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The common root of the geometric conditions in Serrin’s lower semicontinuity theorem. (English) Zbl 1164.49004

The authors consider the functional \(F(u)=\int_{\Omega} f(x,u(x),Du(x))\,dx,\) where \(\Omega\) is an open set of \(\mathbb R^n\), \(u\in W^{1,1}_{loc} (\Omega; \mathbb R^n), Du \) is the approximate differential of u and \(f:\Omega \times \mathbb R \times \mathbb R^n \longrightarrow \mathbb R\) such that \(f(x,u(x),Du(x)) \) is measurable. They are interested in reconsidering the celebrated Serrin’s theorem on the l.s.c. of \(F\) with respect to \(L^1_{loc}(\Omega)-\)convergence, with the aim to obtain weaker condition to ensure the same result for \(F.\) Some results in the spirit of this program are obtained in M. Gori, F. Maggi, P. Marcellini, Differ. Integral Equ. 16, No. 1, 51-76 (2003; Zbl 1028.49012) and M. Gori, P. Marcellini, J. Convex Anal. 9, No.2, 475-502 (2002, Zbl 1019.49021). In this paper the authors consider the following condition (NCL condition): for all \((x,s)\in \Omega \times \mathbb R\) and every \(\xi \in \mathbb R^n, \nu \in S^{n-1}\) fixed, the function: \(\rho \longrightarrow f(x,s, \xi + \rho\nu)\) is a non constant function, and they prove (theorem 3) a result of l.s.c. with respect to \(L^1_{loc}(\Omega)-\)convergence for \(F\) when \(f(s,x,.)\) is convex, \(f(x,s,\xi)\geq 0\) and NCL condition is true. One of the main tecnical tool for the previous result is an approximation theorem for NCL and convex function (theorem 5). As the authors say, the NCL condition is equivalent to demicoercivity condition considered in G. Anzellotti, G. Buttazzo, G. Dal Maso, Nonlinear Anal., Theory Methods Appl. 10, 603-613 (1986; Zbl 0612.49008). The NCL condition is weaker than the following conditions: -\(f(x.s,.)\) strictly convex; -\(f_x, f_{\xi}, f_{x \xi}\) exist and are continuous, which enter (separately) in Serrin’s theorem.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
52A41 Convex functions and convex programs in convex geometry
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