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The summability of solutions to variational problems since Guido Stampacchia. (English) Zbl 1129.35371

Summary: Inequalities concerning the integral of \(|\nabla u|^2\) on the subsets where \(|u(x)|\) is greater than \(k\) can be used in order to prove regularity properties of the function \(u\). This method was introduced by Ennio De Giorgi e Guido Stampacchia for the study of the regularity of the solutions of Dirichlet problems.

MSC:

35J60 Nonlinear elliptic equations
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35D10 Regularity of generalized solutions of PDE (MSC2000)
35J20 Variational methods for second-order elliptic equations
49N60 Regularity of solutions in optimal control
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