Local continuity of weak solutions to the Stefan problem involving the singular $p$-Laplacian

Naian Liao

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arXiv · 2103.00412

Local continuity of weak solutions to the Stefan problem involving the singular $p$-Laplacian

Naian Liao

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Abstract

We establish the local continuity of locally bounded weak solutions (temperatures) to the doubly singular parabolic equation modeling the phase transition of a material: \[ \partial_t \beta(u)-\Delta_p u\ni 0\quad\text{ for }\tfrac{2N}{N+1}<p<2, \] where $\beta$ is a maximal monotone graph with a jump at zero and $\Delta_p$ is the $p$-Laplacian. Moreover, a logarithmic type modulus of continuity is quantified, which has been conjectured to be optimal.

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Naian Liao. 2021-02-28. Local continuity of weak solutions to the Stefan problem involving the singular $p$-Laplacian. https://arxiv.org/abs/2103.00412

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