Gradient bounds for p-harmonic systems with vanishing neumann data in a convex domain

TL;DR

This paper establishes gradient bounds for solutions to p-harmonic systems with vanishing Neumann boundary conditions in convex domains, using subsolution techniques instead of level set methods.

Contribution

It introduces a new approach employing subsolution arguments to obtain gradient bounds, replacing previous level set based methods in similar elliptic problems.

Findings

Gradient of solutions is bounded in the domain segment.

Constants depend only on dimension, p, N, and domain measure.

Method simplifies previous approaches based on level sets.

Abstract

Let $ \ti \Om $ be a bounded convex domain in Euclidean $ n $ space, $ \hat x \in \ar \ti \Om, $ and $ r > 0. $ Let $ \ti u = (\ti u^1, \ti u^2, \dots, \ti u^N) $ be a weak solution to \[\nabla \cdot \left (|\nabla \ti u |^{p-2} \nabla \ti u \right) = 0 \mbox{in} \ti \Om \cap B (\hat x, 4 r) \mbox{with} |\nabla \ti u|^{p-2} \, \ti u_\nu = 0 \mbox{on} \ar \ti \Om \cap B (\hat x, 4 r). \] We show that sub solution type arguments for certain uniformly elliptic systems can be used to deduce that $ | \nabla \ti u | $ is bounded in $ \ti \Om \cap B (\hat x, r)$ with constants depending only on $ n, p, N. $ and $ \frac{r^n}{| \ti \Om \cap B (\hat x, r) |}.$ Our argument replaces an argument based on level sets in recent important work of [CM], [CM1], [GS], [M], [M1], involving similar problems.