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.
In this article we study eigenvalues and minimizers of a fractional non-standard growth problem. We prove several properties on this quantities and their corresponding eigenfunctions.
We consider a homogeneous fractional Sobolev space obtained by completion of the space of smooth test functions, with respect to a Sobolev-Slobodeckiȋ norm. We compare it to the fractional Sobolev space obtained by the K−method in real interpolation theory. We show that the two spaces do not always coincide and give some sufficient conditions on the open sets for this to happen. We also highlight some unnatural behaviors of the interpolation space. The treatment is as self-contained as possible.2010 Mathematics Subject Classification. 46E35, 46B70.
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