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

MSC:

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

Zbl 0682.49002
Full Text:

References:

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