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Stable Solutions to Reaction-Diffusion Elliptic ProblemsThe talk will concern stable solutions to reaction-diffusion elliptic PDEs. We will begin with regularity questions, first addressing the classical Laplacian. In joint work with Figalli, Ros-Oton, and Serra, we proved that stable solutions are smooth up to the optimal dimension 9, thereby solving an open problem posed by Brezis in the mid-1990s. I will describe this result and also discuss related progress and open problems for the fractional Laplacian -arising naturally in boundary reaction problems-, the p-Laplacian, and minimal surfaces.We will then turn to existence questions, starting with the Casten-Holland and Matano theorem for interior reactions, which states that no nonconstant stable solution exists in convex domains under zero Neumann boundary conditions. I will present a recent result with Cónsul and Kurzke establishing that the analogous statement fails for boundary reactions. This requires the development of a new Ginzburg-Landau theory for real-valued functions and the analysis of the half-Laplacian on the real line, for which I will present new results and open problems.Uploaded 06-10-26 Learn more about your ad choices. Visit megaphone.fm/adchoices
