Ponce A.C., Elliptic PDEs, Measures and Capacities: From the Poisson Equation to Nonlinear Thomas-Fermi Problems

European Mathematical Society, 2016. — 465 p. — (EMS Tracts in Mathematics 23) — ISBN: 3037191406Partial differential equations (PDEs) and geometric measure theory (GMT) are branches of analysis whose connections are usually not emphasized in introductory graduate courses. Yet, one cannot dissociate the notions of mass or electric charge, naturally described in terms of measures, from the physical potential they generate. Having such a principle in mind, this book illustrates the beautiful interplay between tools from PD.Es and GMT in a simple and elegant way by investigating properties like existence and regularity of solutions of linear and nonlinear elliptic PDEs. Inspired by a variety of sources, from the pioneer balayage scheme of Poincaré to more recent results related to the Thomas–Fermi and the Chern–Simons models, the problems covered in this book follow an original presentation, intended to emphasize the main ideas in the proofs. Classical techniques like regularity theory, maximum principles and the method of sub- and supersolutions are adapted to the setting where merely integrability or density assumptions on the data are available. The distinguished role played by capacities and precise representatives is also explainedThe Laplacian
Poisson equation
Integrable versus measure data
Variational approach
Linear regularity theory
Comparison tools
Balayage
Precise representative
Maximal inequalities
Sobolev and Hausdorff capacities
Removable singularities
Obstacle problems
Families of solutions
Strong approximation of measures
Traces of Sobolev functions
Trace inequality
Critical embedding
Quasicontinuity
Nonlinear problems with diffuse measures
Extremal solutions
Absorption problems
The Schrödinger operator
Appendices
Sobolev capacity
Hausdorff measure
Solutions and hints to the exercises