On the existence and H\"older regularity of solutions to some nonlinear Cauchy-Neumann problems

TL;DR

This paper establishes uniform H"older regularity for minimizers of certain parabolic functionals and proves existence and regularity of solutions to nonlinear weighted Cauchy-Neumann problems, relevant in combustion and fractional diffusion.

Contribution

It introduces new uniform H"older estimates for minimizers of a class of nonlinear parabolic functionals and applies these results to prove existence and regularity of solutions to weighted nonlinear Cauchy-Neumann problems.

Findings

Proved uniform parabolic H"older estimates for minimizers.

Established existence of weak solutions to weighted nonlinear Cauchy-Neumann problems.

Demonstrated H"older regularity of these solutions.

Abstract

We prove uniform parabolic H\"older estimates of De Giorgi-Nash-Moser type for sequences of minimizers of the functionals \[ \mathcal{E}_\varepsilon(W) = \int_0^\infty \frac{e^{- t/\varepsilon}}{\varepsilon} \bigg\{ \int_{\mathbb{R}_+^{N+1}} y^a \left( \varepsilon|\partial_t W|^2 + |\nabla W|^2 \right) dX + \int_{\mathbb{R}^N \times\{0\}} \Phi(w) dx \bigg\}dt, \qquad \varepsilon \in (0,1) \] where $a \in (-1,1)$ is a fixed parameter, $\mathbb{R}_+^{N+1}$ is the upper half-space and $dX = dxdy$. As a consequence, we deduce the existence and H\"older regularity of weak solutions to a class of weighted nonlinear Cauchy-Neumann problems arising in combustion theory and fractional diffusion.