The behavior of energy minimizers at the boundary of the domain is of great importance in the Van de Waals-Cahn-Hilliard theory for fluid-fluid phase transitions, since it describes the effect of the container walls on the configuration of the liquid. This problem, also known as the liquid-drop problem, was studied by Modica in [Ann. Inst. Henri Poincaré, Anal. non linéaire 4 (1987) 487–512], and in a different form by Alberti et al. in [Arch. Rational Mech. Anal. 144 (1998) 1–46] for a first-order perturbation model.This work shows that using a second-order perturbation Cahn-Hilliard-type model, the boundary layer is intrinsically connected with the transition layer in the interior of the domain. Precisely, considering the energies

$\[{\cal F}_{\varepsilon}(u) := \varepsilon^{3} \int_{\Omega} |D^{2}u|^{2} + \frac{1}{\varepsilon} \int_{\Omega} W (u) + \lambda_{\varepsilon} \int_{\partial \Omega} V(Tu),\]$

where u is a scalar density function and W and V are double-well potentials, the exact scaling law is identified in the critical regime, when $\varepsilon\lambda_{\varepsilon}^{\frac{2}{3}} \sim 1$.

We prove that any Kantorovich potential for the cost functionc = d2/2 on a Riemannian manifold (M, g) is locally semiconvexin the “region of interest”, without any compactness assumptionon M, nor any assumption on its curvature. Such a region ofinterest is of full μ-measure as soon as the starting measureμ does not charge n – 1-dimensional rectifiable sets.

This paper is mainlyconcerned with a class of optimal control problems of systemsgoverned by the nonlinear dynamic systems on time scales.Introducing the reasonable weak solution of nonlinear dynamicsystems, the existence of the weak solution for the nonlineardynamic systems on time scales and its properties are presented.Discussing L1-strong-weak lower semicontinuity of integralfunctional, we give sufficient conditions for the existence ofoptimal controls. Using integration by parts formula and Hamiltonianfunction on time scales, the necessary conditions of optimality arederived respectively. Some examples on continuous optimal controlproblems, discrete optimal control problems, mathematicalprogramming and variational problems are also presented for demonstration.

The Monge-Kantorovich problem is revisited by means of a variantof the saddle-point method without appealing to c-conjugates. Anew abstract characterization of the optimal plans is obtained inthe case where the cost function takes infinite values. It leadsus to new explicit sufficient and necessary optimality conditions.As by-products, we obtain a new proof of the well-knownKantorovich dual equality and an improvement of the convergence ofthe minimizing sequences.

The topological asymptotic analysis provides the sensitivity of a givenshape functional with respect to an infinitesimal domain perturbation, likethe insertion of holes, inclusions, cracks. In this work we present thecalculation of the topological derivative for a class of shape functionalsassociated to the Kirchhoff plate bending problem, when a circular inclusionis introduced at an arbitrary point of the domain. According to theliterature, the topological derivative has been fully developed for a widerange of second-order differential operators. Since we are dealing here witha forth-order operator, we perform a complete mathematicalanalysis of the problem.

We consider a time optimal control problem arisen from the optimalmanagement of a bioreactor devoted to the treatment ofeutrophicated water. We formulate this realistic problem as astate-control constrained time optimal control problem. Afteranalyzing the state system (a complex system of coupled partialdifferential equations with non-smooth coefficients foradvection-diffusion-reaction with Michaelis-Menten kinetics,modelling the eutrophication processes) we demonstrate theexistence of, at least, an optimal solution. Then we present adetailed derivation of a first order optimality condition(involving the corresponding adjoint systems) characterizing theseoptimal solutions. Finally, a numerical example is shown.

A zero-sum stochastic differential gameproblem on infinite horizon with continuous and impulse controls isstudied. We obtain the existence of the value of the game andcharacterize it as the unique viscosity solution of the associatedsystem of quasi-variational inequalities. We also obtain averification theorem which provides an optimal strategy of the game.

This paper studies the strong unique continuation property for theLamé system of elasticity with variable Lamé coefficientsλ, µ in three dimensions, ${\rm{div}}(\mu(\nabla u+\nablau^t))+ \nabla(\lambda{\rm{div}} u)+Vu=0$where λ and μ are Lipschitz continuous and V ∈ L∞. The method is based on the Carleman estimate with polynomial weights for the Lamé operator.

In this paper, we carry out the numerical analysis of adistributed optimal control problem governed by a quasilinearelliptic equation of non-monotone type. The goal is to prove thestrong convergence of the discretization of the problem by finiteelements. The main issue is to get error estimates for thediscretization of the state equation. One of the difficulties inthis analysis is that, in spite of the partial differentialequation has a unique solution for any given control, theuniqueness of a solution for the discrete equation is an openproblem.

This paper is devoted to the study of a coupled system which consists ofa wave equation and a heat equation coupled through a transmission conditionalong a steady interface. This system is a linearized model forfluid-structure interaction introduced by Rauch, Zhang and Zuazuafor a simple transmission condition and by Zhang and Zuazua for anatural transmission condition. Using an abstract theorem of Burq and a new Carleman estimate proved near the interface, wecomplete the results obtained by Zhang and Zuazua and by Duyckaerts.We prove, without a Geometric Control Condition, a logarithmic decayof the energy.

In the setting ofa real Hilbert space ${\cal H}$, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order evolutionequations

          ü(t) + γ$\dot{u}$(t) + ∇ϕ(u(t)) + A(u(t)) = 0,

where ∇ϕ is the gradient operator of a convexdifferentiable potential function ϕ : ${\cal H}\to \R$, A : ${\cal H}\to {\cal H}$ is a maximal monotone operator which is assumed to beλ-cocoercive, and γ > 0 is a damping parameter.Potential and non-potential effects are associated respectively to∇ϕ and A. Under condition λγ2 > 1, it is proved that each trajectory asymptotically weaklyconverges to a zero of ∇ϕ + A. This condition, whichonly involves the non-potential operator and the dampingparameter, is sharp and consistent with time rescaling. Passingfrom weak to strong convergence of the trajectories is obtained byintroducing an asymptotically vanishing Tikhonov-like regularizingterm. As special cases, we recover the asymptotic analysis of theheavy ball with friction dynamic attached to a convex potential, thesecond-order gradient-projection dynamic, and the second-orderdynamic governed by the Yosida approximation of a general maximalmonotone operator. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of constrained optimization,dynamical approach to Nash equilibria for noncooperative games, and asymptotic stabilization in the case of a continuum of equilibria.

Optimization problems with convex but non-smooth cost functional subject to an elliptic partial differential equation are considered.The non-smoothness arises from a L1-norm in the objective functional. The problem is regularized to permit the use of the semi-smooth Newtonmethod. Error estimates with respect to the regularization parameter are provided. Moreover, finite element approximations are studied. A-priori as well as a-posteriori error estimates are developed and confirmed by numerical experiments.

We consider the Dirichlet Laplacian in a thin curvedthree-dimensional rod. The rod is finite. Its cross-section isconstant and small, and rotates along the reference curve in anarbitrary way. We find a two-parametric set of the eigenvalues ofsuch operator and construct their complete asymptotic expansions. Weshow that this two-parametric set contains any prescribed number ofthe first eigenvalues of the considered operator. We obtain thecomplete asymptotic expansions for the eigenfunctions associatedwith these first eigenvalues.