PDE Seminar Abstracts

Comparison results for a nonlocal singular elliptic problem

Barbara Brandolini
University of Palermo, Italy
Mon 28th Nov 2022, 2-3pm, Carslaw Room 829 (AGR)

Abstract

We prove comparison results for the solution \(u\) to the following nonlocal singular problem \begin {align*} \left (-\Delta \right )^s u &=\dfrac {f}{u^\gamma } &&\text {in }\Omega \\ u&>0 && \text {in } \Omega \\ u&=0 && \text {in }\mathbb R^n\setminus \Omega . \end {align*}

Here \(\Omega \) is a bounded domain in \(\mathbb R^n\), \(0<s<1\), \(\gamma >0\) and \(f \in L^1(\Omega )\), \(f \ge 0\). Some interesting consequences are \(L^p\) regularity results and energy estimates for \(u\) depending on the value of \(\gamma \).

This talk is based on joint work with I. De Bonis, V. Ferone and B. Volzone.