Connected perimeter of planar sets

We introduce a notion of connected perimeter for planar sets defined as the lower semicontinuous envelope of perimeters of approximating sets which are measure-theoretically connected. A companion notion of simply connected perimeter is also studied. We prove a representation formula which links the connected perimeter, the classical perimeter, and the length of suitable Steiner trees. We also discuss the application of this notion to the existence of solutions to a nonlocal minimization problem with connectedness constraint.

Strong approximation in h-mass of rectifiable currents under homological constraint

Let {h:\mathbb{R}\to\mathbb{R}_{+}} be a lower semicontinuous subbadditive and even function such that {h(0)=0} and {h(\theta)\geq\alpha|\theta|} for some {\alpha>0} . If {T=\tau(M,\theta,\xi)} is a k-rectifiable chain, its h-mass is defined as \mathbb{M}_{h}(T):=\int_{M}h(\theta)\,d\mathcal{H}^{k}. Given such a rectifiable flat chain T with {\mathbb{M}_{h}(T)<\infty} and {\partial T} polyhedral, we prove that for every {\eta>0} , it decomposes as {T=P+\partial V} with P polyhedral, V rectifiable, {\mathbb{M}_{h}(V)<\eta} and {\mathbb{M}_{h}(P)<\mathbb{M}_{h}(T)+\eta} . In short, we have a polyhedral chain P which strongly approximates T in h-mass and preserves the homological constraint {\partial P=\partial T} . When {h^{\prime}(0^{+})} is well defined and finite, the definition of the h-mass extends as a finite functional on the space of finite mass k-chains (not necessarily rectifiable). We prove in this case a similar approximation result for finite mass k-chains with polyhedral boundary. These results are motivated by the study of approximations of {\mathbb{M}_{h}} by smoother functionals but they also provide explicit formulas for the lower semicontinuous envelope of {T\mapsto\mathbb{M}_{h}(T)+\mathbb{I}_{\partial S}(\partial T)} with respect to the topology of the flat norm.

Relaxation of nonlinear elastic energies involving the deformed configuration and applications to nematic elastomers

We start from a variational model for nematic elastomers that involves two energies: mechanical and nematic. The first one consists of a nonlinear elastic energy which is influenced by the orientation of the molecules of the nematic elastomer. The nematic energy is an Oseen–Frank energy in the deformed configuration. The constraint of the positivity of the determinant of the deformation gradient is imposed. The functionals are not assumed to have the usual polyconvexity or quasiconvexity assumptions to be lower semicontinuous. We instead compute its relaxation, that is, the lower semicontinuous envelope, which turns out to be the quasiconvexification of the mechanical term plus the tangential quasiconvexification of the nematic term. The main assumptions are that the quasiconvexification of the mechanical term is polyconvex and that the deformation is in the Sobolev space W1,p (with p > n − 1 and n the dimension of the space) and does not present cavitation.

The L1L^{1} gradient flow of a generalized scale invariant Willmore energy for radially non-increasing functions

We use the minimizing movement theory to study the gradient flow associated to a non-regular relaxation of a geometric functional derived from the Willmore energy. Thanks to the coarea formula, we can define a Willmore energy on regular functions of L^{1}(\mathbb{R}^{d}). This functional is extended to every {L^{1}} function by taking its lower semicontinuous envelope. We study the flow generated by this relaxed energy for radially non-increasing functions (functions with balls as superlevel sets). In the first part of the paper, we prove a coarea formula for the relaxed energy of such functions. Then, we show that the flow consists of an erosion of the initial data. The erosion speed is given by a first order ordinary equation.

A necessary and sufficient condition for the weak lower semicontinuity of one-dimensional non-local variational integrals

In this short note we prove that the functional I : W1,p(J;R) → R defined by is sequentially weakly lower semicontinuous in W1,p(J,R) if and only if the symmetric part W+ of W is separately convex. We assume that W is real valued, continuous and bounded below by a constant, and that J is an open subinterval of R. We also show that the lower semicontinuous envelope of I cannot in general be obtained by replacing W by its separately convex hull Wsc.