In this paper, we show how under the continuum hypothesis one can
obtain an integral representation for elements of the topological dual of the space
of functions of bounded variation in terms of Lebesgue and Kolmogorov-Burkill
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 L1 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.
In this paper we prove several formulae that enable one to capture the singular portion of the measure derivative of a function of bounded variation as a limit of non-local functionals. One special case shows that rescalings of the fractional Laplacian of a function $u \in SBV$ converge strictly to the singular portion of $Du$.