A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners

AbstractFourth order hinged plate type problems are usually solved via a system of two second order equations. For smooth domains such an approach can be justified. However, when the domain has a concave corner the bi-Laplace problem with Navier boundary conditions may have two different types of solutions, namely u1 with u1,Δu1∈H˚1 and u2∈H2∩H˚1. We will compare these two solutions. A striking difference is that in general only the first solution, obtained by decoupling into a system, preserves positivity, that is, a positive source implies that the solution is positive. The other type of solution is more relevant in the context of the hinged plate. We will also address the higher-dimensional case. Our main analytical tools will be the weighted Sobolev spaces that originate from Kondratiev. In two dimensions we will show an alternative that uses conformal transformation. Next to rigorous proofs the results are illustrated by some numerical experiments for planar domains

Asymptotics of solutions to Joukovskii-Kutta type problems at infinity

We investigate the behavior at infinity of solutions to Joukovskii-Kutta-type problems, arising in the linearized lifting surface theory. In these problems one looks for the perturbation velocity potential induced by the presence of a wing in a basic flow within the scope of a linearized theory and for the wing circulation. We consider at first the pure two-dimensional case, then the three-dimensional case, and finally we show in the case of a time-harmonically oscillating wing in ℝ3 in a weakly damping gas the exponential decay of solutions of the Joukovskii-Kutta problem

Topological derivatives for semilinear elliptic equations

International audienceThe form of topological derivatives for integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in the $L^{\infty}$ norm are obtained. Results of numerical experiments which confirm the theoretical convergence rate are presented

Spectrum for a small inclusion of negative material

International audienceWe study a spectral problem (P δ) for a diffusion like equation in a 3D domain Ω. The main originality lies in the presence of a parameter σ δ , whose sign changes on Ω, in the principal part of the operator we consider. More precisely, σ δ is positive on Ω except in a small inclusion of size δ > 0. Because of the sign-change of σ δ , for all δ > 0 the spectrum of (P δ) consists of two sequences converging to ±∞. However, at the limit δ = 0, the small inclusion vanishes so that there should only remain positive spectrum for (P δ). What happens to the negative spectrum? In this paper, we prove that the positive spectrum of (P δ) tends to the spectrum of the problem without the small inclusion. On the other hand, we establish that each negative eigenvalue of (P δ) behaves like δ −2 µ for some constant µ < 0. We also show that the eigenfunctions associated with the negative eigenvalues are localized around the small inclusion. We end the article providing 2D numerical experiments illustrating these results