# Increasing Set Functions

• Gianni Dal Maso
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 8)

## Abstract

Chapters 14–20 are devoted to questions connected with the problem of the integral representation of Γ-limits. Let Ω be an open subset of R n and let (F h ) be a sequence of integral functionals on L p (Ω), 1 ≤ p < +∞, of the form
$${{F}_{h}}(u) = \left\{ {\begin{array}{*{20}{c}} {\int_{\Omega } {{{f}_{n}}(x,Du)dx,\;\;\,if\:u \in {{W}^{{1,p}}}(\Omega )\} ,} } \hfill \\ { + \infty ,otherwise,} \hfill \\ \end{array} } \right.$$
where f h :Ω × R n → [0,+∞[ are non-negative Borel functions. Suppose that (F h ) Γ-converges to a functional F in L p (Ω). We want to establish conditions on the sequence (f h ) under which the limit functional F can be represented as
$$F(u) = \left\{ {\begin{array}{*{20}{c}} {\int_{\Omega } {f(x,Du)dx,\;\;\,if\:u \in {{W}^{{1,p}}}(\Omega )\} ,} } \hfill \\ { + \infty ,otherwise,} \hfill \\ \end{array} } \right.$$
(14.1)
for a suitable non-negative Borel function f h :Ω × R n → [0,+∞[.

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