On Lavrent'ev's effect (Q1374907)

For Lagrangians \(f(x,\xi)\) that are convex in \(\xi \in \mathbb{R}^n\) and measurable with respect to \(x \in \mathbb{R}^n\) an inequality of the form \[ \min_{u \in W_0^{1,p}(\Omega)} F(u) < \inf_{u \in C_0^{\infty}(\Omega)} F(u), \qquad F(u) = \int_{\Omega}f(x,\nabla u)dx \] is called Lavrent'ev's phenenomenon. In the paper sufficient conditions are given for two special types of Lagrangians \(f\) to be regular, i.e., there is no Lavrent'ev's phenomenon.