The aim of this paper is to provide a rigorous variational formulation forthe detection of points in 2-d biological images. To this purposewe introduce a new functional whose minimizers give the points we want to detect. Then we define an approximating sequence of functionals forwhich we prove the Γ-convergence to the initial one.

The Cahn-Hilliard variational inequality is a non-standard parabolic variational inequality of fourth order for which straightforward numerical approaches cannot be applied. We propose a primal-dual active set method which can be interpreted as a semi-smooth Newton method as solution technique for the discretized Cahn-Hilliard variational inequality. A (semi-)implicit Euler discretization is used in time and a piecewise linear finite element discretization of splitting type is used in space leading to a discrete variational inequality of saddle point type in each time step. In each iteration of the primal-dual active set method a linearized system resulting from the discretization of two coupled elliptic equations which are defined on different sets has to be solved. We show local convergence of the primal-dual active set method and demonstrate its efficiency with several numerical simulations.

This paper deals with a model describing damage processes in a (nonlinear) elastic body which is in contact with adhesion with a rigid support. On the basis of phase transitions theory, we detailthe derivation of the model written in terms of a PDE system, combined with suitable initial and boundary conditions. Some internal constraints on the variables are introduced in the equations and on the boundary, to get physical consistency. We prove theexistence of global in time solutions (to a suitable variational formulation) of therelated Cauchy problem by means of a Schauder fixed point argument, combinedwith monotonicity and compactness tools. We also perform an asymptotic analysis of the solutions as the interfacial damage energy (between the body and the contact surface) goes to +∞.

An optimal control problem forsemilinear parabolic partial differential equations is considered.The control variable appears in the leading term of the equation.Necessary conditions for optimal controls are established by themethod of homogenizing spike variation. Results for problems withstate constraints are also stated.

In the present paper, we consider nonlinear optimal control problems with constraints on the state of the system. We are interested in the characterization of the value function without any controllability assumption. In the unconstrained case, it is possible to derive a characterization of the value function by means of a Hamilton-Jacobi-Bellman (HJB) equation. This equation expresses the behavior of the value function along the trajectories arriving or starting from any position x. In the constrained case, when no controllability assumption is made, the HJB equation may have several solutions. Our first result aims to give the precise information that should be added to the HJB equation in order to obtain a characterization of the value function. This result is very general and holds even when the dynamics is not continuous and the state constraints set is not smooth. On the other hand we study also some stability results for relaxed or penalized control problems.

We consider the numerical solution, in two- and three-dimensionalbounded domains, of the inverse problem for identifying the locationof small-volume, conductivity imperfections in a medium with homogeneousbackground. A dynamic approach, based on the wave equation, permitsus to treat the important case of “limited-view” data. Our numericalalgorithm is based on the coupling of a finite element solution ofthe wave equation, an exact controllability method and finally a Fourierinversion for localizing the centers of the imperfections. Numericalresults, in 2- and 3-D, show the robustness and accuracy of the approachfor retrieving randomly placed imperfections from both complete andpartial boundary measurements.

Using the tool of two-scale convergence, we provide a rigorous mathematical setting for the homogenization result obtained by Fleck and Willis [J. Mech. Phys. Solids 52 (2004) 1855–1888] concerning the effective plastic behaviour of a strain gradient composite material. Moreover, moving from deformation theory to flow theory, we prove a convergence result for the homogenization of quasistatic evolutions in the presence of isotropic linear hardening.

In this paper, a $W^{-1,N'}$ estimate of the pressure is derived when its gradient is the divergence of a matrix-valued measure on $\mathbb R^N$, or on a regular bounded open set of $\mathbb R^N$. The proof is based partially on the Strauss inequality [Strauss, Partial Differential Equations: Proc. Symp. Pure Math. 23 (1973) 207–214] in dimension two, and on a recent result of Bourgain and Brezis [J. Eur. Math. Soc. 9 (2007) 277–315] in higher dimension. The estimate is used to derive a representation result for divergence free distributions which read as the divergence of a measure, and to prove an existence result for the stationary Navier-Stokes equation when the viscosity tensor is only in L1.

This work studies the null-controllability of a class of abstract parabolic equations. The main contribution in the general caseconsists in giving a short proof of an abstract version of a sufficient condition for null-controllability which has been proposed by Lebeau and Robbiano. We do not assume that the control operator is admissible. Moreover, we give estimates of the control cost.In the special case of the heat equation in rectangular domains, we provide an alternative way to check the Lebeau-Robbiano spectral condition. We then show that the sophisticated Carleman and interpolation inequalities used in previous literature may be replaced by a simple result of Turán. In this case, we provide explicit values for the constants involved in the above mentioned spectral condition. As far as we are aware, this is the first proof of the null-controllability of the heat equation with arbitrary control domain in a n-dimensional open set which avoids Carleman estimates.

Exact controllabilityresults for a multilayer plate system are obtained from the method of Carleman estimates.The multilayer plate system is a natural multilayer generalization of a classical three-layer “sandwichplate” system due to Rao and Nakra. The multilayer version involves a number ofLamé systems for plane elasticity coupled with a scalar Kirchhoff plate equation. The plate is assumed to be either clamped or hinged and controlsare assumed to be locally distributed in a neighborhood of a portion of the boundary. The Carleman estimates developed for thecoupled system are based on some new Carleman estimates for the Kirchhoff plate as well as some known Carleman estimates due to Imanuvilov and Yamamoto for the Lamé system.

Let $u\in\phi+ W_0^{1,1}(\Omega)$ be a minimum for

$\[I(v)=\int_{\Omega}g(x,v(x))+f(\nabla v(x))\,{\rm d}x\]$

where f is convex, $v\mapsto g(x,v)$ is convex for a.e. x. We prove that u shares the same modulus of continuity of ϕ whenever Ω is sufficiently regular, the right derivative of g satisfies a suitable monotonicity assumption and the following inequality holds

$\forall \gamma\in\partial\Omega\qquad |u(x)-\phi(\gamma)|\le\omega(|x-\gamma|) \quad\text{a.e. }x\in\Omega.$

This result generalizes the classical Haar-Rado theorem forLipschitz functions.

This paper deals with feedback stabilization of second order equations ofthe form

ytt + A0y + u (t) B0y (t) = 0, t ∈ [0, +∞[,

where A0 is a densely defined positive selfadjoint linear operator on areal Hilbert space H, with compact inverse and B0 is a linear map in diagonal form. It isproved here that the classical sufficient ad-condition of Jurdjevic-Quinn andBall-Slemrod with the feedback control u = ⟨yt, B0y⟩Himplies thestrong stabilization. This result is derived from a general compactnesstheorem for semigroup with compact resolvent and solves several open problems.

Recall that a smooth Riemannian metric on a simply connected domain canbe realized as the pull-back metric of an orientation preserving deformation ifand only if the associated Riemann curvature tensor vanishes identically.When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem byintroducing the non-Euclidean version of the nonlinear elasticity functional, and establish its Γ-convergence under the properscaling. As a corollary, we obtain new necessary and sufficient conditions for existence of a W2,2 isometric immersion of a given 2d metricinto $\mathbb R^3$.

The maximum principle for optimal control problems of fully coupledforward-backward doubly stochastic differential equations (FBDSDEs in short)in the global form is obtained, under the assumptions that the diffusioncoefficients do not contain the control variable, but the control domainneed not to be convex. We apply our stochastic maximum principle (SMP inshort) to investigate the optimal control problems of a class of stochasticpartial differential equations (SPDEs in short). And as an example of theSMP, we solve a kind of forward-backward doubly stochastic linear quadraticoptimal control problems as well. In the last section, we use the solutionof FBDSDEs to get the explicit form of the optimal control for linearquadratic stochastic optimal control problem and open-loop Nash equilibriumpoint for nonzero sum stochastic differential games problem.

In this paper we analyse an approximate controllability result for a nonlinear population dynamics model. In this model the birth term is nonlocal and describes the recruitment process in newborn individuals population, and the control acts on a small open set of the domain and corresponds to an elimination or a supply of newborn individuals. In our proof we use a unique continuation property for the solution of the heat equation and the Kakutani-Fan-Glicksberg fixed point theorem.