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

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.