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Minimization of functionals on classes of Lipschitz domains and $$BV$$ functions. (English) Zbl 0791.49029
Chadam, John M. (ed.) et al., Emerging applications in free boundary problems. Proceedings of the international colloquium ”Free boundary problems: theory and applications”, held in Montreal, Canada, June 13 - June 22, 1990. Harlow, Essex: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 280, 229-234 (1993).
Let $$D$$ be a Lipschitz domain of $$\mathbb{R}^ n$$ of class $${\mathcal C}^{0,1}$$ and fix $$\theta\in]0,{\pi\over 2}[$$. The class $${\mathcal C}^{0,1}_ \theta(D)$$ of Lipschitz subdomains $$\Omega$$ of $$D$$ which have a uniform cone property, with fixed parameter $$\theta$$ (angle and height) is considered, following D. Chenais [J. Math. Anal. Appl. 52, 189-219 (1975; Zbl 0317.49005)]. Also considered is the class $$\hbox{\mathit SBV}(\Omega)$$ of special real functions of bounded variation $$u$$ which have distributional derivatives $$Du$$ which are sums of $$(n-1)$$- dimensional measures with absolutely continuous measures with respect to Lebesgue measure [see e.g. L. Ambrosio, Arch. Ration. Mech. Anal. 111, No. 4, 291-322 (1990; Zbl 0711.49064)]. For $$\Omega\in{\mathcal C}^{0,1}_ \theta(D)$$ and $$u\in \hbox{\mathit SBV}(\Omega)$$, we define $F(\Omega,u):= \int_ \Omega f(x,u,\nabla u) dx+ \int_{S_ u} \psi(u^ +- u^ -) d{\mathcal H}_{n-1},$ where $$f$$ is a normal convex integrand and $$\psi$$ is a concave nondecreasing function with $$\psi(0)=0$$ and $$\psi'(0^ +)= +\infty$$. Here $$\nabla u$$ denotes the approximate differential of $$u$$, $$u^ +$$ and $$u^ -$$ are the approximate upper and lower limits of $$u$$, $$S_ u$$ is the set of jump points where $$u^ -< u^ +$$ and $${\mathcal H}_{n-1}$$ is the $$(n-1)$$-dimensional Hausdorff measure.
It is proved that, given a function $$h\in W^{1,\infty}(D)$$ with $$0\leq h\leq M$$ and $$\int_ D f(x,h,\nabla h)dx<+ \infty$$, then the relaxed functional $F(\Omega,u)+ \int_{\partial\Omega} \psi\bigl(|\gamma_ \Omega u- h|\bigr) d{\mathcal H}_{n-1}$ has a minimum $$(\Omega,u)$$, subject to the restriction $$0\leq u\leq M$$.
For the entire collection see [Zbl 0781.00010].

##### MSC:
 49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)