Workshop on Pdes 2022 - Ponce

A new type of connected component created by singular potentials

Prof. Augusto Ponce
Université Catholique de Louvain

Date: 16th February 11:00 am - 11:45 am (UTC -3) Link: PPGM/UFSCar Youtube channel

Abstract: The Schrödinger operator −∆ + V in Rhas been extensively studied for potentials V in L e Lwith any exponent p > N/2. In a joint work with L. Orsina, we have investigated what happens to potentials V:Ω → [0, +∞] that are merely Borel-measurable in an open subset Ω⊂Rand discovered that, correctly interpreted, the strong maximum principle is still available no matter how singular the potential is. The heart of the matter consists in splitting Ω as a combination of Sobolev-connected components that exclude the set of points where the Green's function is not available. Surprisingly, such a decomposition propagates to any function in W1,2(Ω)∩L1(Ω;V dx).