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On the Dirichlet problem for variational integrals in BV. (English) Zbl 1260.49079
In this outstanding paper, the authors consider multidimensional variational integrals. The authors are interested in the minimization problem for \(F\) in Dirichlet classes, and in particular in existence, uniqueness and regularity of (generalized) minimizers. Firstly the authors study the model integrals. These integrals can be understood as a limit case of the \(p\)-energies where \(p \geq 1\) is another parameter. The boundary integral presented need not vanish for generalized minimizers. The authors present this integration as a penalization term (the non-attainment of the boundary values is not generally ruled out but instead be penalized by an increase of energy). The uniqueness of generalized minimizers is not immediate, and there are two potential sources of non-uniqueness, namely the possible occurrence of singular parts of the derivative and the possible non-attainment of the boundary values. The authors prove that \(u \in W^{1,1} (\Omega, R^N)\) holds for every generalized minimizer \(u\) and not just that there exists a suitable one as stated in the presented assertion.
The paper provides complete investigation to the theory of the Dirichlet problem for multidimensional variational integrals with linear growth which is formulated in a generalized way in the space of functions of bounded variation. It covers existence, uniqueness and convergence results of the Dirichlet problem for multidimensional variational integrals. The authors results (a new type) extend classical results from the scalar case \(N=1\).

49Q20 Variational problems in a geometric measure-theoretic setting
49N60 Regularity of solutions in optimal control
49J20 Existence theories for optimal control problems involving partial differential equations
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