De Giorgi, Ennio; Buttazzo, Giuseppe; Dal Maso, Gianni On the lower semicontinuity of certain integral functionals. (English) Zbl 0554.49006 Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 74, 274-282 (1983). 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 Cited in 1 ReviewCited in 31 Documents 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 Keywords:integral functionals; lower semicontinuity; subdifferential PDFBibTeX XMLCite \textit{E. De Giorgi} et al., Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Nat. 74, 274--282 (1983; Zbl 0554.49006)