Antônio Luiz Pereira

Perturbation of the boundary in boundary-value problems of partial differential equations

Introduction 1. Geometrical preliminaries 2. Differential calculus of boundary perturbations 3. Examples using the implicit function theorem 4. Bifurcation problems 5. The transversality theorem 6. Generic perturbation of the boundary 7. Boundary operators for second-order elliptic equations 8. The method of rapidly-oscillating solutions.

Eigenvalues of the Laplacian on symmetric regions

It was shown by Micheletti [11] that the eigenvalues of the Laplacian with Dirichlet boundary conditions are simple for most boundedC3 regions. We show that this is not true if we restrict our attention to the subset of regions which are invariant under a group of symmetries. We also prove, for some groups, that generically the eigenspaces are irreducible spaces for the action of the group.

A nonlinear elliptic problem with terms concentrating in the boundary

In this paper, we investigate the behavior of a family of steady-state solutions of a nonlinear reaction diffusion equation when some reaction and potential terms are concentrated in a e-neighborhood of a portion Γ of the boundary. We assume that this e-neighborhood shrinks to Γ as the small parameter e goes to zero. Also, we suppose the upper boundary of this e-strip presents a highly oscillatory behavior. Our main goal here was to show that this family of solutions converges to the solutions of a limit problem, a nonlinear elliptic equation that captures the oscillatory behavior. Indeed, the reaction term and concentrating potential are transformed into a flux condition and a potential on Γ, which depends on the oscillating neighborhood. Copyright

A generic property for the eigenfunctions of the Laplacian

In this work we show that, generically in the set of

Generic Simplicity for the Solutions of a Nonlinear Plate Equation

\mathcal{C}^2

Existence, regularity and upper semicontinuity of pullback attractors for the evolution process associated to a neural field model

bounded regions of

Continuity of attractors for a family of \({\mathcal {C}}^1\) perturbations of the square

\mathbb R^n

Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations: Frontmatter

,

Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations: Contents

n \geq 2

Perturbation of the Boundary in Boundary-Value Problems of Partial Differential Equations: Index

, the inequality