Lower semicontinuity and continuity of functions of measures with respect to the strict convergence

Abstract

SynopsisLet Ω be an open subset of Rn. It is well known that, given a suitable real-valued function f on Ω × Rk and a Rk -valued Borel measure µ on Ω, then one can define a real-valued measurefµ on Ω. The object of this note is to define the Ψ-strict convergence of the Rk-valued Borel measures µj to the Rk-valued Borel measure µ, where Ψ: Ω × Rk → [0, + ∞] is a continuous function which is positively homogeneous and convex in the Rk-variable, and to investigate the lower semicontinuity and continuity of the map µ → fμ with respect to the Ψ-strict convergence; here f is positively homogeneous in the Rk-variable and satisfies one suitable convexity condition (related to Ψ).

Relaxation of signed integral functionals in BV

“…We will employ the following result (see Theorem 4), which constitutes an extension of the classical Reshetnyak Continuity Theorem [36] (also see [15] for related results) and which may be interesting in its own right:…”

“…The reason why we need this topology will be clear in the proof of a version of the Reshetnyak approximation theorem in the appendix (which also contains a counterexample for the strict topology). The paper [15] contains more on related topics.…”

Limiting Aspects of Nonconvex ${TV}^{\phi}$ Models

“…Strong convergence in L 2 , which we could in fact obtain from strict convergence for Ω ⊂ R 2 (see [21,28]), is also not enough, as a stronger form of gradient convergence is the important part. A suitable mode of convergence is the so-called area-strict convergence [10,20]. For our purposes, the following definition is the most appropriate one.…”

“…Our associated principal approach is given in Section 6. It utilizes the (stronger) notion of area-strict convergence [10,20], which -as will be showncan be obtained using the multiscale analysis functional η from [32,33]. In Section 7 we also discuss alternative remedies which are related to compact operators and the space SBV(Ω) of special functions of bounded variation.…”

Minimizing functionals depending on surfaces and their curvatures: A class of variational problems in the setting of generalized Gauss graphs

“…In order that E be lower semi-continuous, one can assume, for example, the "adequateness" condition on f (introduced in [5], see also [7]). …”