Everywhere differentiability of viscosity solutions to a class of Aronsson's equations

  • Juhana Siljander

    Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland
  • Changyou Wang

    Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907, USA
  • Yuan Zhou

    Department of Mathematics, Beijing University of Aeronautics and Astronautics, Beijing 100191, PR China

Abstract

We show the everywhere differentiability of viscosity solutions to a class of Aronsson equations in for , where the coefficient matrices are assumed to be uniformly elliptic and . Our result extends an earlier important theorem by Evans and Smart [18] who have studied the case which correspond to the -Laplace equation. We also show that every point is a Lebesgue point for the gradient.

In the process of proving the results we improve some of the gradient estimates obtained for the infinity harmonic functions. The lack of suitable gradient estimates has been a major obstacle for solving the problem in this setting, and we aim to take a step towards better understanding of this problem, too.

A key tool in our approach is to study the problem in a suitable intrinsic geometry induced by the coefficient matrix . Heuristically, this corresponds to considering the question on a Riemannian manifold whose the metric is given by the matrix .

Cite this article

Juhana Siljander, Changyou Wang, Yuan Zhou, Everywhere differentiability of viscosity solutions to a class of Aronsson's equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 1, pp. 119–138

DOI 10.1016/J.ANIHPC.2015.10.003