We study a parameter (σ)dependent relaxation of the Travelling Salesman Problem on  $\mathbb{R}^2$ .The relaxed problem is reduced to the Travelling Salesman Problemas $\sigma\rightarrow$ 0. For increasing σ it is also anordered clustering algorithm for a set of points in $\mathbb{R}^2$ .A dual formulation is introduced, which reduces the problem to aconvex optimization, provided the minimizer is in the domain ofconvexity of the relaxed functional. It is shown that this lastcondition is generically satisfied, provided σ is largeenough.

We propose a necessary and sufficient condition about the existence of variations, i.e.,of non trivial solutions $\eta\in W^{1,\infty}_0(\Omega)$ to the differential inclusion $\nabla\eta(x)\in-\nabla u(x)+{\bf D}$ .

In this paper we prove that every weakand strong localminimizer $u\in{W^{1,2}(\Omega,\mathbb{R}^3)}$ of the functional $I(u)=\int_\Omega|Du|^2+f({\rm Adj}Du)+g({\rm det}Du),$ where $ u:\Omega\subset\mathbb{R}^3\to \mathbb{R}^3$ , f grows like $|{\rm Adj}Du|^p$ , g growslike $|{\rm det}Du|^q$ and1<q<p<2, is $C^{1,\alpha}$ on an opensubset $\Omega_0$ of Ω such that ${\it meas}(\Omega\setminus \Omega_0)=0$ . Suchfunctionals naturally arise from nonlinear elasticity problems. The keypoint in order to obtain the partial regularity result is toestablish an energy estimate of Caccioppoli type, which is based onan appropriate choice of the test functions. The limit case $p=q\le 2$ is also treated for weak local minimizers.

The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form $\textrm{div}\,a(\nabla u)+F[u](x)=0,$ over the functions $u\in W^{1,1}(\Omega)$ that assume given boundary values ϕ on ∂Ω. The vector field $a:{\mathbb R}^n\to {\mathbb R}^n$ satisfies an ellipticity condition and for a fixed x, F[u](x) denotes a non-linear functional of u. In considering the same problem, Hartman and Stampacchia [Acta Math.115 (1966) 271–310] have obtained existence results in the space of uniformly Lipschitz continuous functions when ϕ satisfies the classical bounded slope condition. In a variational context, Clarke [Ann. Sc. Norm. Super. Pisa Cl. Sci.4 (2005) 511–530] has introduced a new type of hypothesis on the boundary condition ϕ: the lower (or upper) bounded slope condition. This condition, which is less restrictive than the previous one, is satisfied if ϕ is the restriction to ∂Ω of a convex function. We show that if a and F satisfy hypotheses similar to those of Hartman and Stampacchia, the lower bounded slope condition implies the existence of solutions in the space of locally Lipschitz continuous functions on Ω.

We show that local minimizers of functionals of the form $\int_{\Omega} \left[f(Du(x)) + g(x\,,u(x))\right]\,{\rm d}x$ ,  $u \in u_0 + W_0^{1,p}(\Omega)$ ,are locally Lipschitz continuous provided f is a convex function with $p-q$ growth satisfying a condition of qualified convexity at infinity and g is Lipschitz continuous in u. As a consequence of this, we obtain an existence result for a related nonconvex functional.

We approximate, in the sense ofΓ-convergence, free-discontinuity functionals with lineargrowth in the gradient by a sequence of non-local integralfunctionals depending on the average of the gradients on smallballs. The result extends to higher dimension what we already proved inthe one-dimensional case.

In [Progress Math.233 (2005)], David suggested the existence of a new type of global minimizers for the Mumford-Shah functional in $\mathbf{R}^3$ . The singular set of such a new minimizer belongs to a three parameters family of sets $(0<\delta_1,\delta_2,\delta_3<\pi)$ . We first derive necessary conditions satisfied by global minimizers of this family. Then we are led to study the first eigenvectors of the Laplace-Beltrami operator with Neumann boundary conditions on subdomains of $\mathbf{S}^2$ with three reentrant corners. The necessary conditions are constraints on the eigenvalue and on the ratios between the three singular coefficients of the associated eigenvector. We use numerical methods (Singular Functions Method and Moussaoui's extraction formula) to compute the eigenvalues and the singular coefficients. We conclude that there is no $(\delta_1,\delta_2,\delta_3)$ for which the necessary conditions are satisfied and this shows that the hypothesis was wrong.


We consider, in an open subset Ω of ${\mathbb R}^N$ , energies depending on the perimeter of a subset $E\subset\Omega$ (or some equivalent surface integral) and on a function u which isdefined only on $\Omega\setminus E$ . We compute the lower semicontinuous envelopeof such energies. This relaxation has to take intoaccount the fact that in the limit, the “holes” E maycollapse into a discontinuity of u, whose surface will be countedtwice in the relaxed energy. We discuss some situations where suchenergies appear, and give, as an application, a new proofof convergence for an extensionof Ambrosio-Tortorelli's approximation to the Mumford-Shah functional.


We consider the homogenization of both the parabolic and eigenvalue problems for a singularly perturbedconvection-diffusion equation in a periodic medium. All coefficients of the equation may vary both on themacroscopic scale and on the periodic microscopic scale. Denoting by ε the period, the potential or zero-orderterm is scaled as $\varepsilon^{-2}$ and the drift or first-order term is scaled as $\varepsilon^{-1}$ . Under a structuralhypothesis on the first cell eigenvalue, which is assumed to admit a unique minimum in the domain withnon-degenerate quadratic behavior, we prove an exponential localization at this minimum point. The homogenizedproblem features a diffusion equation with quadratic potential in the whole space.

This work deals with a two-dimensional inverse problem in the field of tomography. The geometry of an unknown inclusion has to be reconstructed from boundary measurements. In this paper, we extend previous results of R. Kress and his coauthors: the leading idea is to use the conformal mapping function as unknown. We establish an integrodifferential equation that the trace of the Riemann map solves. We write it as a fixed point equation and give conditions for contraction. We conclude with a series of numerical examples illustrating the performance of the method.

In this article, we build a mathematical model to understand theformation of a tree leaf. Our model is based on the idea that a leaftends to maximize internal efficiency by developing an efficienttransport system for transporting water and nutrients. The meaningof “the efficient transport system” may vary as the type of thetree leave varies. In this article, we will demonstrate that treeleaves have different shapes and venation patterns mainly becausethey have adopted different efficient transport systems.The efficient transport system of a tree leaf built here is amodified version of the optimal transport path, which was introducedby the author in [Comm. Cont. Math.5 (2003) 251–279; Calc. Var. Partial Differ. Equ.20 (2004) 283–299; Boundary regularity of optimal transport paths,Preprint] to study thephenomenon of ramifying structures in mass transportation. In thepresent paper, the cost functional on transport systems iscontrolled by two meaningful parameters. The first parameterdescribes the economy of scale which comes with transporting large quantities together, while thesecond parameter discourages the direction of outgoing veins at eachnode from differing much from the direction of the incoming vein.Under the same initial condition, efficient transport systemsmodeled by different parameters will provide tree leaves withdifferent shapes and different venation patterns.Based on this model, we also provide some computer visualization oftree leaves, which resemble many known leaves including the mapleand mulberry leaf. It demonstrates that optimal transportation playsa key role in the formation of tree leaves.

We prove uniform local energy estimates of solutions to the damped Schrödinger equation in exterior domains under the hypothesis of the ExteriorGeometric Control. These estimates are derived from the resolvent properties.

We are concerned with the optimal control of a nonlinear stochastic heat equation on a bounded real interval with Neumann boundary conditions. The specificity here is that both the control and the noise act on the boundary. We start by reformulating the state equation as an infinite dimensional stochastic evolution equation. The first main result of the paper is the proof of existence and uniqueness of a mild solution for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The C 1 regularity of such a solution is then used to construct the optimal feedback for the control problem. In order to overcome the difficulties arising from the degeneracy of the second order operator and from the presence of unbounded terms we study the HJB equation by introducing a suitable forward-backward system of stochastic differential equations as in the appraoch proposed in [Fuhrman and Tessitore, Ann. Probab.30 (2002) 1397-1465; Pardoux and Peng, Lect. Notes Control Inf. Sci.176 (1992) 200-217] for finite dimensional and infinite dimensional semilinear parabolic equations respectively.

We consider a control constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions occur during themodeling of diffuse-gray conductive-radiative heat transfer. After stating first-order necessary conditions, second-order sufficient conditions are derived that account for strongly active sets. These conditions ensure local optimality in an Ls -neighborhood of a reference function whereby the underlying analysis allows to use weaker norms than $L^\infty$ .

This paper provides KKT and saddle point optimality conditions, dualitytheorems and stability theorems for consistent convex optimization problemsposed in locally convex topological vector spaces. The feasible sets ofthese optimization problems are formed by those elements of a given closedconvex set which satisfy a (possibly infinite) convex system. Moreover, allthe involved functions are assumed to be convex, lower semicontinuous andproper (but not necessarily real-valued). The key result in the paper is thecharacterization of those reverse-convex inequalities which are consequenceof the constraints system. As a byproduct of this new versions of Farkas'lemma we also characterize the containment of convex sets in reverse-convexsets. The main results in the paper are obtained under a suitableFarkas-type constraint qualifications and/or a certain closedness assumption.

This paper considers the inversion problem related to themanipulation of quantumsystems using laser-matter interactions. The focusis on the identification of the field free Hamiltonian and/orthe dipole moment of aquantum system. The evolution of the system is given by the Schrödingerequation. The available data are observations as a function of timecorresponding to dynamics generated by electric fields. Thewell-posedness of the problem is proved, mainly focusing on the uniqueness ofthe solution. A numerical approach is also introduced with anillustration of its efficiency on a test problem.

An open-loop system of a multidimensional wave equationwith variable coefficients, partial boundary Dirichlet control andcollocated observation is considered. It is shown that the system iswell-posed in the sense of D. Salamon and regular in the sense of G.Weiss. The Riemannian geometry method is used in the proof ofregularity and the feedthrough operator is explicitly computed.

Given a one-parameter family $\{g_\lambda\colon\lambda\in [a,b]\}$ of semi Riemannian metrics on ann-dimensional manifold M, a family of time-dependent potentials $\{ V_\lambda\colon \lambda\in [a,b]\}$ and a family $\{\sigma_\lambda\colon \lambda\in [a,b]\} $ of trajectories connecting two points of the mechanical system defined by $(g_\lambda, V_\lambda)$ , we show that there are trajectories bifurcating from the trivial branch $\sigma_\lambda$ if the generalized Morse indices $\mu(\sigma_a)$ and $\mu(\sigma_b)$ are different. If the data are analytic we obtain estimates for the number of bifurcation points on the branch and, in particular, for the number of strictly conjugate points along a trajectory using an explicit computation of the Morse index inthe case of locally symmetric spaces and a comparison principle of Morse Schöenberg type.

We prove a relaxation theorem in BV for a non coercive functional with linear growth. No continuity of the integrand with respect to the spatial variable is assumed.

This errata corrects one error in the 2004 version of this paper [Mennucci, ESAIM: COCV10 (2004) 426–451].