We study the Landau-Lifshitz model for the energy of multi-scale transition layers – called “domain walls” – in soft ferromagnetic films. Domain walls separate domains of constant magnetization vectors $m_{\alpha }^{\pm }\in {𝕊}^2$ that differ by an angle $2\alpha$. Assuming translation invariance tangential to the wall, our main result is the rigorous derivation of a reduced model for the energy of the optimal transition layer, which in a certain parameter regime confirms the experimental, numerical and physical predictions: The...

Shape optimization is described by finding the geometry of a structure which is optimal in the sense of a minimized cost function with respect to certain constraints. A Newton’s mesh independence principle was very efficiently used to solve a certain class of optimal design problems in [6]. Here motivated by optimization considerations we show that under the same computational cost an even finer mesh independence principle can be given.

In this paper we prove a regularity result for local minimizers of functionals of the Calculus of Variations of the type$${\int }_{\Omega }f(x,Du)\mathrm{d}x$$where Ω is a bounded open set in ${ℝ}^n$, u∈$W_{\mathrm{loc}}^{1,p}$(Ω; ${ℝ}^N$), p> 1, n≥ 2 and N≥ 1. We use the technique of difference quotient without the usual assumption on the growth of the second derivatives of the function f. We apply this result to give a bound on the Hausdorff dimension of the singular set of minimizers.

Starting from Giaquinta’s counterexample [12] we introduce the class of splitting functionals being of $(p,q)$-growth with exponents $p\le q\<\infty$ and show for the scalar case that locally bounded local minimizers are of class $C^{1,\mu }$. Note that to our knowledge the only $C^{1,\mu }$-results without imposing a relation between $p$ and $q$ concern the case of two independent variables as it is outlined in Marcellini’s paper [15], Theorem A, and later on in the work of Fusco and Sbordone [10], Theorem 4.2.

Let $L:{ℝ}^N\times {ℝ}^N\to ℝ$ be a borelian function and consider the following problems$$inf\left\{F\left(y\right)={\int }_a^{b}L(y\left(t\right),y^{\text{'}}\left(t\right))\mathrm{d}t:y\in AC([a,b],{ℝ}^N),y\left(a\right)=A,y\left(b\right)=B\right\}\left(P\right)$$$$inf\left\{F^{...}\right\}$$

Let $L:{ℝ}^N\times {ℝ}^N\to ℝ$ be a Borelian function and consider the following problems $$inf\left\{F\left(y\right)={\int }_a^{b}L(y\left(t\right),y^{\text{'}}\left(t\right))\mathrm{d}t:y\in AC([a,b],{ℝ}^N),y\left(a\right)=A,y\left(b\right)=B\right\}\left(P\right)$$
 $$inf\left\{F^{**}\left(y\right)={\int }_a^b{L}^{**}(y\left(t\right),y^{\text{'}}\left(t\right))\mathrm{d}t:y\in AC([a,b],{ℝ}^N),y\left(a\right)=A,y\left(b\right)=B\right\}·\left(P^{**}\right)$$ We give a sufficient condition, weaker then superlinearity, under which $infF=inf{F}^{**}$ if L is just continuous in x. We then extend a result of Cellina on the Lipschitz regularity of the minima of (P) when L is not superlinear.


We consider, in an open subset Ω of ${ℝ}^N$, energies depending on the perimeter of a subset $E\subset \Omega$ (or some equivalent surface integral) and on a function u which is defined only on $\Omega \setminus E$. We compute the lower semicontinuous envelope of such energies. This relaxation has to take into account the fact that in the limit, the “holes” E may collapse into a discontinuity of u, whose surface will be counted twice in the relaxed energy. We discuss some situations where such energies appear, and give, as an application,...

We prove a relaxation theorem in BV for a non coercive functional with linear growth. No continuity of the integrand with respect to the spatial variable is assumed.

In this paper, we present a result on relaxability of partially observed control problems for infinite dimensional stochastic systems in a Hilbert space. This is motivated by the fact that measure valued controls, also known as relaxed controls, are difficult to construct practically and so one must inquire if it is possible to approximate the solutions corresponding to measure valued controls by those corresponding to ordinary controls. Our main result is the relaxation theorem which states that...

We prove higher integrability for the gradient of bounded minimizers of some variational integrals with anisotropic growth.

We give a characterization of $K$-weakly precompact sets in terms of uniform Gateaux differentiability of certain continuous convex functions.