Nonlinear potential theory for degenerate elliptic PDE

  • Lecturer: Lukas Koch
  • Date: Thursdays, 09.00-10.30
  • Room: MPI MiS A3 01
  • Keywords: nonlinear elliptic PDE, potential theory, Wolff potentials
  • Prerequisites: Knowledge of basic properties of Sobolev spaces is assumed. Knowledge of basic elliptic regularity theory will be useful, but is not essential.

Abstract

Nonlinear potential theory studies properties of solutions of nonlinear elliptic equations in analogy to the study of solutions to Laplace equation in harmonic analysis. In recent years, this perspective has gained increased interest for questions regarding the regularity theory of such equations and has been used to prove a number of surprising and sharp regularity statements. We will first explore important notions of nonlinear potential theory such as capacity, the comparison principle and polar sets. We will then use these tools to prove a-priori estimates, culminating in a nonlinear-Stein theorem - solutions to a general class of nonlinear elliptic PDEs have continuous gradient if the data is in L(n,1).

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Regular lectures: Winter semester 2022/2023

27.11.2022, 02:30