Notes on the Dirichlet problem of a class of second order elliptic partial differential equations on a Riemannian manifold

TL;DR

This paper investigates the Dirichlet problem for a class of second order elliptic PDEs on Riemannian manifolds, providing a unified approach that covers classical examples like the p-Laplacian and minimal surface equations.

Contribution

It introduces a novel method using a-priori C^1 estimates without relying on the direct calculus of variations, applicable to degenerate elliptic equations on manifolds.

Findings

Unified approach to Dirichlet problems on Riemannian manifolds

A-priori C^1 estimates for solutions of Euler-Lagrange equations

Applicability to classical equations like p-Laplacian and minimal surface

Abstract

In these notes we study the Dirichlet problem for critical points of a convex functional of the form \[ F(u)=\int_{\Omega}\phi\left( \left\vert \nabla u\right\vert \right) , \] where $\Omega$ is a bounded domain of a complete Riemannian manifold $\mathcal{M}.$ We also study the asymptotic Dirichlet problem when $\Omega=\mathcal{M}$ is a Cartan-Hadamard manifold. Our aim is to present a unified approach to this problem which comprises the classical examples of the $p-$Laplacian ($\phi(s)=s^{p}$, $p>1)$ and the minimal surface equation ($\phi(s)=\sqrt{1+s^{2}}$). Our approach does not use the direct method of the Calculus of Variations which seems to be common in the case of the $p-$Laplacian. Instead, we use the classical method of a-priori $C^{1}$ estimates of smooth solutions of the Euler-Lagrange equation. These estimates are obtained by a coordinate free calculus. Degenerate elliptic equations like the $p-$Laplacian are dealt with by an approximation argument. These notes address mainly researchers and graduate students interested in elliptic partial differential equations on Riemannian manifolds and may serve as a material for corresponding courses and seminars.