An interpolation approach to $L^{\infty}$ a priori estimates for elliptic problems with nonlinearity on the boundary

TL;DR

This paper introduces a new interpolation method to derive explicit $L^ abla$ a priori bounds for solutions to elliptic boundary problems with nonlinear boundary conditions, enhancing understanding of solution behavior.

Contribution

It presents a novel combination of Moser estimates, elliptic regularity, and Gagliardo--Nirenberg inequalities to establish explicit bounds for subcritical elliptic problems with boundary nonlinearity.

Findings

Established explicit $L^ abla$ bounds for weak solutions.

Applied the method to problems satisfying Ambrosetti-Rabinowitz condition.

Demonstrated the effectiveness of the interpolation approach.

Abstract

We establish an explicit $L^\infty(\Om)$ a priori estimate for weak solutions to subcritical elliptic problems with nonlinearity on the boundary, in terms of the powers of their $H^1(\Om)$ norms. To prove our result, we combine in a novel way Moser type estimates together with elliptic regularity and Gagliardo--Nirenberg interpolation inequality. We illustrate our result with an application to subcritical problems satisfying Ambrosetti-Rabinowitz condition.