Motivated by the direct method in the calculus of variations in L∞, our main result identifies the notion of convexity characterizing the weakly∗ lower semicontinuity of nonlocal supremal functionals: Cartesian level convexity. This new concept coincides with separate level convexity in the one-dimensional setting and is strictly weaker for higher dimensions. We discuss relaxation in the vectorial case, showing that the relaxed functional will not generally maintain the supremal form. Apart from illustrating this fact with examples of multi-well type, we present precise criteria for structure-preservation. When the structure is preserved, a representation formula is given in terms of the Cartesian level convex envelope of the (diagonalized) original supremand. This work does not only complete the picture of the analysis initiated in Kreisbeck and Zappale (2020), but also establishes a connection with double integrals. We relate the two classes of functionals via an Lp-approximation in the sense of Γ-convergence for diverging integrability exponents. The proofs exploit recent results on nonlocal inclusions and their asymptotic behavior, and use tools from Young measure theory and convex analysis. ©
Cartesian convexity as the key notion in the variational existence theory for nonlocal supremal functionals / Kreisbeck, Carolin; Ritorto, Antonella; Zappale, Elvira. - In: NONLINEAR ANALYSIS. - ISSN 0362-546X. - (2022). [10.1016/j.na.2022.113111]
Cartesian convexity as the key notion in the variational existence theory for nonlocal supremal functionals
Elvira Zappale
2022
Abstract
Motivated by the direct method in the calculus of variations in L∞, our main result identifies the notion of convexity characterizing the weakly∗ lower semicontinuity of nonlocal supremal functionals: Cartesian level convexity. This new concept coincides with separate level convexity in the one-dimensional setting and is strictly weaker for higher dimensions. We discuss relaxation in the vectorial case, showing that the relaxed functional will not generally maintain the supremal form. Apart from illustrating this fact with examples of multi-well type, we present precise criteria for structure-preservation. When the structure is preserved, a representation formula is given in terms of the Cartesian level convex envelope of the (diagonalized) original supremand. This work does not only complete the picture of the analysis initiated in Kreisbeck and Zappale (2020), but also establishes a connection with double integrals. We relate the two classes of functionals via an Lp-approximation in the sense of Γ-convergence for diverging integrability exponents. The proofs exploit recent results on nonlocal inclusions and their asymptotic behavior, and use tools from Young measure theory and convex analysis. ©File | Dimensione | Formato | |
---|---|---|---|
Kreisbeck_Cartesian convexity_2022.pdf
accesso aperto
Tipologia:
Documento in Post-print (versione successiva alla peer review e accettata per la pubblicazione)
Licenza:
Tutti i diritti riservati (All rights reserved)
Dimensione
947.9 kB
Formato
Adobe PDF
|
947.9 kB | Adobe PDF |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.