【Analysis and Partial Differential Equations Seminar】Regularity Results for Double Phase Problems on Metric Measure Spaces

Speaker: Dr. Cintia Pacchiano(Calgary University)

Title: Regularity Results for Double Phase Problems on Metric Measure Spaces

Abstract:

In this talk, we present boundedness, H¨older continuity and Harnack inequality results for local quasiminima to elliptic double phase problems of p-Laplace type in the general context of metric measure spaces. The proofs follow a variational approach, based on the De Giorgi method and a careful phase analysis. The main novelty is the use of an intrinsic approach, based on a double phase Sobolev-Poincar´e inequality.

Furthermore, we present boundary regularity results for quasiminimizers of double-phase functionals. We again use a variational approach to give a pointwise estimate near a boundary point, as well as a sufficient condition for H¨older continuity and a Wiener type regularity condition for continuity up to the boundary. This in an on-going project, together with Prof. Dr. Antonella Nastasi from University of Palermo.

During the past two decades, a theory of Sobolev functions and first degree calculus has been developed in this abstract setting. A central motivation for developing such a theory has been the desire to unify the assumptions and methods employed in various specific spaces, such as weighted Euclidean spaces, Riemannian manifolds, Heisenberg groups, graphs, etc.

Analysis on metric spaces is nowadays an active and independent field, bringing together researchers from different parts of the mathematical spectrum. It has applications to disciplines as diverse as geometric group theory, nonlinear PDEs, and even theoretical computer science. This can offer us a better understanding of the phenomena and also lead to new results, even in the classical Euclidean case.