## Abstract and Applied Analysis

### A New Extension of Serrin's Lower Semicontinuity Theorem

#### Abstract

We present a new extension of Serrin's lower semicontinuity theorem. We prove that the variational functional ${\int }_{\mathrm{\Omega}}^{}f(x,u,{u}^{\prime })dx$ defined on ${W}_{loc}^{1,1}(\mathrm{\Omega})$ is lower semicontinuous with respect to the strong convergence in ${L}_{loc}^{1}$, under the assumptions that the integrand $f(x,s,\xi )$ has the locally absolute continuity about the variable $x$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 368610, 7 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393444383

Digital Object Identifier
doi:10.1155/2013/368610

Mathematical Reviews number (MathSciNet)
MR3070186

Zentralblatt MATH identifier
1292.49016

#### Citation

Hu, Xiaohong; Zhang, Shiqing. A New Extension of Serrin's Lower Semicontinuity Theorem. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 368610, 7 pages. doi:10.1155/2013/368610. https://projecteuclid.org/euclid.aaa/1393444383

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