  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.
