Virginie Bonnaillie-Noël

Interactions between moderately close inclusions for the Laplace equation

The presence of small inclusions modifies the solution of the Laplace equation posed in a reference domain Ω0. This question has been studied extensively for a single inclusion or well-separated inclusions. In two-dimensional situations, we investigate the case where the distance between the holes tends to zero but remains large with respect to their characteristic size. We first consider two perfectly insulated inclusions. In this configuration we give a complete multiscale asymptotic expansion of the solution to the Laplace equation. We also address the situation of a single inclusion close to a singular perturbation of the boundary ∂Ω0. We also present numerical experiments implementing a multiscale superposition method based on our first order expansion.

SUPERCONDUCTIVITY IN DOMAINS WITH CORNERS

We study the two-dimensional Ginzburg-Landau functional in a domain with corners for exterior magnetic field strengths near the critical field where the transition from the superconducting to the normal state occurs. We discuss and clarify the definition of this field and obtain a complete asymptotic expansion for it in the large

Aharonov–Bohm Hamiltonians, isospectrality and minimal partitions

\kappa

Numerical Analysis of Nodal Sets for Eigenvalues of Aharonov–Bohm Hamiltonians on the Square with Application to Minimal Partitions

regime. Furthermore, we discuss nucleation of superconductivity at the boundary.

On Generalized Ventcel's Type Boundary Conditions for Laplace Operator in a Bounded Domain

The spectral analysis of Aharonov–Bohm Hamiltonians with flux 1 leads surprisingly to a new insight on some questions of isospectrality appearing for example in Jakobson et al (2006 J. Comput. Appl. Math. 194 141–55) and Levitin et al (J. Phys. A: Math. Gen. 39 2073–82) and of minimal partitions (Helffer et al 2009 Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire 26 101–38). We will illustrate this point of view by discussing the question of spectral minimal 3-partitions for the rectangle − a , a × − b , b , with 0 <a b. It has been observed in Helffer et al (2009 Ann. Inst. H. Poincar´ e Anal. Non Lin´ 26 101–38) that when 0 < a < √ 3 the minimal 3-partition is obtained by the three nodal domains of the third eigenfunction corresponding to the three rectangles − a , a × − b , − b 6 , − a , a × − b , b and − a , a × b , b . We will describe a possible mechanism of transition for increasing a between these nodal minimal 3-partitions and non-nodal minimal 3-partitions at the value √ 3 8 and discuss the existence of symmetric candidates for giving minimal 3partitions when √ 3 < a 1. Numerical analysis leads very naturally to nice questions of isospectrality which are solved by the introduction of Aharonov– Bohm Hamiltonians or by going on the double covering of the punctured rectangle.

Computing the steady states for an asymptotic model of quantum transport in resonant heterostructures

This paper is devoted to presenting numerical simulations and a theoretical interpretation of results for determining the minimal k-partitions of a domain Ω as considered in [Helffer et al. 09]. More precisely, using the double-covering approach introduced by B. Helffer, M. and T. Hoffmann-Ostenhof, and M. Owen and further developed for questions of isospectrality by the authors in collaboration with T. Hoffmann-Ostenhof and S. Terracini in [Helffer et al. 09, Bonnaillie-Noël et al. 09], we analyze the variation of the eigenvalues of the one-pole Aharonov–Bohm Hamiltonian on the square and the nodal picture of the associated eigenfunctions as a function of the pole. This leads us to discover new candidates for minimal k-partitions of the square with a specific topological type and without any symmetric assumption, in contrast to our previous works [Bonnaillie-Noël et al. 10, Bonnaillie-Noël et al. 09]. This illustrates also recent results of B. Noris and S. Terracini; see [Noris and Terracini 10]. This finally supports or disproves conjectures for the minimal 3- and 5-partitions on the square.

Interactions between moderately close circular inclusions: the Dirichlet-Laplace equation in the plane

Ventcel boundary conditions are second order differential conditions that appear in asymptotic models. Like Robin boundary conditions, they lead to well-posed variational problems under a sign condition of a coefficient. Nevertheless, situations where this condition is violated appeared in several works. The well-posedness of such problems was still open. This manuscript establishes, in the generic case, the existence and uniqueness of the solution for the Ventcel boundary value problem without the sign condition. Then we consider perforated geometries and give conditions to remove the genericity restriction.

Spectral minimal partitions for a family of tori

In this article, we propose a rapid method to compute the steady states, including bifurcation diagrams, of resonant tunneling heterostructures in the far from equilibrium regime. Those calculations are made on a simplified model which takes into account the characteristic quantities which arise from an accurate asymptotic analysis of the nonlinear Schrodinger-Poisson system. After a summary of the existing theoretical results, the asymptotic model is explicitly adapted to physically realistic situations and numerical results are shown in various cases.

Small defects in mechanics

The presence of small inclusions or of a surface defect modifies the solution of the Laplace equation posed in a reference domain. If the characteristic size of the perturbation is small, then one can expect that the solution of the problem posed on the perturbed geometry is close to the solution of the reference shape. Asymptotic expansion with respect to that small parameter -the characteristic size of the perturbation- can then be performed. We consider in the present work the case of two defects with Dirichlet boundary conditions in a bidimensional domain. For the simplicity of the presentation, we assume that the defects we are considering are disks. We build an asymptotic expansion of the solution of the Laplace problem in perturbed domains. We will consider two unstudied cases: In the first case, we are considering two small holes around two fixed points (the distance between both is hence fixed). For the cases of Neumann boundary condition or of Dirichlet boundary conditions in dimension at least three, this cases can be treated by separating each hole through cut-off functions and hence reducing it to the single inclusion case. Here, the presence of the logarithmic term prohibits this approach and the interaction between the holes has to be studied. In the second case, the distance between the centers collapses to 0 slower than the size of the inclusions. The interaction between the two holes are then stronger and we will prove that the leading order of the asymptotic expansion is then modified.

Artificial conditions for the linear elasticity equations

ABSTRACT We study partitions of the two-dimensional flat torus into k domains, with b a real parameter in (0, 1] and k an integer. We look for partitions which minimize the energy, defined as the largest first eigenvalue of the Dirichlet Laplacian on the domains of the partition. We are in particular interested in the way these minimal partitions change when b is varied. We present here an improvement, when k is odd, of the results on transition values of b established by B. Helffer and T. Hoffmann-Ostenhof (2014) and state a conjecture on those transition values. We establish an improved upper bound of the minimal energy by explicitly constructing hexagonal tilings of the torus. These tilings are close to the partitions obtained from a systematic numerical study based on an optimization algorithm adapted from B. Bourdin, D. Bucur, and É. Oudet (2009). These numerical results also support our conjecture concerning the transition values and give better estimates near those transition values.