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An extension of the Serrin’s lower semicontinuity theorem. (English) Zbl 1019.49021

Let \[ F(u,\Omega)=\int_\Omega f(x,u(x),Du(x)) dx, \] where \(\Omega\) is an open subset of \({\mathbb R}^n\), \(u\in W^{1,1}_{\text{loc}}(\Omega)\), and \(f : (x,s,z)\in\Omega\times{\mathbb R}\times{\mathbb R}^n\to f(x,s,z)\in [0,+\infty[\) is continuous and such that \(f(x,s,\cdot)\) is convex for all \((x,s)\in\Omega\times{\mathbb R}^n\).
In 1961, J. Serrin proved various \(L^1_{\text{loc}}(\Omega)\)-lower semicontinuity theorems for \(F\) under additional sets of assumptions on \(f\), like behaviour at infinity, strict convexity, existence and continuity of the derivatives \(f_x\), \(f_z\), and \(f_{zx}\). He also proved the same result by assuming uniform continuity assumptions on \(f\) with respect to the variables \(x\) and \(s\).
In the present paper the above lower semicontinuity result is extended under only an additional local Lipschitz continuity assumption. More precisely, it is assumed that for every compact subset \(K\) of \(\Omega\times{\mathbb R}\times{\mathbb R}^n\) there exists \(L=L(K)\geq 0\) such that \[ |f(x_1,s,z)-f(x_2,s,z)|\leq L|x_1-x_2| \] for every \((x_1,s,z)\), \((x_2,s,z)\in K\).
The proof of the theorem is achieved by using, among other things, an approximation procedure of \(f\), due to De Giorgi, by means of integrands linear with respect to the last variable.
As corollary, the \(L^1_{\text{loc}}(\Omega)\)-lower semicontinuity of \(F\) is obtained by assuming only existence and continuity (or only local boundedness) of \(f_x\).
An example is also discussed showing that the above local Lipschitz continuity assumption cannot be replaced by a local Hölder continuity one with exponent strictly less than 1.
Some remarks in the vector valued case are also developed.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49J10 Existence theories for free problems in two or more independent variables
49K10 Optimality conditions for free problems in two or more independent variables
49J53 Set-valued and variational analysis
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