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Converse symmetry and intermediate energy values in rearrangement optimization problems. (English) Zbl 1371.49004

In this paper the authors discuss three rearrangement optimization problems where the energy functional is connected with the Dirichlet or Robin boundary value problems. First, they consider a simple model of Dirichlet type, derive a symmetry result, and prove an intermediate energy theorem. It is shown for this model that if the optimal domain (or its complement) is a ball centered at the origin, then the original domain must be a ball. As for the intermediate energy theorem, it is shown that if \(\alpha,\beta\) denote the optimal values of corresponding minimization and maximization problems, respectively, then every \(\gamma\) in \((\alpha,\beta)\) is achieved by solving a max-min problem. Second, the authors investigate a similar symmetry problem for the Dirichlet problems where the energy functional is nonlinear. Finally, the existence and uniqueness of rearrangement minimization problems associated with the Robin problems is proved. The article is concluded with a result about asymptotic convergence. In this result, it is shown that as \(\beta\) tends to infinity, the corresponding minimizer of the Robin problem converges to the minimizer of the Dirichlet problem in \(L^2(D)\).

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
49J40 Variational inequalities
49N99 Miscellaneous topics in calculus of variations and optimal control
35J20 Variational methods for second-order elliptic equations
49J35 Existence of solutions for minimax problems
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