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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