Abstract. In this paper we continue to advance the theory regarding the Riesz fractional gradient in the calculus of variations and fractional partial differential equations begun in an earlier work of the same name. In particular we here establish an L 1 Hardy inequality, obtain further regularity results for solutions of certain fractional PDE, demonstrate the existence of minimizers for integral functionals of the fractional gradient with non-linear dependence in the field, and also establish the existence of solutions to corresponding Euler-Lagrange equations obtained as conditions of minimality. In addition we pose a number of open problems, the answers to which would fill in some gaps in the theory as well as to establish connections with more classical areas of study, including interpolation and the theory of Dirichlet forms.
A one-dimensional Ginzburg-Landau model that describes a superconducting closed thin wire with an arbitrary cross-section subject to a large applied magnetic field is derived from the three-dimensional Ginzburg-Landau energy in the spirit of Γ -convergence. Our result proves the validity of the formal result of Richardson and Rubinstein, which reveals the double limit of a large field and a thin domain. An additional magnetic potential related to the applied field is found in the limiting functional, which yields a parabolic background for the oscillatory phase transition curve between the normal and superconducting states.
In this paper, we study a new class of fractional partial differential equations which are obtained by minimizing variational problems in fractional Sobolev spaces. We introduce a notion of fractional gradient which has the potential to extend to many classical results in the Sobolev spaces to the nonlocal and fractional setting in a natural way.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.