H\"older regularity for quasilinear parabolic equations with anisotropic $p$-Laplace nonlinearity -- Announcement

TL;DR

This paper introduces a novel technique to prove H"older continuity of solutions to anisotropic quasilinear parabolic equations, avoiding traditional intrinsic scaling methods and employing a new linearisation approach.

Contribution

It presents a new, elementary linearisation method for establishing regularity of solutions, independent of existing intrinsic scaling techniques.

Findings

Proves H"older continuity for anisotropic p-Laplace type equations.

Develops a new linearisation technique for nonlinear PDEs.

Applicable to equations with variable exponents p_i.

Abstract

We announce some new results for proving H\"older continuity of weak solutions to quasilinear parabolic equations whose prototype takes the form $$u_t - div (|\nabla u|^{p-2}\nabla u)= 0 \qquad \text{or} \qquad u_t - div (|u_{x_1}|^{p_1-2}u_{x_1},|u_{x_2}|^{p_2-2}u_{x_2},\ldots |u_{x_N}|^{p_N-2}u_{x_N})=0$$ and $1<\{p_1,p_2,\ldots,p_N\}<\infty$. We develop a new technique which is independent of the "method of intrinsic scaling" developed by E.DiBenedetto in the degenerate case ($p\geq 2$) and E.DiBenedetto and Y.Z.Chen in the singular case ($p\leq 2$) and instead uses a new and elementary linearisation procedure to handle the nonlinearity.