## On some sharp conditions for lower semicontinuity in $$L^1$$.(English)Zbl 1028.49012

Summary: Let $$\Omega$$ be an open set of $$\mathbb{R}^n$$ and let $$f: \Omega\times\mathbb{R}\times\mathbb{R}^n$$ be a nonnegative continuous function, convex with respect to $$\xi\in\mathbb{R}^n$$. Following the well-known theory originated by J. Serrin [Trans. Am. Math. Soc. 101, 139-167 (1961; Zbl 0102.04601)], we deal with the lower semicontinuity of the integral $F(u,\Omega)= \int_\Omega \bigl(x,u(x), Du(x)\bigr)dx$ with respect to the $$L^1_{\text{loc}} (\Omega)$$ strong convergence. Only recently it has been discovered that dependence of $$f(x,s,\xi)$$ on the $$x$$ variable plays a crucial role in the lower semicontinuity. In this paper we propose a mild assumption on $$x$$ that allows us to consider discontinuous integrands, too. More precisely, we assume that $$f(x, s,\xi)$$ is a nonnegative Carathéodory function, convex with respect to $$\xi$$, continuous in $$(s,\xi)$$ and such that $$f(\cdot,s,\xi) \in W^{1,1}_{\text{loc}} (\Omega)$$ for every $$s\in\mathbb{R}$$ and $$\xi\in\mathbb{R}^n$$, with the $$L^1$$ norm of $$f_x (\cdot,s, \xi)$$ locally bounded. We also discuss some other conditions on $$x$$; in particular we prove that Hölder continuity of $$f$$ with respect to $$x$$ is not sufficient for lower semicontinuity, even in the one dimensional case, thus giving an answer to a problem posed by the authors in [J. Convex Anal. 9, No. 2, 475-502 (2002; Zbl 1019.49021)]. Finally, we investigate the lower semicontinuity of the integral $$F(u,\Omega)$$, with respect to the strong norm topology of $$L^1_{\text{loc}} (\Omega)$$, in the vector-valued case, i.e., when $$f:\Omega\times \mathbb{R}^m \times\mathbb{R}^{m\times n}\to \mathbb{R}$$ for some $$n\geq 1$$ and $$m>1$$.

### MSC:

 49J45 Methods involving semicontinuity and convergence; relaxation 49J10 Existence theories for free problems in two or more independent variables

### Keywords:

lower semicontinuity; Hölder continuity

### Citations:

Zbl 0102.04601; Zbl 1019.49021