Abstract. 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.

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The main results of this paper are new characterizations of W 1,p (Ω), 1 < p < ∞, and BV (Ω) for Ω ⊂ R N an arbitrary open set. Using these results, we answer some open questions of Brezis [11] and Ponce [32].

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.

In this paper we establish new L 1 -type estimates for the classical Riesz potentials of order α ∈ (0, N ):This sharpens the result of Stein and Weiss on the mapping properties of Riesz potentials on the real Hardy space H 1 (R N ) and provides a new family of L 1 -Sobolev inequalities for the Riesz fractional gradient.

In this paper we prove the following optimal Lorentz embedding for the Riesz potentials: Let α ∈ (0, d). There exists a constantWe then show how this result implies optimal Lorentz regularity for a Div-Curl system (which can be applied, for example to obtain new estimates for the magnetic field in Maxwell's equations), as well as for a vector-valued Poisson equation in the divergence free case.

The uniqueness of equilibrium for a compressible, hyperelastic body subject to dead-load boundary conditions is considered. It is shown, for both the displacement and mixed problems, that there cannot be two solutions of the equilibrium equations of Finite (Nonlinear) Elasticity whose nonlinear strains are uniformly close to each other. This result is analogous to the result of Fritz John (Comm. Pure Appl. Math. 25, 617-634, 1972) who proved that, for the displacement problem, there is a most one equilibrium solution with uniformly small strains. The proof in this manuscript utilizes Geometric Rigidity; a new straightforward extension of the Fefferman-Stein inequality to bounded domains; and, an appropriate adaptation, for Elasticity, of a result from the Calculus of Variations. Specifically, it is herein shown that the uniform positivity of the second variation of the energy at an equilibrium solution implies that this mapping is a local minimizer of the energy among deformations whose gradient is sufficiently close, in BMO ∩ L 1 , to the gradient of the equilibrium solution.In the absence of body forces and surface tractions, the total energy of a deformation u : Ω → R n of a compressible, hyperelastic body is given bywhere W : Ω × M n×n + → [0, ∞) denotes the stored-energy density and we write M n×n + for the set of n by n matrices with positive determinant. We require that u = d on D, where d is prescribed and D ⊂ ∂Ω is nonempty and relatively open. The pure-displacement problem can then be expressed as the condition D = ∂Ω, while the genuine-mixed problem is the condition D ∂Ω. We here consider both problems. With this notation, we call u e an equilibrium solution if it is a weak solution of the corresponding Euler-Lagrange equations:for all w ∈ W 1,2 (Ω; R n ) that satisfy w = 0 on D, while the uniform positivity of the second variation of E at u e is then the condition that δ 2 E(u e )[w, w] =

We prove the Boxing inequality:is the Hausdorff content of U of dimension d − α and the constant C > 0 depends only on d. We then show how this estimate implies a trace inequality in the fractional Sobolev space W α,1 (R d ) that includes Sobolev's L d d−α embedding, its Lorentz-space improvement, and Hardy's inequality. All these estimates are thus obtained with the appropriate asymptotics as α tends to 0 and 1, recovering in particular the classical inequalities of first order. Their counterparts in the full range α ∈ (0, d) are also investigated.