Fractional order Orlicz-Sobolev spaces

Abstract

In this paper we define the notion of nonlocal magnetic Sobolev spaces with nonstandard growth for Lipschitz magnetic fields. In this context we prove a Bourgain -Brezis -Mironescu type formula for functions in this space as well as for sequences of functions. Finally, we deduce some consequences such as the Γ−convergence of modulars and convergence of solutions for some non-local magnetic Laplacian allowing non-standard growth laws to its local counterpart.On the other hand, when studying phenomena allowing behaviors more general than power laws, such as anisotropic fluids with flows obeying nonstandard rheology [8,17] or capillarity phenomena, 2010 Mathematics Subject Classification. 46E30, 35R11, 45G05.

Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems

“…In this context, the natural setting for studying problem (1.1) are fractional Orlicz-Sobolev spaces. Currently, as far as we know, the only results for fractional Orlicz-Sobolev spaces and fractional M -Laplacian operator are obtained in [3,8,12,13,14,31,37]. In particular, in [12], Bonder and Salort define the fractional Orlicz-Sobolev space associated to an N -function M and a fractional parameter 0 < s < 1 as Motivated by these above results, our first aim is to prove the compact embedding W s,M (Ω) ֒→ L M * (Ω) where M * is the Sobolev conjugate of M and Ω is bounded.…”

“…In this subsection we give a brief overview on the fractional Orlicz-Sobolev spaces studied in [12], and the associated fractional M -laplacian operator.…”

Gamma convergence and asymptotic behavior for eigenvalues of nonlocal problems

“…As a matter of fact, in [1], Theorem 2.12 is never stated even the computations and proofs of that paper give the desired result. For the convenience of the reader, Theorem 2.12 (in the more general framework of Orlicz spaces) is stated and proved in [9,Theorem 6.5] See also [16] for more results regarding the connection of fractional norms and Gamma convergence.…”

On fractional Orlicz–Sobolev spaces

“…where u L A ( ) stands for the Luxemburg norm in the Orlicz space L A ( ). Definitions (1.3) and (1.4) have been introduced in [37], where some basic properties of the space W s,A ( ) are analyzed under the 2 and ∇ 2 conditions on A. Plainly, these definitions recover (1.1) and (1.2) when A(t) = t p for some p ∈ [1, ∞).…”