Partial regularity for optimal design problems involving both bulk and surface energies
The paper, which is a refinement and an improvement of a previous work of the first author [Calc. Var. Partial Differ. Equ. 1, No. 2, 149-168 (1993; Zbl 0794.49038)], deals with the regularity of variational problems involving both a volume and a surface energy. Specifically, one considers pairs \((u,A)\) with \(u\in H^1(\Omega)\) and \(A\subset\Omega\) with finite perimeter in \(\Omega\) minimizing the energy (possibly with volume constraints) \[ (u,A)\mapsto \int_\Omega [F(x,u,\nabla u)+\chi_AG(x,u,\nabla u)] dx+ \int_{\partial^* A}\psi(x,\nu_A) d{\mathcal H}^{n-1} \] where \(\chi_A:\Omega\to\{0,1\}\) is the characteristic function of \(A\), \(\partial^* A\) is the essential boundary of \(A\) and \(\nu_A\) is its inner normal. Under suitable smoothness and ellipticity assumptions on the data and on the energy densities the authors prove that \(\partial^* A\) is closed and \(C^{1,\alpha}\) out of a \({\mathcal H}^{n-1}\)-negligible closed set \(\Sigma\). Related results have been obtained by \textit{L. Ambrosio, N. Fusco} and \textit{D. Pallara} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 24, No. 1, 39-62 (1997; Zbl 0896.49024)] in the more general framework of free discontinuity problems, in which the surface energy density is not necessarily supported on a boundary.
scientific article; zbMATH DE number 1340795