On semicontinuity of some functionals. (Russian) Zbl 0798.46027

Strugov, Yu. F.

Lavrent’ev, M. M. (ed.), Mathematical analysis and discrete mathematics. Interuniversity collection of scientific works. Novosibirsk: Novosibirskij Gosudarstvennyj Universitet, 42-47 (1989).

Let \(D\) be a bounded domain in \(R^ n\). The author gives rather weak conditions on a non-negative function \(L(x,u,v)\), \(x\in D\), \(u\in R^ m\), \(v\in R^ n\), under which the semicontinuity property \[ \int_ D L(x,f,\nabla f)^ r dr \leq \underline{\lim}_{n\nu\to\infty} \int_ D L(x,f_ \nu, \nabla f_ \nu)^ rdx \qquad r\geq 1, \] holds with \(f_ \nu\in W^ 1_ 1(D)\) and \(f_ \nu\to f\) locally uniformly in \(D\). Some close assertions and corollaries are also proved.
For the entire collection see [Zbl 0787.00010].

Reviewer: S.G.Samko (Rostov-na-Donu)

MSC:

46E35

Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Keywords:

semicontinual functionals; quasiconformic mappings; semicontinuity property