Volume 7, Issue 1

Abstract. We introduce n /-harmonic maps as critical points of the energy where pointwise , for the N -sphere and . This energy combines the non-local behaviour of the fractional harmonic maps introduced by Rivière and the first author with the degenerate arguments of the n -Laplacian. In this setting, we will prove Hölder continuity.

Abstract. We derive the Euler–Lagrange equations for minimizers of causal variational principles in the non-compact setting with constraints, possibly prescribing symmetries. Considering first variations, we show that the minimizing measure is supported on the intersection of a hyperplane with a level set of a function which is homogeneous of degree two. Moreover, we perform second variations to obtain that the compact operator representing the quadratic part of the action is positive semi-definite. The key ingredient for the proof is a subtle adaptation of the Lagrange multiplier method to variational principles on convex sets.

Abstract. In this paper we consider integral functionals of the form with convex integrand satisfying p growth conditions with respect to the gradient variable. As a novel feature, the dependence of the integrand on the x -variable is allowed to be through a Sobolev function. We prove local higher differentiability results for local minimizers of the functional 𝔉, establishing uniform higher differentiability estimates for solutions to a class of auxiliary problems, constructed adding singular higher order perturbations to the integrand. Furthermore, we prove a dimension free higher integrability result for the gradient of local minimizers, by the use of a weighted version of the Gagliardo–Nirenberg interpolation inequality.

Abstract. We give a proof that Brakke's mean curvature flow under the unit density assumption is smooth almost everywhere in space-time. More generally, if the velocity is equal in a weak sense to its mean curvature plus some given α-Hölder continuous vector field, then we show the regularity almost everywhere.