Volume 10, Issue 2

In this paper, first we study existence results for a linearly perturbed elliptic problem driven by the fractional Laplacian. Then, we show a multiplicity result when the perturbation parameter is close to the eigenvalues. This latter result is obtained by exploiting the topological structure of the sublevels of the associated functional, which permits to apply a critical point theorem of mixed nature due to Marino and Saccon.

In this article, a proof of the interpolation inequality along geodesics in p -Wasserstein spaces is given. This interpolation inequality was the main ingredient to prove the Borel–Brascamp–Lieb inequality for general Riemannian and Finsler manifolds and led Lott–Villani and Sturm to define an abstract Ricci curvature condition. Following their ideas, a similar condition can be defined and for positively curved spaces one can prove a Poincaré inequality. Using Gigli’s recently developed calculus on metric measure spaces, even a q -Laplacian comparison theorem holds on q -infinitesimal convex spaces. In the appendix, the theory of Orlicz–Wasserstein spaces is developed and necessary adjustments to prove the interpolation inequality along geodesics in those spaces are given.

We study solutions of Euler–Lagrange equations for isotropic energy functionals, generalizing a previous result on p -harmonic mappings. We classify all stored energy functions which give rise to a first-order differential expression whose Laplacian involves no third derivatives of the stationary solution. This classification gives rise to a new technique of finding subharmonicity results for the variational equations, and we also illustrate this technique in two examples. Firstly, we prove a subharmonicity result for the Jacobian determinant in the case of weighted Dirichlet energy. Secondly, we find optimal subharmonicity results in the case of a Neohookean-type stored energy function.

We study the lower semicontinuity of some free discontinuity functionals with linear growth defined on the space of functions with bounded deformation. The volume term is convex and depends only on the Euclidean norm of the symmetrized gradient. We introduce a suitable class of surface terms, which make the functional lower semicontinuous with respect to L1${L^{1}}$ convergence.