We propose a model for segmentation problems involving an energy concentrated on the vertices of an unknown polyhedral set, where the contours of the images to be recovered have preferred directions and focal points. We prove that such an energy is obtained as a Γ-limit of functionals defined on sets with smooth boundary that involve curvature terms of the boundary. The minimizers of the limit functional are polygons with edges either parallel to some prescribed directions or pointing to some fixed...

Controlling growth at crystalline surfaces requires a detailed and quantitative understanding of the thermodynamic and kinetic parameters governing mass transport. Many of these parameters can be determined by analyzing the isothermal wandering of steps at a vicinal [“step-terrace”] type surface [for a recent review see [4]]. In the case of $orthodox$ crystals one finds that these meanderings develop larger amplitudes as the equilibrium temperature is raised (as is consistent with the statistical mechanical...

Controlling growth at crystalline surfaces requires a detailed and quantitative understanding of the thermodynamic and kinetic parameters governing mass transport. Many of these parameters can be determined by analyzing the isothermal wandering of steps at a vicinal [“step-terrace”] type surface [for a recent review see [4]]. In the case of orthodox crystals one finds that these meanderings develop larger amplitudes as the equilibrium temperature is raised (as is consistent with the statistical...

Rate-independent evolution for material models with nonconvex elastic energies is studied without any spatial regularization of the inner variable; due to lack of convexity, the model is developed in the framework of Young measures. An existence result for the quasistatic evolution is obtained in terms of compatible systems of Young measures. We also show as this result can be equivalently reformulated with probabilistic language and leads to the description of the quasistatic evolution in terms...

Rate-independent evolution for material models with nonconvex elastic energies is studied without any spatial regularization of the inner variable; due to lack of convexity, the model is developed in the framework of Young measures. An existence result for the quasistatic evolution is obtained in terms of compatible systems of Young measures. We also show as this result can be equivalently reformulated with probabilistic language and leads to the description of the quasistatic evolution in terms...

Following the $\Gamma$-convergence approach introduced by Müller and Ortiz, the convergence of discrete dynamics for lagrangians with quadratic behavior is established.

Following the Γ-convergence approach introduced by Müller and Ortiz, the convergence of discrete dynamics for Lagrangians with quadratic behavior is established.

∗ The work is partially supported by NSFR Grant No MM 409/94.We develop an abstract subdifferential calculus for lower semicontinuous functions and investigate functions similar to convex functions. As application we give sufficient conditions for the integrability of a lower semicontinuous function.

We establish an approximation theorem for a sequence of linear elastic strains approaching a compact set in $L^1$ by the sequence of linear strains of mapping bounded in Sobolev space $W^{1,p}$. We apply this result to establish equalities for semiconvex envelopes for functions defined on linear strains via a construction of quasiconvex functions with linear growth.

We establish an approximation theorem for a sequence of linear elastic strains approaching a compact set in L1 by the sequence of linear strains of mapping bounded in Sobolev space W1,p . We apply this result to establish equalities for semiconvex envelopes for functions defined on linear strains via a construction of quasiconvex functions with linear growth.

The weak lower semicontinuity of the functional $$F\left(u\right)={\int }_{\Omega }f(x,u,\nabla u)\mathrm{d}x$$ is a classical topic that was studied thoroughly. It was shown that if the function $f$ is continuous and convex in the last variable, the functional is sequentially weakly lower semicontinuous on $W^{1,p}\left(\Omega \right)$. However, the known proofs use advanced instruments of real and functional analysis. Our aim here is to present a proof understandable even for students familiar only with the elementary measure theory.

We prove by giving an example that when $n\ge 3$ the asymptotic behavior of functionals ${\int }_{\Omega }\epsilon |{\nabla }^2{u|}^2{+(1-|\nabla u|}^2{)}^2/\epsilon$ is quite different with respect to the planar case. In particular we show that the one-dimensional ansatz due to Aviles and Giga in the planar case (see [2]) is no longer true in higher dimensions.

We prove by giving an example that when n ≥ 3 the asymptotic behavior of functionals ${\int }_{\Omega }\epsilon |{\nabla }^2{u|}^2{+(1-|\nabla u|}^2{)}^2/\epsilon$ is quite different with respect to the planar case. In particular we show that the one-dimensional ansatz due to Aviles and Giga in the planar case (see [2]) is no longer true in higher dimensions.

Local Lipschitz continuity of minimizers of certain integrals of the Calculus of Variations is obtained when the integrands are convex with respect to the gradient variable, but are not necessarily uniformly convex. In turn, these regularity results entail existence of minimizers of variational problems with non-homogeneous integrands nonconvex with respect to the gradient variable. The $x$-dependence, explicitly appearing in the integrands, adds significant technical difficulties in the proof.

Local Lipschitz continuity of minimizers of certain integrals of the Calculus of Variations is obtained when the integrands are convex with respect to the gradient variable, but are not necessarily uniformly convex. In turn, these regularity results entail existence of minimizers of variational problems with non-homogeneous integrands nonconvex with respect to the gradient variable. The x-dependence, explicitly appearing in the integrands, adds significant technical difficulties in the proof.