## 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

### Keywords:

lower semicontinuity; integral functionals; convexity