A remark on Serrin’s Theorem

Abstract

An approximation theorem for families of weakly coercive convex functions by means of countably many smooth families of convex functions is proved. As a consequence, the classical "three-fold" lower semicontinuity theorem for integral functionals of the Calculus of Variations by James Serrin is unified in one single statement.2000 Mathematics Subject Classification: Primary 49J45, Secondary 52A41.

Weak lower semicontinuity for non coercive polyconvex integrals

“…Some extensions of Serrin's results have been recently obtained by and by several other authors, also in the context of BV -functions (see [18,15,9,10,3]). …”

A Relaxation Result for an Inhomogeneous Functional Preserving Point-Like and Curve-Like Singularities in Image Processing

“…We conclude this section with another approximation result for convex function contained in [12]. Also in this case we state the result in a simpler case and we refer the reader to Lemma 8 of [12] for a more general statement.…”

“…The proof is based on an approximation result for lower semicontinuous functions contained in [12]. This result, under a suitable uniform lower semicontinuity condition with respect to the spatial variable (see Theorem 2.3), permits to write a convex integrand f as a supremum of functions which are split as a product of a function depending only on the spatial variable times a function only depending on the second variable.…”

“…Also in this case we state the result in a simpler case and we refer the reader to Lemma 8 of [12] for a more general statement. Theorem 2.3.…”

Lower semi-continuity for non-coercive polyconvex integrals in the limit case

“…It was established in [20,Theorem 7] that demi-coercive integrands, i.e. coercive up to addition of null Lagrangeans, belong to the latter class.…”