A characterization of relatively compact sets in $L^p(\Omega,B)$
Dedicated to Professor Gheorghe Coman on the occasion of his 80th anniversary
Keywords:
Kolmogorov-Riesz-type compactness result, Banach-space valued functions, homogenization of processes on periodic surfaces.Abstract
We give a characterization of relatively compact sets $F$ in $L^p(\Omega,B)$ for $p\in [1,\infty)$, $B$ a Banach-space, and $\Omega \subset \R^n$. This is a generalization of the results obtained in \cite{Simon} for the space $L^p((0,T),B)$ with $T>0$, first to rectangles $\Omega =(a,b) \subset \R^n$ and, under additional conditions, to arbitrary open and bounded subsets of $\R^n$. An application of the main compactness result to a problem arising in homogenization of processes on periodic surfaces is given.
Mathematics Subject Classification (2010): 35K57, 46E40, 46B50.
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