Some Recent Results on the Mixed Local and Nonlocal Problems

This lecture presents recent advancements in the field of mixed local and nonlocal problems, delivered by Chao Zhang at the Workshop on "Degenerate and Singular PDEs" held at the Erwin Schrödinger International Institute for Mathematics and Physics (ESI) in February 2025. Explore the significant progress made in establishing regularity results for local and nonlocal functionals or equations with non-standard growth. The 33-minute talk begins with a brief review of known regularity results for nonlocal problems before focusing on the regularity of mixed local and nonlocal problems, covering higher Sobolev regularity, Hölder regularity, and weak Harnack type estimates of solutions.