Regularity for minimal boundaries in \(\mathbb{R}^n\) with mean curvature in \(L^n\)
The author studies the regularity of the boundary of a Caccioppoli set \(E_0\) minimizing the functional \[ F_H(E) = P(E,\Omega) + \int_{E\cap\Omega}H, \] where \(\Omega\subset\mathbb R^n\) is a given open set and \(H\) is a given Lebesgue measurable function. Such a minimizer \(E_0\) is called a minimal set with mean curvature \(H\) in \(\Omega\) because if \(H\) and \(\partial E_0\) are sufficiently regular near a point \(x_0\in\partial E_0\cap\Omega,\) then the usual mean curvature of \(\partial E_0\) equals \(-H(x)/(n-1)\) near \(x_0.\) It is known that the Lebesgue class of \(H\) is decisive for the regularity of \(\partial E_0.\) Massari has shown that if \(H\in L^p,\) \(p>n,\) then, except for a set of dimension not greater than \(n-8,\) \(\partial E_0\) is of class \(C^{1,\alpha},\) \(\alpha = (p-n)/4p\) [\textit{U. Massari}, Rend. Sem. Mat. Univ. Padova 53(1975), 37-52 (1976; Zbl 0358.49019)]. By contrast, Barozzi, Gonzalez, and Tamanini showed that if \(H\in L^p\) with \(1\leq p <n,\) then \(\partial E_0\) may have a large singular set. Indeed, the same authors showed that any Caccioppoli set \(E\) is a minimal set with mean curvature \(H\) for an appropriate \(H\in L^1\). The author's work applies to the critical case \(H\in L^n.\) By applying Reifenberg's topological disc theorem [\textit{E. R. Reifenberg}, Acta Math. 104, 1-92 (1960; Zbl 0099.08503)], it is shown that if \(2\leq n\leq 7\) and \(H\in L^n,\) then, for each \(0<\alpha<1\) and each \(x_0\in \partial E_0,\) there exists a neighborhood of \(x_0\) in \(\partial E_0\) in which \(\partial E_0\) can be parametrized by a map of class \(C^{0,\alpha}\) with a \(C^{0,\alpha}\) inverse. To verify that the hypotheses of Reifenberg's topological disc theorem are satisfied, the author develops a number of estimates using interesting geometric arguments.
scientific article; zbMATH DE number 1215547
scientific article