Given the probability measure ν over the given region $\Omega \subset {ℝ}^n$, we consider the optimal location of a set Σ composed by n points in Ω in order to minimize the average distance $\Sigma \mapsto {\int }_{\Omega }\mathrm{dist}(x,\Sigma )\mathrm{d}\nu$ (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all n points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and preserving...

New $L^1$-lower semicontinuity and relaxation results for integral functionals defined in BV($\Omega$) are proved, under a very weak dependence of the integrand with respect to the spatial variable $x$. More precisely, only the lower semicontinuity in the sense of the $1$-capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to $x$. Under this further BV dependence, a...

New L1-lower semicontinuity and relaxation results for integral functionals defined in BV(Ω) are proved, under a very weak dependence of the integrand with respect to the spatial variable x. More precisely, only the lower semicontinuity in the sense of the 1-capacity is assumed in order to obtain the lower semicontinuity of the functional. This condition is satisfied, for instance, by the lower approximate limit of the integrand, if it is BV with respect to x. Under this further BV dependence, a...

A lower semicontinuity result in BV is obtained for quasiconvex integrals with subquadratic growth. The key steps in this proof involve obtaining boundedness properties for an extension operator, and a precise blow-up technique that uses fine properties of Sobolev maps. A similar result is obtained by Kristensen in [Calc. Var. Partial Differ. Equ. 7 (1998) 249–261], where there are weaker asssumptions on convergence but the integral needs to satisfy a stronger growth condition.

Lower semicontinuity results are obtained for multiple integrals of the kind ${\int }_{{ℝ}^n}f(x,{\nabla }_{\mu }u)\mathrm{d}\mu$, where $\mu$ is a given positive measure on ${ℝ}^n$, and the vector-valued function $u$ belongs to the Sobolev space $H_{\mu }^{1,p}({ℝ}^n,{ℝ}^m)$ associated with $\mu$. The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to $\mu$. More precisely, for fully general $\mu$, a notion of quasiconvexity for $f$ along the tangent bundle to $\mu$, turns out to be necessary for lower...

Lower semicontinuity results are obtained for multiple integrals of the kind ${\int }_{{ℝ}^n}f(x,{\nabla }_{\mu }u)\mathrm{d}\mu$, where μ is a given positive measure on ${ℝ}^n$, and the vector-valued function u belongs to the Sobolev space $H_{\mu }^{1,p}({ℝ}^n,{ℝ}^m)$ associated with μ. The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to μ. More precisely, for fully general μ, a notion of quasiconvexity for f along the tangent bundle to μ, turns out to be necessary for lower...

We prove a lower semicontinuity result for variational integrals associated with a given first order elliptic complex, extending, in this general setting, a well known result in the case ${}^{\text{'}}(ℝⁿ,ℝ){\to }^{\nabla }\text{'}(ℝⁿ,ℝⁿ){\to }^{curl}\text{'}(ℝⁿ,{ℝ}^{n\times n})$.

The lower semicontinuity of functionals of the type ${\int }_{\Omega }f(x,u,v,\nabla u)dx$ with respect to the $\left(W^{1,1}\times L^p\right)$-weak* topology is studied. Moreover, in absence of lower semicontinuity, an integral representation in $W^{1,1}\times L^p$ for the lower semicontinuous envelope is also provided.

The study of small magnetic particles has become a very important topic, in particular for the development of technological devices such as those used for magnetic recording. In this field, switching the magnetization inside the magnetic sample is of particular relevance. We here investigate mathematically this problem by considering the full partial differential model of Landau-Lifschitz equations triggered by a uniform (in space) external magnetic field.

The study of small magnetic particles has become a very important topic, in particular for the development of technological devices such as those used for magnetic recording. In this field, switching the magnetization inside the magnetic sample is of particular relevance. We here investigate mathematically this problem by considering the full partial differential model of Landau-Lifschitz equations triggered by a uniform (in space) external magnetic field.

The paper deals with approximations and the numerical realization of a class of hemivariational inequalities used for modeling of delamination and nonmonotone friction problems. Assumptions guaranteeing convergence of discrete models are verified and numerical results of several model examples computed by a nonsmooth variant of Newton method are presented.

The left-invariant sub-Riemannian problem on the group of motions of a plane is considered. Sub-Riemannian geodesics are parameterized by Jacobi's functions. Discrete symmetries of the problem generated by reflections of pendulum are described. The corresponding Maxwell points are characterized, on this basis an upper bound on the cut time is obtained.