Minimization of Functionals of the Gradient by Baire's Theorem

Abstract

We give sufficient conditions for the existence of solutions of the minimum problem Pu 0 :Minimizebased on the structure of the epigraph of the lower convex envelope of g, which is assumed be lower semicontinuous and to grow at infinity faster than the power p with p larger than the dimension of the space. No convexity conditions are required on g, and no assumptions are made on the boundary datum u 0 ∈ W 1,p 0 (Ω, R).

An existence result for a nonconvex variational problem via regularity

“…Theorem 3.1 in Section 3 is specific to the case n ≥ 2, and it is an extension of some analogous results, obtained under more restrictive assumptions, by Marcellini [22], Mascolo-Schianchi [23], Cellina [7] and Friesecke [21]. In particular, Theorem 3.1 is an extension of related results recently proved by Sychev [28] and Zagatti [29] for integrands independent of x and under a strong assumption on the growth of f which ensures the almost everywhere differentiability of minimizers, i.e., p > n, by Celada-Perrotta [5] for p > 1, and by Dacorogna-Marcellini in [12,13].…”

Reconstruction of noisy signals by minimization of non-convex functionals

“…• A novel proof methodology to establish the existence of a minimum in optimization functionals within the signal processing field, which has been adopted from a number of theoretical works developed in different application domains such as material science, physics of phase transitions, and fracture mechanics [27,28,29,30,31].…”

“…where n i ∈ R n and l i ∈ R. This is a standard condition used to prove the existence of minimum of non-convex functionals (see [33,34,27,35,30,31]). …”

Nonconvex Problems of the Calculus of Variations and Differential Inclusions

“…The scalar case (n = 1 or N = 1) has been intensively studied notably by: Aubert-Tahraoui [4], [5], [6], Bauman-Phillips [10], Buttazzo-Ferone-Kawohl [13], Celada-Perrotta [14], [15], Cellina [16], [17], Cellina-Colombo [18], Cesari [20], [21], Cutri [22], Dacorogna [26], Ekeland [39], Friesecke [40], FuscoMarcellini-Ornelas [41], Giachetti-Schianchi [43], Klötzler [47], Marcellini [50], [51], [52], Mascolo [54], Mascolo-Schianchi [56], [57], Monteiro Marques-Ornelas [58], Ornelas [64], Raymond [67], [68], [69], Sychev [75], Tahraoui [76], [77], Treu [78] and Zagatti [80].…”