Hölder regularity for quasilinear parabolic equations with anisotropic $p$-Laplace nonlinearity -- Announcement

Abstract

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

The announcement is withdrawn as the proof holds under additional assumptions