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On the lower semicontinuity of certain integral functionals. (English) Zbl 0554.49006

This paper obtains the lower semicontinuity of the functional \(F(u)=\int_{\Omega}f(u,Du)dx\) on \(W^{1,1}_{loc}(\Omega)\) with respect to the topology induced by \(L^ 1_{loc}(\Omega)\). The significance of the result is that the nonnegative real valued function f(s,p) on \(R\times R^ n\) is assumed to be lower semicontinuous in s only for \(p=0\). Other assumptions on f are measurability in s, convexity in p and that \(\alpha_ f(s)= \limsup_{p\to 0}([f(s,0)- f(s,p)]^+/| p|)\) belongs to \(L^ 1_{loc}(R)\). Here \(a^+\) denotes max(a,0). An example is given to show that the assumption on \(\alpha_ f\) is necessary. The result is first proved for functions f with \(f(s,0)=0\). The general case is then proved by considering \(g(s,p)=f(s,p)-f(s,0)-<b(s),p>\) where b is an element of the subdifferential \(\partial f(s,0)\) with the minimum norm.
Reviewer: M.Sury

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26B25 Convexity of real functions of several variables, generalizations
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
54C08 Weak and generalized continuity
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