Variational mean curvatures in abstract measure spaces
Variational mean curvatures were introduced in [\textit{U. Massari}, Arch. Ration. Mech. Anal. 55, 357--382 (1974; Zbl 0305.49047)] in the Euclidean setting. Later, in 1987, \textit{E. Barozzi}, et al. [Proc. Am. Math. Soc. 99, 313--316 (1987; Zbl 1358.49041)] as well as \textit{E. Barozzi} [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 5, No. 2, 149--159 (1994; Zbl 0809.49038)] have proved that any finite perimeter has a sumable variational mean curvature. The initial Massari's definition of variational mean curvature is: a function \(H\in L^1(\Omega)\), where \(\Omega\) is an open set of \(\mathbb{R}^n\), is a variational mean curvature of the set \(E\subset \Omega\), if \(E\) is a local minimizer of the functional \(\mathcal{F}(F)= \int_\Omega |D\mathbf{1}_F|+ \int_{\Omega\cap F}H(x)dx\), where \(\mathbf{1}_F\) is the characteristic function of \(F\), \(F\) is an arbitary set with finite perimeter in \(\Omega\), and the distributional gradient \(D\mathbf{1}_F\) is a vector Radon measure with finite total variation. In 1987, Barozzi, E., Gonzalez, E.H.A. and Tamanini, I. proved that any finite perimeter set has variational mean curvature in \(L^1(\Omega)\). Next, the result was strongly improved by Barozzi, E. in 1994, who has shown the existence of a variational mean curvature \(H_E \in L^1(\Omega)\), for any set \(E\subset \Omega\) statisfying some appropriate conditions and which is, in a certain sense, minimal with respect to any other variational curvature \(H\) on \(\Omega\). In the present paper, the authors extend the results of Barozzi to a generic measurable space \((\Omega ,\mu)\) equipped with a \(\sigma\)-additive function \(\mu :\mathcal{M}\to (0,\infty)\), with \(\mu(\Omega)<+\infty\). After recalling some preliminary results, the authors presents the definition of a \(P\)-perimeter as a function \(P: \mathcal{M}\to [0,\infty] \), with the properties: (P1)\(P(A\cup B)+P(A\cap B)\leq P(A)+P(B)\), for all \(A,B\in \mathcal{M}\); (P2) \(P(E)= P(\Omega\backslash E)\) for all \(E\in \mathcal{M}\); (P3) \(P(\emptyset) =0\) and then \(P(\Omega)=0\); (P4) \(P\) is lower semicontinuous with respect to \(L^1(\Omega,\mu)\) convergence; (P5) the families \(\{E\in \mathcal{M}:P(E)\leq c\}\) are compact in \(L^1(\Omega,\mu)\). A set with \(P(E)<+\infty\) is called a \(P\)-set. In its essence, the definition gives the list of abstract properties that any perimeter should have in order that the construction of the variational mean curvature works. The authors provide the generalization to an abstract setting of all the results obtained by Barozzi. A simple example shows that the assumption \(\mu(\Omega)< +\infty\) cannot be relaxed. In the final the authors provide a brief outlook of applications of the developed theory.
perimeters in Carnot-Caratheodory spaces
scientific article; zbMATH DE number 6670014
scientific article