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

We announce some new results for proving H"older continuity of weak solutions to quasilinear parabolic equations whose prototype takes the form $$u_t-div(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}\mathrm{\nabla }u)=0\text{or}{u}_t-div(\mathrm{\mid }{u}_{x_1}{\mathrm{\mid }}^{p_1-2}{u}_{x_1},\mathrm{\mid }{u}_{x_2}{\mathrm{\mid }}^{p_2-2}{u}_{x_2},\dots \mathrm{\mid }{u}_{x_N}{\mathrm{\mid }}^{p_N-2}{u}_{x_N})=0$$ and $1<\{p_1,p_2,\dots ,p_N\}<\mathrm{\infty }$. 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.