Functions of least gradient and BV functions. (English) Zbl 0965.46023

Ziemer, William P.

Krbec, Miroslav (ed.) et al., Nonlinear analysis, function spaces and applications. Vol. 6. Proceedings of the spring school held in Prague, Czech Republic, May 31-June 6, 1998. Prague: Czech Academy of Sciences, Mathematical Institute. 270-312 (1999).

The lectures have essentially three parts in which always \(\Omega \subset \mathbb R ^n\) is an open set. First, we look for a minimizer of \(\|\nabla u\|\) among all functions \(u\in BV(\Omega)\) satisfying \(u=g\) on \(\partial \Omega \) in the sense of traces. Here \(\nabla u\) is “full” (i.e. including singular part) distributional derivative of \(u\) and \(\|\nabla u\|\) its variation. The “Dirichlet condition” \(g\) is assumed to be continuous. Denoting by \(P(A,U)\) the perimeter of \(A\) in \(U\), we suppose that \(\Omega \) is a Lipschitz domain satisfying \(P(\Omega ,\mathbb R ^n)\leq P(\Omega \cup A,\mathbb R ^n)\) for any small test set \(A\), and that for each ball \(B\) centered at \(\partial \Omega \) there is a set \(E\subset B\) such that \(P(\Omega \setminus E,B)<P(\Omega ,B)\). It is proven that the solution of the minimization problem exists and is continuous.
Next, given \(v\in \mathbb R \), the problem of finding a set \(E\subset \Omega \) such that \(P(E,\mathbb R ^n)\leq P(F,\mathbb R ^n)\) for all test sets \(F\subset \Omega \), with the volume constraint \(|E|=|F|=v\), is considered. The regularity question is studied, mainly under what additional hypotheses the minimizer is convex.
Finally, let us consider a function \(f\) on \(\Omega \) and denote by \(f^*\) the function which is \(f\) on \(\Omega \) and zero outside \(\Omega \). If \(f\in BV(\Omega ')\) for each \(\Omega '\subset \Omega \) and \(f^*\) is approximately continuous \(H^{n-1}\)-a.e.in \(\mathbb R ^n\), then \(f^*\in BV_{\text{loc}}(\mathbb R ^n)\). If \(f\in W^{1,p}(\Omega)\), \(1<p<\infty \), and \(f^*\) has a Lebesgue point at \(p\)-capacity-quasi every \(x\in \partial \Omega \), then \(f\in W_0^{1,p}(\Omega)\). These extension theorems differ from previously known, because existence of any weakly differentiable extension of \(f\) to \(\mathbb R ^n\) is not a priori assumed.
For the entire collection see [Zbl 0952.00033].

Reviewer: J.Malý (Praha)

MSC:

46E35	Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
49Q20	Variational problems in a geometric measure-theoretic setting
49Q05	Minimal surfaces and optimization

Keywords:

functions of bounded variations; least gradient problem; minimal surfaces; extension theorems; Sobolev spaces; Dirichlet condition