Increasing Set Functions

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


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
$$ F(u) = \left\{ {\begin{array}{*{20}{c}} {\int_{\Omega } {f(x,Du)dx,\;\;\,if\:u \in {{W}^{{1,p}}}(\Omega )\} ,} } \hfill \\ { + \infty ,otherwise,} \hfill \\ \end{array} } \right. $$
for a suitable non-negative Borel function f h :Ω × R n → [0,+∞[.


Open Subset Pairwise Disjoint Dense Subset Borel Measure Radon Measure 
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Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • Gianni Dal Maso
    • 1
  1. 1.International School for Advanced Studies (SISSA)TriesteItaly

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