We generalize to the p-LaplacianΔp a spectral inequality proved by M.-T.Kohler−Jobin. As a particular case of such a generalization, we obtain a sharp lower boundon the first Dirichlet eigenvalue of Δp of aset in terms of its p-torsional rigidity. The result is valid in everyspace dimension, for every1 < p < ∞ and for every openset with finite measure. Moreover, it holds by replacing the first eigenvalue with moregeneral optimal Poincaré-Sobolev constants. The method of proof is based on ageneralization of the rearrangement technique introduced by Kohler−Jobin.

Several kinds of exact synchronizations and the generalized exact synchronization areintroduced for a coupled system of 1-D wave equations with various boundary conditions andwe show that these synchronizations can be realized by means of some boundarycontrols.

The problem of distributing two conducting materials with a prescribed volume ratio in aball so as to minimize the first eigenvalue of an elliptic operator with Dirichletconditions is considered in two and three dimensions. The gap ε between the twoconductivities is assumed to be small (low contrast regime). The main result of the paperis to show, using asymptotic expansions with respect to ε and to small geometricperturbations of the optimal shape, that the global minimum of the first eigenvalue in lowcontrast regime is either a centered ball or the union of a centered ball and of acentered ring touching the boundary, depending on the prescribed volume ratio between thetwo materials.

In this paper we are concerned with questions of multiplicity and concentration behaviorof positive solutions of the elliptic problem

$$ (P_{\var})\hspace*{4cm} \left\{ \begin{array}{rcl} \mathcal{L}_{\var}u=f(u) \ \ \mbox{in} \ \ \R^{3},\\[1.5mm] u>0 \ \ \mbox{in} \ \ \R^{3},\\[1.5mm] u \in H^{1}(\R^3), \end{array} \right. $$$\mathrm{\left(}{\mathit{P}}_{\mathit{\epsilon }}\mathrm{\right)}\left\{\begin{array}{c}\\ {\mathrm{ℒ}}_{\mathit{\epsilon }}\mathit{u}\mathrm{=}\mathit{f}\mathrm{\left(}\mathit{u}\mathrm{\right)}\mathrm{in}{\mathrm{IR}}^{\mathrm{3}}\mathit{,}\\ \mathit{u}\mathit{>}\mathrm{0}\mathrm{in}{\mathrm{IR}}^{\mathrm{3}}\mathit{,}\\ \mathit{u}\mathrm{\in }{\mathit{H}}^{\mathrm{1}}\mathrm{\left(}{\mathrm{IR}}^{\mathrm{3}}\mathrm{\right)}\mathit{,}\end{array}\right.$

where ε is a small positiveparameter, f : ℝ → ℝ is a continuous function, $$ \mathcal{L}_{\var} $$${\mathrm{ℒ}}_{\mathit{\epsilon }}$ is a nonlocaloperator defined by

$$ \mathcal{L}_{\var}u=M\left(\dis\frac{1}{\var}\int_{\R^{3}}|\nabla u|^{2}+\frac{1}{\var^{3}}\dis\int_{\R^{3}}V(x)u^{2}\right)\left[-\var^{2}\Delta u + V(x)u \right], $$${\mathrm{ℒ}}_{\mathit{\epsilon }}\mathit{u}\mathrm{=}\mathit{M}\left(\frac{\mathrm{1}}{\mathit{\epsilon }}{\mathrm{\int }}_{{\mathrm{IR}}^{\mathrm{3}}}\mathrm{|}\mathrm{\nabla }\mathit{u}{\mathrm{|}}^{\mathrm{2}}\mathrm{+}\frac{\mathrm{1}}{{\mathit{\epsilon }}^{\mathrm{3}}}{\mathrm{\int }}_{{\mathrm{IR}}^{\mathrm{3}}}\mathit{V}\mathrm{(}\mathit{x}\mathrm{)}{\mathit{u}}^{\mathrm{2}}\right)\mathrm{[}\mathrm{-}{\mathit{\epsilon }}^{\mathrm{2}}\mathrm{\Delta }\mathit{u}\mathrm{+}\mathit{V}\mathrm{(}\mathit{x}\mathrm{)}{\mathit{u}}^{\mathrm{]}}\mathit{,}$

M : IR+ → IR+ andV : IR3 → IR are continuous functions whichverify some hypotheses.

In this paper, we employ the reduced basis method as a surrogate model for the solutionof linear-quadratic optimal control problems governed by parametrized elliptic partialdifferential equations. We present a posteriori error estimation and dualprocedures that provide rigorous bounds for the error in several quantities of interest:the optimal control, the cost functional, and general linear output functionals of thecontrol, state, and adjoint variables. We show that, based on the assumption of affineparameter dependence, the reduced order optimal control problem and the proposed boundscan be efficiently evaluated in an offline-online computational procedure. Numericalresults are presented to confirm the validity of our approach.

We consider the variational problem

        inf{αλ1(Ω) + βλ2(Ω) + (1 − α − β)λ3(Ω) | Ω open in ℝn, |Ω| ≤ 1},

for α, β ∈ [0, 1], α + β ≤ 1, where λk(Ω) is the kth eigenvalue of the Dirichlet Laplacian acting in L2(Ω) and |Ω| is the Lebesgue measure of Ω. We investigate for which values of α, β every minimiser is connected.

Regularity results for minimal configurations of variational problems involving both bulk and surface energies and subject to a volume constraint are established. The bulk energies are convex functions with p-power growth, but are otherwise not subjected to any further structure conditions. For a minimal configuration (u,E), Hölder continuity of the function u is proved as well as partial regularity of the boundary of the minimal set E. Moreover, full regularity of the boundary of the minimal set is obtained under suitable closeness assumptions on the eigenvalues of the bulk energies.

The adaptive stabilization is investigated for a class of coupled PDE-ODE systems with multiple uncertainties. The presence of the multiple uncertainties and the interaction between the sub-systems makes the systems to be considered more general and representative, and moreover it may result in the ineffectiveness of the conventional methods on this topic. Motivated by the existing literature, an infinite-dimensional backsteppping transformation with new kernel functions is first introduced to change the original system into a target system, from which the control design and performance analysis of the original system will become quite convenient. Then, by certainty equivalence principle and Lyapunov method, an adaptive stabilizing controller is successfully constructed, which guarantees that all the closed-loop system states are bounded while the original system states converging to zero. A simulation example is provided to validate the proposed method.

In this note we prove compactness for the Cahn–Hilliard functional without assumingcoercivity of the multi-well potential.

A dual-weighted residual approach for goal-oriented adaptive finite elements for a class of optimal control problems for elliptic variational inequalities is studied. The development is based on the concept of C-stationarity. The overall error representation depends on primal residuals weighted by approximate dual quantities and vice versa as well as various complementarity mismatch errors. Also, a priori bounds for C-stationary points and associated multipliers are derived. Details on the numerical realization of the adaptive concept are provided and a report on numerical tests including the critical cases of biactivity are presented.

This article aims at studying the controllability of a simplified fluid structure interaction model derived and developed in [C. Conca, J. Planchard and M. Vanninathan, RAM: Res. Appl. Math. John Wiley & Sons Ltd., Chichester (1995); J.-P. Raymond and M. Vanninathan, ESAIM: COCV 11 (2005) 180–203; M. Tucsnak and M. Vanninathan, Systems Control Lett. 58 (2009) 547–552]. This interaction is modeled by a wave equation surrounding a harmonic oscillator. Our main result states that, in the radially symmetric case, this system can be controlled from the outer boundary. This improves previous results [J.-P. Raymond and M. Vanninathan, ESAIM: COCV 11 (2005) 180–203; M. Tucsnak and M. Vanninathan, Systems Control Lett. 58 (2009) 547–552]. Our proof is based on a spherical harmonic decomposition of the solution and the so-called lateral propagation of the energy for 1d waves.

We consider the optimal distribution of several elastic materials in a fixed working domain. In order to optimize both the geometry and topology of the mixture we rely on the level set method for the description of the interfaces between the different phases. We discuss various approaches, based on Hadamard method of boundary variations, for computing shape derivatives which are the key ingredients for a steepest descent algorithm. The shape gradient obtained for a sharp interface involves jump of discontinuous quantities at the interface which are difficult to numerically evaluate. Therefore we suggest an alternative smoothed interface approach which yields more convenient shape derivatives. We rely on the signed distance function and we enforce a fixed width of the transition layer around the interface (a crucial property in order to avoid increasing “grey” regions of fictitious materials). It turns out that the optimization of a diffuse interface has its own interest in material science, for example to optimize functionally graded materials. Several 2-d examples of compliance minimization are numerically tested which allow us to compare the shape derivatives obtained in the sharp or smoothed interface cases.

In this note we show the characteristic function of every indecomposable setF in theplane is BVequivalent to the characteristic function a closed set $\mathrm{See\; Formula\; in\; PDF}$\hbox{See Formula in PDF} .We show by example this is false in dimension three and above. As a corollary to thisresult we show that for every ϵ > 0 a set of finite perimeter S can be approximated by aclosed subset $\mathrm{See\; Formula\; in\; PDF}$\hbox{See Formula in PDF} with finitely many indecomposablecomponents and with the property that $\mathrm{See\; Formula\; in\; PDF}$\hbox{See Formula in PDF} and $\mathrm{See\; Formula\; in\; PDF}$\hbox{See Formula in PDF} .We apply this corollary to give a short proof that locally quasiminimizing sets in theplane are BVlextension domains.