Existence theorem for a Dirichlet problem with free discontinuity set. (English) Zbl 0713.49003

The following result is proved for a Dirichlet type problem with free discontinuities.
Let \(n\in N\), \(n\geq 2\), let \(\Omega \subset {\mathbb{R}}^ n\) be a bounded domain with \({\mathcal H}^{n-1}(\partial \Omega)<+\infty\); assume that a closed set M exists such that \({\mathcal H}^{n-1}(M)=0\) and \(\partial \Omega \setminus M\) is a \(C^ 1\) surface; let \(w\in C^ 1(\partial \Omega \setminus M)\cap L^{\infty}(\partial \Omega \setminus M)\). Then there exists at least one pair \((K_ 0,u_ 0)\) minimizing the functional \({\mathcal G}\) defined for every closed set \(K\subset {\bar \Omega}\) and for every \(u\in C^ 1(\Omega \setminus K)\cap C^ 0({\bar \Omega}\setminus (M\cup K))\) with \(u=w\) on \(\partial \Omega \setminus (M\cup K)\) by \[ {\mathfrak G}\setminus (K,u)=\int_{\Omega \setminus K}| \nabla u|^ 2dy+{\mathcal H}^{n-1}(K). \] Moreover \(K_ 0\) is (\({\mathcal H}^{n-1},n-1)\) rectifiable, and there exists a unique essential minimizing pair \((K',u')\) (i.e. \(K'\) is the smallest closed set contained in \(K_ 0\) for which \(u_ 0\) has an extension \(u'\in C^ 1(\Omega \setminus K')\cap C^ 0({\bar \Omega}\setminus (M\cup K'))\) with \(u'=w\) on \(\partial \Omega \setminus (M\cup K'))\) such that \[ \liminf_{\rho \to 0}\rho^{1-n}{\mathcal H}^{n-1}(K'\cap \bar B_{\rho}(x))>0 \] for every \(x\in K'\setminus M.\)
Notice that the Dirichlet condition is required on \(\partial \Omega\) but for that part of the boundary that might be touched by the singular set K; however there is a price to be paid for this, because \({\mathcal H}^{n-1}(K)\) penalizes both \(K\cap \Omega\) (where \(u\) is discontinuous) and the part of the boundary \(K\cap \partial \Omega\) where \(u\) may not assume the given value.
A Neumann type problem with free discontinuities has been studied by the authors and E. De Giorgi in Arch. Ration. Mech. Anal. 108, No. 3, 195-218 (1989; Zbl 0682.49002).
Reviewer: M.Carriero


49J10 Existence theories for free problems in two or more independent variables


Zbl 0682.49002
Full Text: DOI


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