We present here our most recent results ([1def]) about the definition of non-linear Weiertrass-type integrals over BV varieties, possibly discontinuous and not necessarily Sobolev's.

We discuss variational problems on two-dimensional domains with energy densities of linear growth and with radially symmetric data. The smoothness of generalized minimizers is established under rather weak ellipticity assumptions. Further results concern the radial symmetry of solutions as well as a precise description of their behavior near the boundary.

We provide a detailed analysis of the minimizers of the functional $u\mapsto {\int }_{{ℝ}^n}{\left|\nabla u\right|}^2+D{\int }_{{ℝ}^n}{\left|u\right|}^{\gamma }$, $\gamma \in (0,2)$, subject to the constraint ${\parallel u\parallel }_{L^2}=1$. This problem,e.g., describes the long-time behavior of the parabolic Anderson in probability theory or ground state solutions of a nonlinear Schrödinger equation. While existence can be proved with standard methods, we show that the usual uniqueness results obtained with PDE-methods can be considerably simplified by additional variational arguments. In addition, we investigate qualitative properties...

We provide a detailed analysis of the minimizers of the functional $u\mapsto {\int }_{{ℝ}^n}{\left|\nabla u\right|}^2+D{\int }_{{ℝ}^n}{\left|u\right|}^{\gamma }$, $\gamma \in (0,2)$, subject to the constraint ${\parallel u\parallel }_{L^2}=1$. This problem, e.g., describes the long-time behavior of the parabolic Anderson in probability theory or ground state solutions of a nonlinear Schrödinger equation. While existence can be proved with standard methods, we show that the usual uniqueness results obtained with PDE-methods can be considerably simplified by additional variational arguments. In addition, we investigate qualitative properties...

We study a variational problem which was introduced by Hannon, Marcus and Mizel [ESAIM: COCV9 (2003) 145–149] to describe step-terraces on surfaces of so-called “unorthodox” crystals. We show that there is no nondegenerate intervals on which the absolute value of a minimizer is $\pi /2$ identically.

We consider the problem of minimizing the energy$$E\left(u\right):={\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^2\mathrm{d}x+{\int }_{S_u\cap \Omega }\left(1+\left|\right[u\left]\right(x\left)\right|\right)\mathrm{d}{H}^{N-1}\left(x\right)$$among all functions $u\in SB{V}^2\left(\Omega \right)$ for which two level sets $\{u=l_i\}$ have prescribed Lebesgue measure ${\alpha }_i$. Subject to this volume constraint the existence of minimizers for $E(·)$ is proved and the asymptotic behaviour of the solutions is investigated.

We consider the problem of minimizing the energy $$E\left(u\right):={\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^2\mathrm{d}x+{\int }_{S_u\cap \Omega }\left(1+\left|\right[u\left]\right(x\left)\right|\right)\mathrm{d}{H}^{N-1}\left(x\right)$$ among all functions u ∈ SBV²(Ω) for which two level sets $\{u=l_i\}$ have prescribed Lebesgue measure ${\alpha }_i$. Subject to this volume constraint the existence of minimizers for E(.) is proved and the asymptotic behaviour of the solutions is investigated.

In this paper we analyze a typical shape optimization problem in two-dimensional conductivity. We study relaxation for this problem itself. We also analyze the question of the approximation of this problem by the two-phase optimal design problems obtained when we fill out the holes that we want to design in the original problem by a very poor conductor, that we make to converge to zero.

The closures of the pre-images associated with families of mappings in different topologies of normed spaces are considered. The question of finding a description of these closures by means of families of the same kind as original ones is studied. It is shown that for the case of the weak topology this question may be reduced to finding an appropriate closure of a given family. There are discussed various situations when the description may be obtained for the case of the strong topology. An example...

We study convergence properties of ${\left\{v\left(\nabla u_k\right)\right\}}_{k\in \mathbb{N}}$ if $v\in C\left({ℝ}^{m\times n}\right)$, $\left|v\left(s\right)\right|\le C(1+|s{|}^p)$, $1<p<+\infty$, has a finite quasiconvex envelope, $u_k\to u$ weakly in $W^{1,p}(\Omega ;{ℝ}^m)$ and for some $g\in C\left(\Omega \right)$ it holds that ${\int }_{\Omega }g\left(x\right)v\left(\nabla u_k\left(x\right)\right)\mathrm{d}x\to {\int }_{\Omega }g\left(x\right)Qv\left(\nabla u\left(x\right)\right)\mathrm{d}x$ as $k\to \infty$. In particular, we give necessary and sufficient conditions for $L^1$-weak convergence of ${\{det\nabla u_k\}}_{k\in \mathbb{N}}$ to $det\nabla u$ if $m=n=p$.

The strong closure of feasible states of families of operators is studied. The results are obtained for self-adjoint operators in reflexive Banach spaces and for more concrete case - families of elliptic systems encountered in the optimal layout of $r$ materials. The results show when it is possible to parametrize the strong closure by the same type of operators. The results for systems of elliptic operators for the case when number of unknown functions $m$ is less than the dimension $n$ of the reference...

We characterize some $G$-limits using two-scale techniques and investigate a method to detect deviations from the arithmetic mean in the obtained $G$-limit provided no periodicity assumptions are involved. We also prove some results on the properties of generalized two-scale convergence.

A theory of integral representation, relaxation and homogenization for some types of variational functionals taking extended real values and possibly being not finite also on large classes of regular functions is presented. Some applications to gradient constrained relaxation and homogenization problems are given.