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

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].

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.MalĂ˝ (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 |