PDEA Seminar: Coarse-grained ellipticity and De Giorgi-Nash-Moser theory

Title: Coarse-grained ellipticity and De Giorgi-Nash-Moser theory

Abstract:

In this talk I will present some recent progress using suitably modified coarse-grained ellipticity to prove local boundedness and a Harnack inequality for nonnegative weak solutions of linear divergence form equations. Coarse-grained ellipticity is a scale-dependent condition defined for symmetric coefficient fields a satisfying only a, a^{-1} in L^1, in terms of effective diffusion “matrices" on triadic cubes of all sizes. This recovers Trudinger's classical result under the integrability condition a in L^p, a^{-1} \in L^q with 1/p+1/q<2/d as a special case, and it applies to new classes of degenerate coefficient fields for which a, a^{-1} \notin L^{1+\delta} for all \delta>0, including examples generated by singular fractal measures and Gaussian multiplicative chaos.

This is a seminar in the PDEs and Applications seminar series.