# Eigenvalue problems in optimality conditions, duality, and relations with optimal transport :

## Abstract

In this article we characterize the eigenvalue problem associated to the Rayleigh quotient and relate it to a divergence-form PDE, similarly to what is known for eigenvalue problems and the for -Laplacian Contrary to existing methods, which study . as limits of -problems for -problems we develop a novel framework for analyzing the limiting problem directly using convex analysis and geometric measure theory. For this, we derive a novel fine characterization of the subdifferential of the Lipschitz-constant-functional , We show that the eigenvalue problem takes the form . where , and are non-negative measures concentrated where respectively are maximal, and is the tangential gradient of with respect to Lastly, we investigate a dual Rayleigh quotient whose minimizers solve an optimal transport problem associated to a generalized KantorovichтАУRubinstein norm. Our results apply to all stationary points of the Rayleigh quotient, including infinity ground states, infinity harmonic potentials, distance functions, etc., and generalize known results in the literature. .

## 1. Introduction

### 1.1. Motivation and main contributions

Nonlinear eigenvalue problems for the for -Laplacian have been the subject of extensive research for the last three decadesтАФsee Reference 22Reference 30Reference 31Reference 32Reference 33Reference 34Reference 36Reference 39 for a nonexhaustive listтАФand have applications in data science Reference 8Reference 23. They can be characterized as solutions of a nonlinear divergence-form PDE or as unique minimizers of a Rayleigh quotient involving the energy (we refer to this as the -Dirichlet eigenvalue problem). For minimizers of the Rayleigh quotient, now involving the Lipschitz constant, are no longer unique. We refer to this problem as the eigenvalue problem. A certain class of minimizers, called infinity ground states, can be recovered as limits of eigenfunctions as -Laplacian General minimizers, however, do not admit such a variational principle. .

In this paper we develop a novel analytical framework for studying the eigenvalue problem which does not require taking the limit and instead uses techniques from convex analysis and geometric measure theory. This allows us to generalize various known results about special classes of minimizers and extend them to general minimizers.

Let us fix some notation. Let be a bounded domain. For we denote the , with respect to a measure -spaces as and we write simply when is the Lebesgue measure. These spaces are equipped with standard -norms or where we omit the dependency on , for the sake of a compact notation. For the Sobolev space , is defined as the closure of the space of smooth and compactly supported functions with respect to the norm .

The eigenvalue problem of the (see -LaplacianReference 39 for a detailed study) consists in finding a function which is a weak solution of

The eigenvalue is given by the minimal value of a nonlinear Rayleigh quotient

Solutions of the eigenvalue problem -LaplacianEquation 1.1 for are known to be unique up to normalization and are in one-to-one correspondence with minimizers of the Rayleigh quotient in Equation 1.2.

In this paper we study the following limiting minimization problem of an Rayleigh quotient over the space of Lipschitz functions on , which are zero on the boundary:

We denote by the inradius of defined as maximal value of the distance function: ,

It is very easy to show Reference 28 that the infimal value in Equation 1.3 is given by

which implies that the distance function is always a minimizer of the Rayleigh quotient.

It has been shown in Reference 15Reference 20Reference 29 that certain classes of minimizers of Equation 1.3 satisfy a divergence-form PDE which is structurally similar to Equation 1.1. Furthermore, a connection between infinity ground states and solutions of a certain optimal transport problem was established in Reference 15.

In this paper we ask the following questions:

- (1)
Do all solutions of the nonlinear eigenvalue problem associated to the Rayleigh quotient satisfy a PDE which is structurally similar to the eigenvalue problem -Laplacian ?

- (2)
Can all minimizers of the Rayleigh quotient be related to solutions of an optimal transport problem?

The short answer is yes, see the PDE Equation 1.7 and the optimal transport problem Equation 1.8. To answer these questions we work with general *stationary points* instead of minimizers of the Rayleigh quotient for which we derive a nonlinear eigenvalue problem in the form of a divergence PDE. Then we shall study minimizers of the Rayleigh quotient which we will relate to the distance function and solutions of an optimal transport problem.

The techniques we use to study the eigenvalue problem are also novel: instead of approximating the with -problem and sending -problems to infinity, we mainly rely on elegant and well-established methods of convex analysis. On the one hand, this establishes a new analytical framework to tackle problems without using viscosity solutions or similar technical concepts from PDE analysis. On the other hand, this makes our results more general since the class of minimizers to the -type considered is strictly larger than the class of minimizers which can be approximated with -problem -problems.

Our **main contributions** are the following:

- тАв
We develop a novel analytical framework solely based on convex analysis and geometric measure theory which allows us to prove known and novel results for without the need to take the technical limit -problems .

- тАв
We derive a nonlinear eigenvalue problem, involving duality maps and subdifferentials, which describes stationary points of the Rayleigh quotient .

- тАв
We characterize solutions to the eigenvalue problem as solutions to a fully nonlinear PDE in divergence form,

involving non-negative measures and which are concentrated where respectively are maximal, and the notion of a

*tangential gradient*developed in Reference 5, see also Reference 15Reference 26Reference 40. This is our main result Theorem 2.1.- тАв
We show geometric relations between general minimizers of the Rayleigh quotient, the distance functions to the boundary, and the distance function to a generalized inball.

- тАв
We derive a dual Rayleigh quotient defined on the space of measures on and relate it to an optimal transport problem involving a variant of the KantorovichтАУRubinstein norm. In particular, Proposition 4.6 shows that the measure solves

where and are the spaces of probability measures on and its boundary respectively, and , is the geodesic distance. -Wasserstein

The rest of the paper is organized as follows: Section 1.2 discusses special classes of minimizers of the Rayleigh quotient, namely infinity ground states and infinity harmonic potentials. In Section 1.3 we introduce essential concepts from convex analysis and derive general relations of nonlinear eigenvalue problems and Rayleigh quotients on Banach spaces. In particular, we show equivalence between minimizers of the Rayleigh quotient and those of a dual Rayleigh quotient, which is a new result in its own right. In Section 1.4 we define suitable spaces of continuous functions and measures and their duality relations. Section 2 constitutes the core of our article where we first state our main result and some corollaries, characterizing the eigenvalue problem and minimizers of the Rayleigh quotient, and then characterize subdifferentials to prove the result. In Section 3 we provide some geometric relations between minimizers of the Rayleigh quotient and the distance function. Section 4, where we investigate a dual Rayleigh quotient and provide an optimal transport characterization of the subgradients of minimizers using KantorovichтАУRubinstein theory, is self-contained and does not utilize the subdifferential characterizations from Section 2. Section 5 concludes the paper with a summary of our results and some open questions.

### 1.2. Special solutions of the eigenvalue problem

Besides the distance function Equation 1.4, which is always a minimizer of the Rayleigh quotient in Equation 1.3, there are two other important classes of minimizers: infinity ground states and infinity harmonic potentials. Unless for very specific domains Reference 46, these three different classes of minimizers are different.

In Reference 28 it was shown that in the limit normalized eigenfunctions of the i.e., solutions of -Laplacian,Equation 1.1 with converge (up to a subsequence) uniformly to a continuous function ,

Here

While every solution to Equation 1.9 is a minimizer of the Rayleigh quotient in Equation 1.3, the converse is not true and there are typically many minimizers which do not solve Equation 1.9. Furthermore, this PDE can have solutions which do not arise as limits of solutions of Equation 1.1 for

Apart from the distance function and infinity ground states, another class of minimizers of the Rayleigh quotient is infinity harmonic potentials, defined as solutions to

The set

Also infinity harmonic potentials are in general no infinity ground states; a counterexample on a convex domain can be found in Reference 37. For interesting properties of these potentials and their streamlines we refer to Reference 38.

### 1.3. Nonlinear eigenvalue problems on Banach spaces

Before we specialize the discussion to

We let

The subdifferential is a generalization of the Frechet derivative for nondifferentiable convex functionals. Geometrically,

In the context of nonlinear eigenvalue problems, absolutely homogeneous functionals have particular importance since they can be used to formulate a plethora of eigenvalue problems, e.g., associated to linear operators, or nonlinear differential operators like the

Since absolutely one-homogeneous functionals are seminorms on subspaces of

For the specific choice

We assume without loss of generality that

which can always be achieved by replacing

and the minimal value of the Rayleigh quotient is defined as

Positivity of

In this case, obviously

Indeed, the minimal value

Proposition 1.5тАФthe proof of which is standard and can be found in Reference 2 or Reference 11 in large generalityтАФstates that minimizers of

We conclude this section with a study of the dual eigenvalue problem to Equation 1.21. For this, we define the *dual functional* of

Since

with associated dual eigenvalue problem

The relation to the primal Rayleigh quotient

### 1.4. Functions and measures

Having some abstract theory of nonlinear eigenvalue problems in Banach spaces at hand, we now introduce the setup for the

For a bounded domain

Weak

We denote non-negative measures by

The space of vector-valued Radon measures on

We let