The curvature of a set with finite area
It is shown that, given any measurable set \(E\subset \mathbb{R}^ n\) of finite perimeter \(P(E)\), there always exists a function \(H\in L^ 1(\mathbb{R}^ n)\) such that \(E\) minimizes the functional \[ {\mathcal F}_ H(X)= P(X)+ \int_ X H(x)dx \] among all measurable sets \(X\subset \mathbb{R}^ n\). One should observe that in case \(\partial E\) and \(H\) are smooth the value \(- H(x)/(n-1)\) is the mean curvature of \(\partial E\) at \(x\). Concerning the existence of the function \(H\) which is clearly not unique the author uses a natural construction which yields a specific \(H= H_ E\) having minimal \(L^ 1(\mathbb{R}^ n)\)-norm equal to \(2P(E)\). Moreover, it is shown that if \(E\) is of finite measure and minimizes \({\mathcal F}_ H\) for some \(H\in L^ 1(\mathbb{R}^ n)\cap L^ p(E)\), \(p> 1\), then also \(H_ E\in L^ p(E)\), \(H_ E\) has minimal \(L^ p(E)\)-norm among all such functions \(H\) and is unique with this property. The proofs use only the basic properties of Lebesgue measure and perimeter.
scientific article; zbMATH DE number 650743