On the isoperimetric nature of a rearrangement inequality and its consequences for some variational problems (Q1082606)

As is well known, for functions \(u\in W_ 0^{1,2}(\Omega)\) rearrangement by symmetrization to a function \(u^*\) does not increase the Dirichlet integral: \(D(u^*)\leq D(u)\). This is true for Steiner symmetrization, circular symmetrization, and Schwarz symmetrization, too. The paper is concerned with the following question: Does equality \(D(u^*)=D(u)\) imply that \(u=u^*?\) (The question was originally raised by Polya and Szegö, and a positive answer was conjectured by E. Lieb.) Simple examples (constructed by means of continuous and piecewise linear functions) show that the answer is generally negative in \(W_ 0^{1,2}(\Omega)\). It is, however, positive for analytic functions which form a dense set in \(W_ 0^{1,2}(\Omega)\). Applications to non- quadratic extremal problems are presented.