FALL 2020 Nonlinear Analysis Seminar Series

Takanobu Hara, Postdoctoral Scholar, Hokkaido University　

Title: Trace inequalities of Sobolev type and nonlinear Dirichlet problems

Abstract:

We discuss solvability of Dirichlet problems of the type $- \Delta_{p} u = \mu$ in $\Omega$, $u = 0$ on $\partial \Omega$, where $\Omega$ is a bounded domain, $\Delta_{p}$ is the p-Laplacian, and $\mu$ is a nonnegative locally finite Radon measure on $\Omega$. We do not assume the finiteness of $\mu(\Omega)$ here. We revisit this problem with a potential theoretic viewpoint and give sufficient conditions for the existence of solutions. Our main tools are $L^{p}(dx)-L^{q}(d \mu)$ trace inequalities and capacitary conditions. Also, we derive the trace inequalities using solutions conversely.

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