Gary M. Lieberman | Search arXiv

arXiv2018

Hölder continuity of a bounded weak solution of generalized parabolic $p-$Laplacian equations

Here we generalize quasilinear parabolic $p-$Laplacian type equations to obtain the prototype equation as \[ u_t - \text{div} (g(|Du|)/ |Du| \cdot Du) = 0, \] where a nonnegative, increasing, and continuous function $g$ trapped in between two power functions $|Du|^{g_0 -1}$ and $|Du|^{g_1 -1}$ with $1<g_0 \leq g_1 < \infty$. Through this generalization in the setting from Orlicz spaces, we provide a uniform proof with a single geometric setting that a bounded weak solution is locally Hölder continuous considering $1 < g_0 \leq g_1 \leq 2$ and $2 \leq g_0 \leq g_1 < \infty$ separately. By using geometric characters, our proof does not rely on any of alternatives which is based on the size of solutions.

math.AP↗

arXiv2018

Gradient estimates for oblique derivative problems via the maximum principle

We prove gradient estimates for solutions of the oblique derivative problem for a large class of elliptic and parabolic quasilinear PDEs. In particular, we expand on previous work of the author using a maximum principle argument. In addition, we prove estimates for the parabolic problem of evolution via mean curvature with nonzero Neumann data, which seem to be new.

math.AP↗