Selected problems on elliptic equations involving measures

TL;DR

This monograph explores linear and nonlinear elliptic PDEs with measure data, providing characterizations of measures for solvability and introducing a novel approach to reduced measures, based on capacity and Hausdorff measure concepts.

Contribution

It offers new characterizations of measures allowing solutions to elliptic equations with nonlinearities and presents a different perspective on reduced measures not included in the original book.

Findings

Characterization of measures for polynomial nonlinearities

Characterization of measures for exponential nonlinearities

Introduction of a new approach to reduced measures

Abstract

This monograph is the core of my book "Elliptic PDEs, Measures and Capacities: From the Poisson equation to Nonlinear Thomas-Fermi Problems" which has received the 2014 EMS Monograph Award and is available in the series EMS Tracts in Mathematics published by the European Mathematical Society. Many chapters have been thoroughly rewritten during the book preparation. The manuscript here has kept the original presentation and concerns linear and nonlinear Dirichlet problems involving $L^1$ data and more generally measure data, based on Stampacchia's definition of weak solution. I explain some of the main tools: linear regularity theory, maximum principles, Kato's inequality, method of sub and supersolutions, and the Perron method. The semilinear Dirichlet problem need not have a solution for every finite measure. I give characterizations of measures for which the problem has a solution with polynomial and exponential nonlinearities in connection with capacities and Hausdorff measures. Finally, the reader will find a different approach to the concept of reduced measure introduced in collaboration with H. Brezis and M. Marcus, which has not been retained in my EMS book due to personal time constraints.