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, we assume $\Phi(t)f(t,u)$ to be decreasing in t. Our method combines variational and topological arguments and can be applied to some elliptic problems in annular domains. An $L_\infty$ bound for the solution is provided by the $L_\infty$ norm of any test function with negative energy.

Given the probability measure ν over the given region $\Omega\subset \mathbb{R}^n$, we consider the optimal location of a set Σ composed by n points in Ω in order to minimize the average distance $\Sigma\mapsto \int_\Omega \mathrm{dist}\,(x,\Sigma)\,{\rm d}\nu$ (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 the configuration built at previous steps. We show that the respective optimization problems 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 problem of determining n(n+1)/2 unknown functions aij 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 prove the 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 Carleman estimate.

We consider the Laplacian in a domain squeezed between two parallel curves in the plane, subject to Dirichlet boundary conditions on one of the curves and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the curves tends to zero. The asymptotics are uniform and local in the sense that the coefficients depend only on the extremal points where the ratio of the curvature radii of the Neumann boundary to the Dirichlet one is the biggest. We also show that the asymptotics can be obtained from a form of norm-resolvent convergence which takes into account the width-dependence of the domain of definition of the operators involved.

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

Critical points of a variant of the Ambrosio-Tortorelli functional, for which non-zero Dirichlet boundary conditions replace the fidelity term, are investigated. They are shown to converge to particular critical points of the corresponding variant of the Mumford-Shah functional; those exhibit many symmetries. That Dirichlet variant is the natural functional when addressing a problem 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 pointwise bilateral control-state constraints are considered. A locally superlinearly convergent numerical solution algorithm is proposed and its mesh independence is established. Further, for the efficient numerical solution reduced space and Schur complement based preconditioners are proposed which take into account the active and inactive set structure of the problem. The paper ends by numerical tests illustrating our theoretical findings and comparing 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 electric potential, which can vanish somewhere, f is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution $u\colon \mathbb{R}^N \to \mathbb{C}$ to (0.1), under conditions on 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 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.

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 the 2-Wasserstein distance between probability measures on the 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 state for 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.