Eigenvalue problems in $\mathrm{L}^\infty$: optimality conditions, duality, and relations with optimal transport
By Leon Bungert and Yury Korolev
Abstract
In this article we characterize the $\mathrm{L}^\infty$ eigenvalue problem associated to the Rayleigh quotient $\left.{\|\nabla u\|_{\mathrm{L}^\infty }}\middle /{\|u\|_\infty }\right.$ and relate it to a divergence-form PDE, similarly to what is known for $\mathrm{L}^p$ eigenvalue problems and the $p$-Laplacian for $p<\infty$. Contrary to existing methods, which study $\mathrm{L}^\infty$-problems as limits of $\mathrm{L}^p$-problems for $p\to \infty$, 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 $u\mapsto \|\nabla u\|_{\mathrm{L}^\infty }$. We show that the eigenvalue problem takes the form $\lambda \nu u =-\operatorname {div}(\tau \nabla _\tau u)$, where $\nu$ and $\tau$ are non-negative measures concentrated where $|u|$ respectively $|\nabla u|$ are maximal, and $\nabla _\tau u$ is the tangential gradient of $u$ with respect to $\tau$. 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 $p$-Laplacian for $p<\infty$ 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 $p$-Dirichlet energy (we refer to this as the $\mathrm{L}^p$ eigenvalue problem). For $p=\infty$ minimizers of the Rayleigh quotient, now involving the Lipschitz constant, are no longer unique. We refer to this problem as the $\mathrm{L}^\infty$ eigenvalue problem. A certain class of minimizers, called infinity ground states, can be recovered as limits of $p$-Laplacian eigenfunctions as $p\to \infty$. General minimizers, however, do not admit such a variational principle.
In this paper we develop a novel analytical framework for studying the $\mathrm{L}^\infty$ eigenvalue problem which does not require taking the limit $p\to \infty$ 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 $\Omega \subset \mathbb{R}^n$ be a bounded domain. For $p\in [1,\infty ]$, we denote the $\mathrm{L}^p$-spaces with respect to a measure $\mu$ as $\mathrm{L}^p_\mu (\Omega )$ and we write simply $\mathrm{L}^p(\Omega )$ when $\mu$ is the Lebesgue measure. These spaces are equipped with standard $\mathrm{L}^p$-norms$\left\Vert u \right\Vert _{\mathrm{L}^p_\mu }$ or $\left\Vert u \right\Vert _{\mathrm{L}^p}$, where we omit the dependency on $\Omega$ for the sake of a compact notation. For $p\in [1,\infty )$, the Sobolev space $\mathrm{W}^{1,p}_0(\Omega )$ is defined as the closure of the space of smooth and compactly supported functions with respect to the norm $\left\Vert u \right\Vert _{\mathrm{W}^{1,p}}\coloneq \left\Vert u \right\Vert _{\mathrm{L}^p}+\left\Vert \nabla u \right\Vert _{\mathrm{L}^p}$.
The eigenvalue problem of the $p$-Laplacian (see Reference 39 for a detailed study) consists in finding a function $u\in \mathrm{W}^{1,p}_0(\Omega )$ which is a weak solution of
$$\begin{equation} \lambda _p^p \left\vert u \right\vert ^{p-2}u=-\operatorname {div}(\left\vert \nabla u \right\vert ^{p-2}\nabla u). \cssId{texmlid1}{\tag{1.1}} \end{equation}$$
The eigenvalue $\lambda _p>0$ is given by the minimal value of a nonlinear Rayleigh quotient
$$\begin{equation} \lambda _p\coloneq \inf _{u\in \mathrm{W}^{1,p}_0(\Omega )}\frac{\left\Vert \nabla u \right\Vert _{\mathrm{L}^p}}{\left\Vert u \right\Vert _{\mathrm{L}^p}}. \cssId{texmlid2}{\tag{1.2}} \end{equation}$$
Solutions of the $p$-Laplacian eigenvalue problem Equation 1.1 for $p>1$ 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 $\mathrm{L}^\infty$ Rayleigh quotient over $\mathrm{Lip}_0(\Omega )$, the space of Lipschitz functions on $\Omega$ which are zero on the boundary:
$$\begin{equation} \lambda _\infty \coloneq \inf _{u\in \mathrm{Lip}_0(\Omega )}\frac{\left\Vert \nabla u \right\Vert _{\mathrm{L}^\infty }}{\left\Vert u \right\Vert _{\mathrm{L}^\infty }}. \cssId{texmlid3}{\tag{1.3}} \end{equation}$$
We denote by $r_\Omega >0$ the inradius of $\Omega$, defined as maximal value of the distance function:
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 ${\left\Vert \nabla u \right\Vert _{\mathrm{L}^\infty }}/{\left\Vert u \right\Vert _{\infty }}$ satisfy a PDE which is structurally similar to the $p$-Laplacian eigenvalue problem $\lambda \left\vert u \right\vert ^{p-2}u=-\operatorname {div}(\left\vert \nabla u \right\vert ^{p-2}\nabla u)$?
(2)
Can all minimizers of the Rayleigh quotient ${\left\Vert \nabla u \right\Vert _{\mathrm{L}^\infty }}/{\left\Vert u \right\Vert _{\infty }}$ 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 ${\left\Vert \nabla u \right\Vert _{\mathrm{L}^\infty }}/{\left\Vert u \right\Vert _{\infty }}$ 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 $\mathrm{L}^\infty$ eigenvalue problem are also novel: instead of approximating the $\mathrm{L}^\infty$-problem with $\mathrm{L}^p$-problems and sending $p$ 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 $\mathrm{L}^\infty$-type 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 $\mathrm{L}^\infty$-problem considered is strictly larger than the class of minimizers which can be approximated with $\mathrm{L}^p$-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 $\mathrm{L}^\infty$-problems without the need to take the technical limit $p\to \infty$.
тАв
We derive a nonlinear eigenvalue problem, involving duality maps and subdifferentials, which describes stationary points of the Rayleigh quotient ${\left\Vert \nabla u \right\Vert _{\mathrm{L}^\infty }}/{\left\Vert u \right\Vert _\infty }$.
тАв
We characterize solutions to the eigenvalue problem as solutions to a fully nonlinear PDE in divergence form,$$\begin{equation} \lambda \nu u = -\operatorname {div}(\tau \nabla _\tau u), \cssId{texmlid4}{\tag{1.7}} \end{equation}$$
involving non-negative measures $\nu$ and $\tau$ which are concentrated where $|u|$ respectively $|\nabla u|$ are maximal, and the notion of a tangential gradient$\nabla _\tau u$ 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 $\Omega$ and relate it to an optimal transport problem involving a variant of the KantorovichтАУRubinstein norm. In particular, Proposition 4.6 shows that the measure $\mu \coloneq \nu \left\Vert u \right\Vert _\infty$ solves$$\begin{equation} \max _{\tilde{\mu }\in \mathcal{P}(\Omega )}\inf _{\rho \in \mathcal{P}(\partial \Omega )} W^1(\tilde{\mu },\rho ), \cssId{texmlid5}{\tag{1.8}} \end{equation}$$
where $\mathcal{P}(\Omega )$ and $\mathcal{P}(\partial \Omega )$ are the spaces of probability measures on $\Omega$ and its boundary $\partial \Omega$, respectively, and $W^1(\cdot ,\cdot )$ is the geodesic $1$-Wasserstein distance.
The rest of the paper is organized as follows: Section 1.2 discusses special classes of minimizers of the $\mathrm{L}^\infty$ 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 $\mathrm{L}^\infty$ 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 $p\to \infty$ normalized eigenfunctions $u_p\in \mathrm{W}^{1,p}_0(\Omega )$ of the $p$-Laplacian, i.e., solutions of Equation 1.1 with $\left\Vert u_p \right\Vert _{\mathrm{L}^p}=1$, converge (up to a subsequence) uniformly to a continuous function $u_\infty$, termed infinity ground state. Furthermore, $u_\infty$ is a viscosity solution of the following PDE, which is structurally completely different from Equation 1.1:
Here $\lambda _\infty$ is given by the reciprocal inradius as in Equation 1.6, and $\Delta _\infty u\coloneq \langle \nabla u, D^2 u\nabla u\rangle$ denotes the infinity Laplacian operator, see the seminal work Reference 1 for a detailed study and Reference 27 for intriguing properties.
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 $p\to \infty$ and are hence called nonvariational ground states, see Reference 25 for an example. Only for a very specific class of domains $\Omega$, namely stadium-like sets as classified in Reference 46, these ambiguities do not occur and the distance function is the unique minimizer of the Rayleigh quotient and viscosity solution of the PDE.
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 $\mathcal{R}_\Omega \subset \Omega$ is the so-called high ridge of $\Omega$, defined as the set of all points with maximal distance to the boundary:
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 $\mathrm{L}^\infty$ eigenvalue problems, this section contains a short primer on nonlinear eigenvalue problems in Banach spaces. We introduce some important concepts from convex analysis, e.g., subdifferentials and duality maps, introduce nonlinear eigenvalue problems, and discuss their dual versions. The presentation follows the lines of Reference 9Reference 11.
We let $\mathcal{X}$ be a Banach space over $\mathbb{R}$ with topological dual space $\mathcal{X}^*$. The duality product is denoted by $\langle \cdot ,\cdot \rangle$ and the norm on $\mathcal{X}^*$ is given by
The subdifferential is a generalization of the Frechet derivative for nondifferentiable convex functionals. Geometrically, $\partial J(u)$ contains all slopes such that the linerarization of $J$ in $u$ with this slope lies below the graph of $J$. By definition, $\partial J(u)$ is a subset of the dual space $\mathcal{X}^*$.
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 $p$-Laplacian or the porous medium operator (see, e.g., Reference 10Reference 12Reference 24).
Since absolutely one-homogeneous functionals are seminorms on subspaces of $\mathcal{X}$, their subdifferential can be characterized as Reference 2Reference 14
which can always be achieved by replacing $\mathcal{X}$ with the quotient space $\mathcal{X}/\mathcal{N}(J)$, see Reference 10. Then we can define a nonlinear Rayleigh quotient
$$\begin{equation} R(u)=\frac{J(u)}{\left\Vert u \right\Vert _\mathcal{X}},\quad u\in \mathcal{X}\setminus \{0\} \cssId{texmlid39}{\tag{1.18}} \end{equation}$$
and the minimal value of the Rayleigh quotient is defined as
In this case, obviously $C=\lambda _{\min }$ is the optimal constant in Equation 1.20.
Indeed, the minimal value $\lambda _{\min }$ of the Rayleigh quotient can be interpreted as smallest eigenvalue. To see this we define a doubly nonlinear eigenvalue problem as follows:
Proposition 1.5тАФthe proof of which is standard and can be found in Reference 2 or Reference 11 in large generalityтАФstates that minimizers of $R$ coincide with eigenfunctions with eigenvalue $\lambda _{\min }$.
We conclude this section with a study of the dual eigenvalue problem to Equation 1.21. For this, we define the dual functional of $J$тАФnot to be confused with the convex conjugateтАФas follows:
Since $J$ is a seminorm when being absolutely one-homogeneous, the dual functional is nothing but the dual seminorm, see Reference 12. In particular, it is also absolutely one-homogeneous and we can define the dual Rayleigh quotient
The relation to the primal Rayleigh quotient $R(u)$ and the eigenvalue problem Equation 1.21 becomes clear in Proposition 1.8, which states that a solution of the primal problem gives rise to a dual solution.
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 $\mathrm{L}^\infty$-type problem that we are studying.
For a bounded domain $\Omega \subset \mathbb{R}^n$ we let $\mathrm{C}_0({\Omega })$ denote the space of all continuous functions on ${\Omega }$ which vanish on $\partial \Omega$. Equipped with the norm $\left\Vert u \right\Vert _\infty \coloneq \max _{\Omega }|u|$ this is a Banach space. We note that $\mathrm{C}_0(\Omega )=\mathrm{C}_0(\overline{\Omega })$ and is hence a closed subspace of $\mathrm{C}(\overline{\Omega })$. Its dual space is given by the space of finite and signed Radon measures $\mathcal{M}(\Omega )$ on $\Omega$ equipped with the total variation norm $\left\Vert \mu \right\Vert _{\mathcal{M}(\Omega )}\coloneq \left\vert \mu \right\vert (\Omega )$, and the duality pairing is
Weak$^*$ convergence of measures $\{\mu _n\}_{n\in \mathbb{N}}\subset \mathcal{M}(\Omega )$ to $\mu \in \mathcal{M}(\Omega )$ is denoted by $\mu _n\rightharpoonup ^\ast \mu$ and means $\langle u,\mu _n\rangle \to \langle u,\mu \rangle$ for all $u\in \mathrm{C}_0({\Omega })$. Because of the weak$^*$ lower semicontinuity of the total variation, one has $\left\vert \mu \right\vert ({\Omega })\leq \liminf _{n\to \infty }\left\vert \mu _n \right\vert ({\Omega })$.
We denote non-negative measures by $\mathcal{M}_+(\Omega )$ and abbreviate by $\mathcal{P}(\Omega )$ the set of probability measures which consists of all measures $\mu \in \mathcal{M}_+(\Omega )$ with $\mu (\Omega )=1$.
The space of vector-valued Radon measures on $\Omega$ is denoted as $\mathcal{M}(\Omega ,\mathbb{R}^n)$ and can be equipped with the same notion of convergence. The so-called divergence-measure fields Reference 16Reference 17Reference 18Reference 41 constitute an important subclass of vector-valued Radon measures, which will turn out to be essential for studying $\mathrm{L}^\infty$ variational problems.
We let $\mathrm{Lip}(\overline{\Omega })$ denote the space of all Lipschitz continuous functions on $\overline{\Omega }$ and let $\mathrm{Lip}_0(\overline{\Omega })$ be the subspace of Lipschitz-functions vanishing on $\partial \Omega$. A norm on $\mathrm{Lip}(\overline{\Omega })$ is given by $\left\Vert u \right\Vert _{\mathrm{Lip}(\overline{\Omega })}\coloneq \max (\left\Vert u \right\Vert _\infty ,\mathrm{Lip}(u))$, where $\mathrm{Lip}(u)$ denotes the Lipschitz constant of $u\in \mathrm{Lip}(\overline{\Omega })$. An equivalent norm on $\mathrm{Lip}_0(\overline{\Omega })$ is given by