Selected problems on elliptic equations involving measures

Details Selected problems on elliptic equations involving measures Selected problems on elliptic equations involving measures

2012-04-03

arxiv.org/abs/1204.0668v1

Mathematics

Analysis of PDEs

143 pages; manuscript submitted to the concours annuel 2012 of the Acad\'emie royale de Belgique

Scientific paper

This monograph concerns linear and nonlinear Dirichlet problems involving L^1 data and more generally measure data, based on Stampacchia's definition of weak solution. We explain some of the main tools: linear regularity theory, maximum principles, Kato's inequality, method of sub and supersolutions, and the Perron method. The nonlinear Dirichlet problem need not have a solution for every finite measure. We give characterizations of measures for which the problem has a solution with polynomial and exponential nonlinearities in connection with capacities and Hausdorff measures. Finally, we give a different approach to study the concept of reduced measure introduced by Brezis, Marcus and Ponce.

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