed such that 2 ≤ κ ≤ N, we assume the following property: where β is a positive constant and σ κ−1 is the (κ − 1)th elementary symmetric function of the principal curvatures of ∂D with respect to its inner normal (see [13,14]). If we denote by τ 1 , τ 2 , . . . , τ N−1 the principal curvatures of the surface ∂D, we have Note that condition (P N ) means that is strictly convex. Moreover, if enjoys property (P κ ) then also D t = (I + tV)(D), for |t| small, enjoys the same property (possibly with a smaller constant β). Finally, f (t) is a positive smooth function such that problem (11) has a unique admissible solution. As usual, a solution is admissible if the operator S κ (u) is positive definite. In this situation, the solution u is negative in D and ν = ∇u |∇u| on the boundary ∂D. We refer to [13,14] for a careful discussion of this problem.
It is convenient to define the matrix We put T 0 (u) = I, the identity matrix. The matrix T κ (u) is known as the κth Newton tensor associated with H. We have [15] T Since H is symmetric, T κ is also symmetric. It has several nice properties. For example, we have ∂x i , and here and in what follows, we use the summation convention over repeated indices from 1 to N. To prove (13), we recall the definition of the generalized Kronecker symbol where i 1 , . . . , i q are distinct integers between 1 and N, and also j 1 , . . . , j q are distinct integers between 1 and N. The value of the symbol is 1 (respectively −1) if (j 1 , . . . , j q ) is an even (respectively an odd) permutation of (i 1 , . . . , i q ), and is 0 in all other cases. If 1 ≤ κ ≤ N − 1 we have (see [16]) We find Simplifying we can write We note that u i 1 j 1 i is symmetric with respect to i 1 i, while the Kronecker symbol is skewsymmetric with respect to those indices. Thus, the sum over i 1 i vanish and (13) follows. The proof above can be extended to prove that, if v is also a smooth function, we have We refer to Proposition 2.1 of [16] for details. Another very interesting property is the following (see [15,16]) We are now ready to find the equation for v defined as in (6) with L(u) = S κ (u). Let u t be the (admissible) solution to problem (11) corresponding to D t . Using (16) and (13), we have We have Using (13) again we find Moreover, using (14), we have Hence, By using (15) and changing the indices conveniently, we find where (14) has been used once more. From (17), (18) and the latter result, we find Hence, recalling (9), we find the equation for v:

Duality results
In this section, we extend Theorem 1.1 to p-Laplace equations and to generalized Monge-Ampére equations.

p-Laplace equations
Let D ⊂ R N be a bounded smooth domain, and let f : R → R be a C 1 positive function such that the problem has a unique (negative) solution u ∈ C 1 (D) ∩ W 1,p (D). For example, one can take, for τ < 0, f (τ ) = (−τ ) α , 0 ≤ α < p (see [17]). We have (i) There is a constant c such that for all solutions v to Equation (10).

Generalized Monge-Ampére equations
Let κ be an integer such that 1 ≤ κ ≤ N. Let D ⊂ R N be a bounded smooth domain satisfying property (P κ ), and let f : R → R be a C 1 positive function such that the problem has a unique admissible solution u ∈ C 3 (D) ∩ C 1 (D). We have (i) There is a constant c such that

hold for all solutions v to Equation (19).
Proof: Multiplying (19) by −u, integrating over D, using (13) and recalling that u = 0 on ∂D we find Integrating by parts and using (13) again, we find Since T ij κ−1 (u)u ij = κf (u) in D and ν i |∇u| = u i on ∂D, from the latter equation we find From (29) and (30) it follows that Finally, using boundary condition (7), we get Since V is arbitrary, (27) follows with c 2 = d. The theorem is proved.
Let us recall a result from [18]. such that u satisfies the additional condition then D must be an ellipse.

Corollary 3.4: Let D be a bounded convex domain in the plane and let c be a constant. If there exists a convex solution u to problem (33) such that the integral equations
hold for all solutions v to the equation then, D is an ellipse.

Minimization of functionals
In this section, we present motivation of overdetermined conditions (2) and (27).

p-Laplace equations
Let D ⊂ R N be a bounded smooth domain, and recall problem (20) below where f is a positive function such that problem (37) has a unique (negative) solution.
Given D and the corresponding solution u to problem (37), we consider the functional We compute first dJ(D, V). Let u be the solution of problem (37) with D =D, and let u t be the solution of problem (37) corresponding to D t . We have Since u = 0 on ∂D, we find Therefore, we find where v is defined as Integrating the equation Recalling that ∇u = |∇u| ν on ∂D and using boundary condition (7), from the latter equation, we find By (40) and the latter equation, we find (41) On the other hand (see [10, p.652] formula (12) with C(u) = 1), we have Insertion of (41) and (42) into (39) yields Since dI(D, V) = 0 for every vector field V, it follows that |∇u| p = λ p−1 . Therefore, |∇u| is a constant on ∂D, and the theorem is proved.

Generalized Monge-Ampére equations
Now we prove a similar result for generalized Monge-Ampére equations. Assume the domain D bounded, smooth and having the property (P κ ). Let u be an admissible solution to the problem Here 1 ≤ κ ≤ N and f (t) > 0. Consider the functional where u is an admissible solution to problem (43). Proof: Let us find a different formulation for E(D). If we multiply (43) by u and use (16), we have Integration over D yields Hence, the functional defined by (44) can be rewritten as From now on, we shall use this formula for E(D).
By the well-known Lagrange principle,D is a stationary point of the functional where λ is a real parameter. For a smooth vector field V, let D t = (I + tV)D be a deformation ofD. We must have dI(D, V) = 0 for every vector field V. Clearly, If u t is the solution to problem (43) corresponding to D t , we compute Since u = 0 on ∂D, the first integral vanishes. Hence, As usual, the function v is defined as Since dI(D, V) = 0 for every vector field V, it follows that T ij κ−1 (u)u i u j = λ κ . Therefore, T ij κ−1 (u)u i u j is a constant on ∂D, and the theorem is proved.

Disclosure statement
No potential conflict of interest was reported by the author(s).