Band 11, Heft 4

We characterize the continuity of prototypical functionals acting on finite Caccioppoli partitions and prove that it is equivalent to convergence of the perimeter of the jump set.

In this paper, starting from a non-convex and nonlocal 3D-variational model for the electric polarization in a ferroelectric material, and using an asymptotic process based on dimensional reduction, we analyze junction phenomena for two orthogonal joined ferroelectric thin films. We obtain three different 2D-variational models for joined thin films, depending on how the reduction happens. Indeed, a memory effect of the reduction process appears, and it depends on the competition of the relative thickness of the two films. The guide parameter is the limit of the ratio between these two small thickness.

In this work we study a family of shape optimization problem where the state equation is given in terms of a nonlocal operator. Examples of the problems considered are monotone combinations of fractional eigenvalues. Moreover, we also analyze the transition from nonlocal to local state equations.

The variational capacity capp{\operatorname{cap}_{p}} in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every E⊂ℝn{E\subset{\mathbb{R}}^{n}}, infx∈ℝn⁡capp⁡(E∩B⁢(x,r),B⁢(x,2⁢r))capp⁡(B⁢(x,r),B⁢(x,2⁢r))\inf_{x\in{\mathbb{R}}^{n}}\frac{\operatorname{cap}_{p}(E\cap B(x,r),B(x,2r))}% {\operatorname{cap}_{p}(B(x,r),B(x,2r))} is either zero or tends to 1 as r→∞{r\to\infty}. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a p -Poincaré inequality, but that it can fail in nongeodesic metric spaces and also for the Sobolev capacity in ℝn{{\mathbb{R}}^{n}}. It turns out that the shape of balls impacts the validity of the density dichotomy. Even in more general metric spaces, we construct families of sets, such as John domains, for which the density dichotomy holds. Our arguments include an exact formula for the variational capacity of superlevel sets for capacitary potentials and a quantitative approximation from inside of the variational capacity.

We consider the partial differential equation u-f=div⁡(um⁢∇⁡u|∇⁡u|)u-f=\operatornamewithlimits{div}\biggl{(}u^{m}\frac{\nabla u}{|\nabla u|}% \biggr{)} with f nonnegative and bounded and m∈ℝ{m\in\mathbb{R}}. We prove existence and uniqueness of solutions for both the Dirichlet problem (with bounded and nonnegative boundary datum) and the homogeneous Neumann problem. Solutions, which a priori belong to a space of truncated bounded variation functions, are shown to have zero jump part with respect to the ℋN-1{{\mathcal{H}}^{N-1}}-Hausdorff measure. Results and proofs extend to more general nonlinearities.