A sharp global estimate and an overdetermined problem for Monge-Ampère type equations

1 Introduction

About forty years ago, G. Talenti [14] pioneered a method for establishing the following result. Let Ω be a convex domain in ℝ², and let u = u(x, y) be a (negative) convex solution to the problem

Moreover, let Ω(1)* be the disc centered at the origin, having the same perimeter as Ω, and let v = v(x, y) be the negative solution to the problem

Then

where u^* is a suitable symmetrization of u which is defined in Ω(1)* . Clearly, the solution u depends on the shape of Ω, and cannot be written explicitly in general, whereas v is radially symmetric and can be computed explicitly as a solution to an ordinary differential equation. The previous result has been extended to arbitrary dimension n by K. Tso [18, 19] and by N. Trudinger [15, 16], by using a suitable (depending on the dimension n) symmetrization of u. See also [1, 10, 17]. One purpose of the present paper is to extend a similar result to equations which also depend on the gradient Du.

The second problem we consider in this paper is inspired by the following well known result. Let Ω be a bounded smooth domain in ℝⁿ, and let u be the solution to the problem

If there is a constant c > 0 such that |Du| = c on ∂Ω, then it is well-known that Ω must be a ball (see [13]). As far as we know, a problem of this kind, usually referred to as an overdetermined problem, was first considered by J. Serrin, [13]. The above result is a special case of Serrin's problem. The paper [13] has inspired numerous investigations involving linear, nonlinear and fully nonlinear operators. We refer to [3, 4, 7, 20] and the references therein.

We now introduce the Monge-Ampère type equations considered in this paper. Let Ω ⊂ ℝⁿ be a bounded convex domain with a smooth boundary ∂Ω. For x ∈ Ω, we write x = (x¹, ⋯, xⁿ). We use subscripts to denote partial differentiation. For example, we write ui=∂u∂xi, uij=∂2u∂xi∂xj , etc. The familiar Monge-Ampère operator can be written as

where D²u denotes the Hessian matrix of the function u, det(D²u) is the determinant of D²u, and T_(n−1) = T_(n−1)(D²u) is the (n − 1)-order Newton tensor of D²u ([11, 12]). Here and in what follows, the summation convention from 1 to n over repeated indices is in effect. Note that T_(n−1) corresponds to the cofactor matrix of D²u.

Let g : [0, ∞) → (0, ∞) be a smooth real function satisfying

Since

the function g(s²)s is positive and strictly increasing for s > 0. Note that we are assuming that g(0) and G(0) are positive numbers.

We define the g-Monge-Ampère operator as

Recalling that T_(n−1)(D²u) is divergence free (see [11, 12]), we have

Moreover, since T_(n−1)(D²u) is the cofactor matrix of D²u we have

where I denotes the n × n identity matrix. By using (1) and (2), we find

Since the operator det(D²u) is elliptic (in the framework of convex functions), our g-Monge-Ampère operator is elliptic.

A motivation for the definition of the g-Monge-Ampère operator is the following. Using the Kronecker delta δ^i𝓁, define the n × n matrix 𝒜 = [𝒜^ij] with

The trace of the matrix 𝒜 is the familiar operator (g(|Du|²)u_i)_i. We claim that the determinant of the matrix 𝒜 coincides with our operator (3). Indeed, the eigenvalues Λⁱ, i = 1, ⋯ , n, of the n × n matrix

are the following

Since det𝒜=detℬ·det(D²u), we find

The claim follows from the latter equation and (3).

In this paper we show that many results which hold for Monge-Ampère equations can be extended to the corresponding g-Monge-Ampère equations. To state our main results we begin by introducing two assumptions. We need the following notations.

We make the following assumptions.

There is a constants γ > 0 such that

Let f (t) be a non-decreasing function. With γ as in (5), suppose there is 0 ≤ q < γ such that

In stating our results we will consider functions from the class

We now state our first result. We refer to Section 2 for the definitions of (n − 1)-symmetrand of a function on Ω, the one-mean radius η₍₁₎(Ω), and the Gauss curvature ℋ_(n−1) of ∂Ω.

For our second main result we consider an overdetermined system that may admit a convex solution only when the underlying domain is a ball. More specifically we have the following. Recall that functions in Φ(Ω) vanish on the boundary ∂Ω.

We organize the rest of the paper as follows. Section 2 provides a brief preliminary in which basic notions are recalled. In Section 3 we give an estimate of (n − 1)-symmetrands of supersolutions v ∈ Φ(Ω) to (7). This will be followed by a comparison principle for the equation (7), which may be of independent interest. In the same section we will also present the proof of Theorem 1.1. As an application of Theorem 1.1, we give an estimate of an Hessian integral. In Section 4, we will give the proof of Theorem 1.2.

2 Preliminaries

Let κ¹, ⋯ , κⁿ⁻¹ be the principal curvatures of ∂Ω. For j = 1, ⋯ , n − 1 we define the j-th mean curvature of ∂Ω by

where S_(j) denotes the elementary symmetric function of order j of κ¹, ⋯ , κⁿ⁻¹. Let u ∈ Φ(Ω). From Sard's theorem it follows that for almost all t ∈ (m₀, 0), m₀ = min_Ω u, the sub-level sets

will have smooth boundary ∑_t given by the level surface

Clearly, Ω₀ = Ω. Furthermore, we denote with ω_n the volume of the unit ball in ℝⁿ.

For an open set 𝒪 ⊂ ℝⁿ, the quermassintegral V₍₁₎ = V₍₁₎(𝒪) is defined by

where dσ denotes the (n − 1)-dimensional Hausdorff measure in ℝⁿ.

Following [16] we define the 1-mean radius of 𝒪, denoted by η₍₁₎(𝒪), as

In case 𝒪 is a ball, η₍₁₎(𝒪) is the radius of 𝒪. For a general 𝒪, we denote by 𝒪(n−1)* the ball with radius η₍₁₎(𝒪).

As usual, we denote by |E| the Lebesgue measure of a subset E ⊂ ℝⁿ. The following isoperimetric inequality is well known.

It follows that

We define the rearrangement of u with respect to the quermassintegral V₍₁₎ as

The function u^⋆(s) is negative, increasing and satisfies

We also define

The function u^*(x) is called the (n − 1)-symmetrand of u, and can also be defined by (see [16])

Since u^*(x) is radially symmetric we often write u^*(x) = u^*(r) for |x| = r. We have u^*(0) = min _Ω u(x) and u^*(η₍₁₎(Ω)) = 0.

3 Estimates via symmetrization

If Ω is a ball and v is a (radial) solution of (15) with equality sign, then, all the inequalities used in the proof of the latter theorem are equalities. In this situation we have v^*(r) = v(r) and

Now we prove a comparison principle.

Proof of Theorem 1.1. Let w < 0 be a (radial) solution of

Recalling the definition of 𝒦 given in (4), and since

from (27) we find

Integration over (0, r) yields

Comparing this equation with inequality (16) we find

which implies

Integrating over (r, R) and making use of the conditions v^*(R) = w(R) = 0, we find

With v^*(x) = v^*(r) and w(x) = w(r) for r = |x| we have

By (27) and (28) we find

In summary, we see that w and z satisfy

By Lemma 3.3 we have,

Thus, from (28) and the latter inequality we conclude

The theorem follows.

Special cases.

If β = 1, recalling (4) and (3), we find

If β = 0, recalling (4), (3) and (13), we find

We end this section with the following remark. In [5] (Section 5) and in [8] (Chapter 17), the following equation has been investigated.

We note that this equation coincides with our equation (7) with

Since

we find

and

Using the latter function g(s²) we can apply our previous results to the present equation. For instance, (16) reads as

4 Overdetermined problem

Proof of Theorem 1.2. We note that in case of g(s²) = 1 and c = 1, this theorem is proved in [4]. On using (13) we find

From this and (9) we find

Now we prove that for a solution u to (8) we have

Indeed, in view of (8), inequality (30) can be written as

Recall that, if 𝒜 = [𝒜^ij] with 𝒜^ij = g(|Du|²u_i)_j, we have

By the latter equation and (8), (31) can be written as

To prove (32), we first apply the arithmetic-geometric mean inequality

We have equality in (33) if and only if

Now we apply the following Hadamard inequality to the positive definite matrix 𝒜 (see Theorem 7.8.1 of [9])

with equality sign if and only if

With 𝒜^ij = g(|Du|²)u_i)_j, (35) reads as

Conditions (36) become

Inequality (32) follows by (33) and (37). Hence (30) holds. Then, by (29) we must have

But, as equality holds in (30), also equations (34) and (38) must hold.

To finish the proof, we note that by (39) and (8) we find

Recall that the summation convention is in effect in the latter equation. Therefore, on using (34), for j fixed we find

Integrating and using (38) we get

where x¯=(x¯1,⋯,x¯n) is a point of minimum for u(x). It follows that

Adding from j = 1 to j = n we find

and