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

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.