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\(S^ k\)-valued maps minimizing the \(L^ p\) norm of the gradient with free discontinuities. (English) Zbl 0753.49018
We prove existence and partial regularity of a solution of a free discontinuity problem in the vector case with a constraint. More precisely for every \(p>1\) and \(q\geq 1\) we prove the existence of a minimizing pair for the functional defined for every closed set \(K\subset{\mathbf R}^ n\) and for every \(u\in C^ 1(\Omega\setminus K; S^ k)\) by \[ \int_{\Omega\setminus K} |\nabla u|^ p dy+\int_{\Omega\setminus K}| u-g|^ q dy+{\mathcal H}^{n- 1}(K\cap\Omega), \] where \(n\geq 2\), \(\Omega\subset{\mathbf R}^ n\) is a bounded open set, \(k\in N\) and \(S^ k=\{z\in{\mathbf R}^{k+1}; | z|=1\}\), \(g\in L^ \infty(\Omega; {\mathbf R}^{k+1})\) and \({\mathcal H}^{n-1}\) is the \((n-1)\)-dimensional Hausdorff measure. Moreover for an optimal closed set \(K\) an estimate of the lower \(n\)-dimensional density is given. Analogous results hold under Dirichlet type conditions. The interest in considering the constraint \(u(x)\in S^ k\) is connected to the recent studies of minima of non-convex functionals and on possible applications of these studies to the static theory of liquid crystals.
Reviewer: M.Carriero

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
26A45 Functions of bounded variation, generalizations
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