We obtain a measure representation for a functional arising in the context of optimal design problems under linear growth conditions. The functional in question corresponds to the relaxation with respect to a pair $(\chi,u)$, where $\chi$ is the characteristic function of a set of finite perimeter and $u$ is a function of bounded deformation, of an energy with a bulk term depending on the symmetrized gradient as well as a perimeter term.

In this paper we analyze the first-order behavior (that is, the right-sided derivative) of the volume of the dilation A⊕tQ as t converges to 0. Here A and Q are subsets of n-dimensional Euclidean space, A has finite perimeter, and Q is finite. If Q consists of two points only, n and n+u, say, this derivative coincides up to a sign with the directional derivative of the covariogram of A in direction u. By known results for the covariogram, this derivative can therefore be expressed by the cosine transform of the surface area measure of A. We extend this result to finite sets Q and use it to determine the derivative of the contact distribution function with finite structuring element of a stationary random set at 0. The proofs are based on an approximation of the indicator function of A by smooth functions of bounded variation.

We establish that the intrinsic distance dE associated with an indecomposable plane set E of finite perimeter is infinitesimally Euclidean; namely, in E. By this result, we prove through a standard argument that a conservative vector field in a plane set of finite perimeter has a potential. We also provide some applications to complex analysis. Moreover, we present a collection of results that would seem to suggest the possibility of developing a De Rham cohomology theory for integral currents.

In this note we show the characteristic function of every indecomposable setF in theplane is BVequivalent to the characteristic function a closed set $\mathrm{See\; Formula\; in\; PDF}$\hbox{See Formula in PDF} .We show by example this is false in dimension three and above. As a corollary to thisresult we show that for every ϵ > 0 a set of finite perimeter S can be approximated by aclosed subset $\mathrm{See\; Formula\; in\; PDF}$\hbox{See Formula in PDF} with finitely many indecomposablecomponents and with the property that $\mathrm{See\; Formula\; in\; PDF}$\hbox{See Formula in PDF} and $\mathrm{See\; Formula\; in\; PDF}$\hbox{See Formula in PDF} .We apply this corollary to give a short proof that locally quasiminimizing sets in theplane are BVlextension domains.

We prove existence of minimizing movements for thedislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscositysolutions of the corresponding level-set equation. We also prove theconsistency of this approach, by showing that any minimizing movementcoincides with the smooth evolution as long as the latter exists. Inrelation with this, we finally prove short time existence and uniqueness of a smoothfront evolving according to our law, provided the initial shape issmooth enough.