Concavity of solutions to degenerate elliptic equations on the sphere

This paper is about classical solutions to fully nonlinear degenerate equations of the form

on a domain $\Omega\subset\mathbb{S}^{n}$, $n\geq 2$, of the unit sphere whose closure $\bar{\Omega}$ is geodesically convex. (We recall that a subset $C$ of $\mathbb{S}^{n}$ is geodesically convex if no two points of $C$ are antipodal and the unique minimizing geodesic segment between any two points of $C$ lies entirely in $C$.) Here

where $D$ is the Levi-Civita connection of the round metric $\sigma$ on $\mathbb{S}^{n}$ and the $\sharp$-operator is the usual index raising operator with respect to $\sigma$, so that $\nabla u$ is the gradient vector field and $\nabla^{2}u$ the Hessian endomorphism field on $\Omega$.

We want to prove concavity of solutions $u\in C^{2}(\Omega)\cap C^{1}(\bar{\Omega})$ to (1.1) satisfying suitable boundary conditions. The question of concavity of solutions to elliptic equations on Euclidean domains is widely studied, in particular in connection with Laplacian eigenvalues. For example, Brascamp and Lieb [4] proved that the first Dirichlet eigenfunction of the Laplacian over a convex domain is log-concave. A recent highlight in this area was, with many partial results preceding it, the proof of the fundamental gap conjecture by Andrews and Clutterbuck [2]. Using a refined log-concavity estimate for the first eigenfunction, they obtained a sharp estimate for the difference between the first and second Dirichlet eigenvalues (the “fundamental gap”). An analogous result on the sphere was obtained by Seto, Wang and Wei [12].

Our main tool will be the “two-point maximum principle” approach introduced by Korevaar [9]. The idea is to estimate the function

using a first and second derivative test from below by zero, where $u$ is the solution to a certain quasilinear equation with some structure properties. Refinements of this approach for $C^{2}$-solutions are due to Kennington [8] and Kawohl [7]. Using regularization, Sakaguchi [11] has transferred this method to solutions of certain $p$-Laplace equations, the solutions of which may be non-smooth. Similar extensions to viscosity solutions are the content of [1] and [6]. The argument was used to obtain proofs of differential Harnack inequalities for hypersurface flows in [3].

To our knowledge, there is no such concavity maximum principle for non-Euclidean domains. The reason for this seems to arise from a major technical obstruction: in order to make the application of the maximum principle to (1.3) work, it is necessary to differentiate a family of geodesics with respect to their end-points. This leads to the consideration of a family of Jacobi fields and their variations. In order to exploit the first order conditions, one needs detailed knowledge about the nonlinear Jacobi fields along the $\tilde{Z}$-minimizing geodesic. Even worse, the second order condition contains a further space derivative of the Jacobi equation and in general it seems impossible to extract the correct sign, which is required to apply the maximum principle, from these conditions.

The purpose of this paper is to initiate the study of concavity maximum principles for degenerate elliptic equations on a Riemannian manifold and we start this journey on the unit sphere. Here we have managed to deal with the pretty well known Jacobi fields and their maybe not so well known derivatives. One first key observation is that, for technical reasons, the full three-parameter function $\tilde{Z}$ is no longer suitable. The concavity of a solution is equivalent to the concavity of

for every geodesic

cf. [13]. By a well known result due to Jensen [5], for continuous functions this is equivalent to asking this function to be midpoint-concave, i.e.

Hence it will suffice to prove that the function

is non-negative, where $\gamma_{x,y}$ is the unique minimizing geodesic from $x$ to $y$. The mid-point approach has also been taken for equations on Euclidean domains in [6, 7].

Before we state our main theorem, we need to introduce some more objects. We denote by $(\mathcal{V},\mathcal{L})$ the category of $n$-dimensional real vector spaces with linear transformations as its objects. A function $f$ on that category is called isotropic if it acts on the subclass $\mathcal{E}$ of endomorphisms in a $\mathrm{GL}_{n}$-invariant way:

where $\mathcal{L}_{\mathrm{inv}}$ denotes the class of all invertible linear maps, understood to map to the correct space such that (1.8) makes sense. From this property it is clear that the action of $f$ on real diagonalizable endomorphism is only determined through the action on the ordered set of eigenvalues $(\kappa_{1},\dots,\kappa_{n})$,

The properties we will require from $f$ will in general only be satisfied if $f$ is restricted to a subclass of $\mathcal{E}$ and the best way to describe the domain of definition of $f$ is via $\tilde{f}$. Therefore we suppose the domain of definition for $\tilde{f}$, $\Gamma\subset\mathbb{R}^{n}$, is a symmetric, open and convex cone which contains the positive cone

Denote by $\mathrm{Sym}^{n\times n}(\mathbb{R})$ the vector space of symmetric $(n\times n)$-matrices. Then there exists an open domain of definition $\mathcal{D}_{\Gamma}\subset\mathrm{Sym}^{n\times n}(\mathbb{R})$ for $f$ consisting of matrices with eigenvalues in $\Gamma$, such that (1.9) holds for all $\mathcal{W}\in\mathcal{D}_{\Gamma}$.

Abusing notation, the orthogonal invariance of $f$ allows us to understand $\mathcal{D}_{\Gamma}$ to be a subset of any other space of $\bar{g}$-self-adjoint endomorphisms of an arbitrary $n$-dimensional inner product space $(E,\bar{g})$. Here is our main theorem:

For the proof of Theorem 1.1 we need an easy result from linear algebra and some tedious calculations of Jacobi fields and their derivatives. We address these issues in the next two sections, before we can proceed to the proof of the theorem.

The nonlinear ambient space will introduce an algebraic problem. In order deal with it, we need the following lemma from linear algebra. It should be standard, but we did not find a proper reference, so let us give the proof for the reader’s convenience.

We have to differentiate (1.7) twice and in this section we collect the geometric ingredients of this process. The derivatives of $\gamma_{x,y}$ will be Jacobi fields and variations of them and we will make extensive use of the Jacobi equation. Define

For the moment choose arbitrary coordinates $(x^{i},y^{i})$ in $\Omega\times\Omega$. The vector fields

are, for given $x,y$, Jacobi fields along $\gamma_{x,y}$ with

where we identify $\partial_{x^{i}}$ with its pushforward under the identity map $\Gamma(\cdot,y,-1)$ and likewise for $\partial_{y^{i}}$. We will have to deal with the second derivatives of $\Gamma$ which are given by (neglecting the distinction between $D_{\partial_{x^{j}}}$ and $D_{\frac{\partial\Gamma}{\partial x^{j}}}$, since later we will work in Riemannian normal coordinates)

Those satisfy a differentiated Jacobi equation, which we derive in the following. We use the following convention for the Riemannian curvature tensor.

i.e. on $\Omega\subset\mathbb{S}^{n}$ we have

where we recall that $\sigma$ denotes the round metric. With this convention the Jacobi equation, which for example $J_{x^{i}}$ satisfies for fixed $x,y$, is

see e.g. [10, Ch. 10]. The same holds for $J_{y^{i}}$ and now we have to differentiate this equation with respect to $x^{j}$ and $y^{j}$. We consider it to be easier to follow those calculations if we do not explicitly put in the specific form of the Riemann tensor just yet. However we will already use $D\operatorname{Rm}=0$. For $K_{x^{i}x^{j}}$ we calculate:

In order to get more information about these quantities, first we note that at $t=\pm 1$ all $K$ are zero. This is due to the fact that

and the second covariant derivative of an isometry is always zero. Also it is in order to pick the right coordinates. In the following, latin indices range from $0$ to $n-1$ and greek indices range from $1$ to $n-1$. At a given point $x_{0}$ pick Riemannian normal coordinates, where with a rotation we arrange

while at $y_{0}$ we do the same. Since we have Riemannian normal coordinates at $x_{0}$ and $y_{0}$, the $2n$ vectors $(\partial_{x^{i}},\partial_{y^{i}})$ form an orthonormal basis at $(x_{0},y_{0})$, while $\partial_{y^{0}}$ is the parallel transport of $\partial_{x^{0}}$. Furthermore, the Christoffel symbols vanish at $(x_{0},y_{0})$ in the product space. After a possible orthogonal transformation at $y_{0}$ and reordering of basis vectors, there exists a set of vector fields $(E_{i}(t))_{t\in[-1,1]}$ parallel along $\gamma_{x_{0},y_{0}}$, such that

To prove Theorem 1.1, we need to prove that the two-point function

defined in $\bar{\Omega}\times\bar{\Omega}$ is non-negative at minimal points. Therefore we may assume that $x\neq y$. If one of the minimizing points, say $x$, is at the boundary, then we prove that by moving $x$ and $y$ towards each other at the same speed (i.e. while fixing the mid-point $z$), we could decrease the value of $Z$. To see this, consider the function

There holds

Hence $\dot{c}(0)$ is strictly negative and $Z$ must achieve its minimum at interior points.

Let $z=\gamma_{x,y}(0)$. The first order conditions on $Z$ are (suppressing the spatial arguments for $J$)

and

with

and the latter solves (3.24) and hence

We use the bases

which are orthonormal at the respective points $x,z,y$, to identify all of the three tangent spaces canonically with one single Euclidean $\mathbb{R}^{n}$. In these coordinates, define an endomorphism $V$ through its matrix representation,

Then