We prove the existence of a positive solution to the BVP $$(\Phi(t)u'(t))'=f(t,u(t)),\,\,\,\,\,\,\,\,\,\,\,u'(0)=u(1)=0, $$ imposing some conditions on Φ and f. In particular, weassume $\Phi(t)f(t,u)$ to be decreasing in t. Our methodcombines variational and topological arguments and can be appliedto some elliptic problems in annular domains. An $L_\infty$ boundfor the solution is provided by the $L_\infty$ norm of any testfunction with negative energy.

Given the probability measure ν over the given region $\Omega\subset \mathbb{R}^n$ , we consider the optimal location of a setΣ composed by n points in Ω in order to minimize theaverage distance $\Sigma\mapsto \int_\Omega \mathrm{dist}\,(x,\Sigma)\,{\rm d}\nu$ (theclassical optimal facility location problem). The paper compares twostrategies to find optimal configurations: the long-term one whichconsists 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 addingat each step at most one point and preserving the configurationbuilt at previous steps. We show that the respective optimizationproblems exhibit qualitatively different asymptotic behavior as $n\to\infty$ , although the optimization costs in both cases have the same asymptotic orders of vanishing.

Let y(h)(t,x) be one solution to \[\partial_t y(t,x) - \sum_{i, j=1}^{n}\partial_{j} (a_{ij}(x)\partial_i y(t,x))= h(t,x), \thinspace 0<t<T, \thinspace x\in \Omega\] with a non-homogeneous term h, and $y\vert_{(0,T)\times\partial\Omega} = 0$ ,where $\Omega \subset\Bbb R^n$ is a bounded domain. We discuss an inverse problemof determining n(n+1)/2 unknown functions a ij by $\{ \partial_{\nu}y(h_{\ell})\vert_{(0,T)\times \Gamma_0}$ , $y(h_{\ell})(\theta,\cdot)\}_{1\le \ell\le \ell_0}$ after selecting input sources $h_1, ...,h_{\ell_0}$ suitably, where $\Gamma_0$ is an arbitrary subboundary, $\partial_{\nu}$ denotes the normal derivative, $0 < \theta < T$ and $\ell_0 \in \Bbb N$ . In the case of $\ell_0 = (n+1)^2n/2$ , we provethe Lipschitz stability in the inverse problem if we choose $(h_1, ...,h_{\ell_0})$ from a set ${\cal H} \subset \{ C_0^{\infty}((0,T)\times \omega)\}^{\ell_0}$ with an arbitrarily fixed subdomain $\omega \subset \Omega$ . Moreover we can take $\ell_0 = (n+3)n/2$ by making special choices for $h_{\ell}$ , $1 \le \ell \le \ell_0$ . The proof is based on a Carlemanestimate.

We consider the Laplacian in a domain squeezedbetween two parallel curves in the plane,subject to Dirichlet boundary conditions on one of the curvesand Neumann boundary conditions on the other.We derive two-term asymptotics for eigenvaluesin the limit when the distance between the curves tends to zero.The asymptotics are uniform and local in the sense thatthe coefficients depend only on the extremal points wherethe ratio of the curvature radii of the Neumann boundaryto the Dirichlet one is the biggest.We also show that the asymptotics can be obtainedfrom a form of norm-resolvent convergencewhich takes into account the width-dependenceof the domain of definition of the operators involved.

In this note we provide a new geometric lower bound on theso-called Grad's number of a domain Ω in terms of how far Ωis from being axisymmetric. Such an estimate is important in thestudy of the trend to equilibrium for the Boltzmann equation fordilute gases.

Critical points of a variant of the Ambrosio-Tortorelli functional,for which non-zero Dirichlet boundary conditions replace thefidelity term, are investigated. They are shown to converge toparticular critical points of the corresponding variant of theMumford-Shah functional; those exhibit many symmetries. ThatDirichlet variant is the natural functional when addressing aproblem of brittle fracture in an elastic material.

For a general time-varying system, we prove that existence of an “Output Robust Control Lyapunov Function” implies existence of continuous time-varying feedback stabilizer, which guarantees output asymptotic stability with respect to the resulting closed-loop system. The main results of the present work constitute generalizations of a well known result due to Coron and Rosier [J. Math. Syst. Estim. Control4 (1994) 67–84] concerning stabilization of autonomous systems by means of time-varying periodic feedback.

Optimal control problems for the heat equation with pointwisebilateral control-state constraints are considered. A locallysuperlinearly convergent numerical solution algorithm is proposedand its mesh independence is established. Further, for theefficient numerical solution reduced space and Schur complementbased preconditioners are proposed which take into account theactive and inactive set structure of the problem. The paper endsby numerical tests illustrating our theoretical findings andcomparing the efficiency of the proposed preconditioners.

In this work we consider the magnetic NLS equation $$ ( \frac{\hbar}{i} \nabla -A(x))^2 u + V(x)u - f(|u|^2)u \, = 0 \, \qquad \mbox{ in } \mathbb{R}^N\qquad\qquad(0.1)$$ where $N \geq 3$ , $A \colon \mathbb{R}^N \to \mathbb{R}^N$ is a magnetic potential,possibly unbounded, $V \colon \mathbb{R}^N \to \mathbb{R}$ is a multi-well electricpotential, which can vanish somewhere, f is a subcriticalnonlinear term. We prove the existence of a semiclassical multi-peaksolution $u\colon \mathbb{R}^N \to \mathbb{C}$ to (0.1), under conditionson the nonlinearity which are nearly optimal.

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 insidethe magnetic sample is of particular relevance. We here investigate mathematicallythis problem by considering the full partial differential model of Landau-Lifschitzequations triggered by a uniform (in space) external magnetic field.

We study existence and approximation of non-negative solutions of partial differential equations of the type 
 $$\partial_t u - \div (A(\nabla (f(u))+u\nabla V )) = 0 \qquad \mbox{in } (0,+\infty )\times \mathbb{R}^n,\qquad\qquad (0.1)$$ where A is a symmetric matrix-valued function of the spatial variable satisfying a uniform ellipticity condition, $f:[0,+\infty) \rightarrow[0,+\infty)$ is a suitable non decreasing function, $V:\mathbb{R}^n \rightarrow\mathbb{R}$ is a convex function.Introducing the energy functional $\phi(u)=\int_{\mathbb{R}^n} F(u(x))\,{\rm d}x+\int_{\mathbb{R}^n}V(x)u(x)\,{\rm d}x$ ,where F is a convex function linked to f by $f(u) = uF'(u)-F(u)$ ,we show that u is the “gradient flow” of ϕ with respect to the2-Wasserstein distance between probability measures onthe space $\mathbb{R}^n$ , endowed with the Riemannian distance induced by $A^{-1}.$ In the case of uniform convexity of V, long time asymptotic behaviour and decay rate to the stationary statefor solutions of equation (0.1) are studied.A contraction property in Wasserstein distance for solutions of equation (0.1)is also studied in a particular case.

A corrected version of [P. Grabowski and F.M. Callier, ESAIM: COCV12 (2006) 169–197], Theorem 4.1, p. 186, and Example, is given.