Global boundedness of the gradient for a class of nonlinear elliptic systems

TL;DR

This paper proves that solutions to certain nonlinear elliptic systems have bounded gradients up to the boundary, even with minimal regularity and complex nonlinearities, including in convex domains.

Contribution

It establishes boundary gradient boundedness for elliptic systems with Uhlenbeck type structures under minimal regularity assumptions, extending previous results to more general nonlinearities and domain shapes.

Findings

Gradient boundedness up to the boundary is achieved.

Results apply to nonlinearities of non-polynomial type.

Includes arbitrary bounded convex domains.

Abstract

Gradient boundedness up to the boundary for solutions to Dirichlet and Neumann problems for elliptic systems with Uhlenbeck type structure is established. Nonlinearities of possibly non-polynomial type are allowed, and minimal regularity on the data and on the boundary of the domain is assumed. The case of arbitrary bounded convex domains is also included.