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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].

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