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Eigenvalue Inequalities for Fully Nonlinear Elliptic Equations via the Alexandroff-Bakelman-Pucci Method

Dimitrios Gazoulis

TL;DR

This work develops eigenvalue inequalities for elliptic operators by exploiting the Alexandroff-Bakelman-Pucci (ABP) method and extends these ideas to fully nonlinear equations. Starting from the ABP estimate for subsolutions, the authors derive a bound on the normal derivative at the boundary and then establish lower bounds for the $L^n$ (and $L^p$ to infinity) norms of the Laplacian in terms of boundary data, with sharp cases linking equality to symmetry. These ABP-driven bounds yield eigenvalue inequalities for the Laplacian under Dirichlet and Robin conditions, and they are extended to the Monge-Ampère operator, providing analogous eigenvalue lower bounds. The paper then broadens the framework to fully nonlinear elliptic equations via Pucci operators, obtaining an $L^{\infty}$ gradient bound and a $C^3$ regularity estimate, and establishing corresponding Dirichlet and Robin eigenvalue inequalities in the viscosity-solution setting. Collectively, the results demonstrate that ABP-type estimates are powerful tools for controlling nonlinear elliptic operators and yield sharp, symmetry-attained bounds in several classical eigenvalue problems, with clear implications for fully nonlinear theory and regularity.

Abstract

In this work we establish eigenvalue inequalities for elliptic differential operators either for Dirichlet or for Robin eigenvalue problems, by using the technique introduced by Alexandroff, Bakelman and Pucci. These inequalities can be extended for fully nonlinear elliptic equations, such as for the Monge-Ampère equation and for Pucci's equations. As an application we establish a lower bound for the $ L^p -$norm of the Laplacian and this bound is sharp, in the sense that, when equality is achieved then a symmetry property is obtained. In addition, we obtain an $ L^{\infty} $ bound for the gradient of solutions to fully nonlinear elliptic equations and as a result, a $ C^3 $ estimate.

Eigenvalue Inequalities for Fully Nonlinear Elliptic Equations via the Alexandroff-Bakelman-Pucci Method

TL;DR

This work develops eigenvalue inequalities for elliptic operators by exploiting the Alexandroff-Bakelman-Pucci (ABP) method and extends these ideas to fully nonlinear equations. Starting from the ABP estimate for subsolutions, the authors derive a bound on the normal derivative at the boundary and then establish lower bounds for the (and to infinity) norms of the Laplacian in terms of boundary data, with sharp cases linking equality to symmetry. These ABP-driven bounds yield eigenvalue inequalities for the Laplacian under Dirichlet and Robin conditions, and they are extended to the Monge-Ampère operator, providing analogous eigenvalue lower bounds. The paper then broadens the framework to fully nonlinear elliptic equations via Pucci operators, obtaining an gradient bound and a regularity estimate, and establishing corresponding Dirichlet and Robin eigenvalue inequalities in the viscosity-solution setting. Collectively, the results demonstrate that ABP-type estimates are powerful tools for controlling nonlinear elliptic operators and yield sharp, symmetry-attained bounds in several classical eigenvalue problems, with clear implications for fully nonlinear theory and regularity.

Abstract

In this work we establish eigenvalue inequalities for elliptic differential operators either for Dirichlet or for Robin eigenvalue problems, by using the technique introduced by Alexandroff, Bakelman and Pucci. These inequalities can be extended for fully nonlinear elliptic equations, such as for the Monge-Ampère equation and for Pucci's equations. As an application we establish a lower bound for the norm of the Laplacian and this bound is sharp, in the sense that, when equality is achieved then a symmetry property is obtained. In addition, we obtain an bound for the gradient of solutions to fully nonlinear elliptic equations and as a result, a estimate.
Paper Structure (9 sections, 21 theorems, 116 equations)

This paper contains 9 sections, 21 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. A Bound for the normal derivative of subsolutions to Linear Elliptic equations
  3. A Lower bound for the norm of the Laplacian
  4. Applications: Lower bounds for eigenvalue problems
  5. A Lower bound for eigenvalue problems of the Laplacian
  6. Eigenvalue inequalities for the Monge-Ampère equation
  7. The case of Fully Nonlinear Elliptic Equations
  8. An $L^{\infty}$ bound for the gradient
  9. Applications to Eigenvalue problems

Key Result

Lemma 2.1

Let $u : \Omega \rightarrow \mathbb{R}$ and $u \in C^1( \overline{\Omega} )$ such that Then either or

Theorems & Definitions (47)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Corollary 2.6
  • ...and 37 more