Fernando Cardoso

Exponential dichotomy and boundedness for retarded functional difference equations

We characterize the exponential dichotomy of difference equations with infinite delay. We apply the results to study the robustness of exponential dichotomy. This kind of dichotomy gives us relevant information about boundedness of solutions for several perturbed quasi linear systems with infinite delay. Applications to Volterra difference equations are shown.

ASYMPTOTICS OF THE NUMBER OF RESONANCES IN THE TRANSMISSION PROBLEM

We consider the resonances for the transmission problem associated with a strictly convex transparent obstacle. Under some natural assumptions we show that there is a free of resonances region in the complex upper half plane given by {C ≤ Im λ ≤ C 1|λ|1/3 − C 2}, where C, C 1 and C 2 are positive constants. Moreover, we obtain asymptotics for the number of resonances counted with multiplicities in the region {0 < Im λ ≤ C, 0 < Re λ ≤ r} as r → ∞, where C > 0 is the same constant as above. *Partially supported by CNPq (Brazil).

High Frequency Resolvent Estimates and Energy Decay of Solutions to the Wave Equation

Let (M, g) be an n-dimensional non-compact, connected Riemannian manifold with a Riemannian metric g of class C(M) and a compact C-smooth boundary ∂M (which may be empty), of the form M = X0 ∪X, where X0 is a compact, connected Riemannian manifold with a metric g|X0 of class C (X0) with a compact boundary ∂X0 = ∂M ∪ ∂X, ∂M ∩ ∂X = ∅, X = [r0,+∞)× S, r0 1, with metric g|X := dr + σ(r). Here (S, σ(r)) is an n − 1 dimensional compact Riemannian manifold without boundary equipped with a family of Riemannian metrics σ(r) depending smoothly on r which can be written in any local coordinates θ ∈ S in the form

Optimal Dispersive Estimates for the Wave Equation with Potentials in Dimensions 4 ≤ n ≤ 7

We prove optimal dispersive estimates for the wave group for a class of real-valued potentials , 4 ≤ n ≤ 7, such that , , .

Boundary stabilization of transmission problems

We study the transmission problem in bounded domains with dissipative boundary conditions. Under some natural assumptions, we prove uniform bounds of the corresponding resolvents on the real axis at high frequency and, as a consequence, we obtain regions free of eigenvalue. To this end, we extend the result of Cardoso et al. [“Distribution of resonances and local energy decay in the transmission problem. II,” Math. Res. Lett. 6, 377 (1999)] under more general assumptions. As an application, we get exponential decay of the energy of the solutions of the corresponding mixed boundary value problems.

Resolvent estimates for perturbations by large magnetic potentials

We prove optimal high-frequency resolvent estimates for self-adjoint operators of the form G = −Δ + ib(x) · ∇ + i∇ · b(x) + V(x) on L2(Rn), n ⩾ 3, where b(x) and V(x) are large magnetic and electric potentials, respectively. No continuity of the magnetic potential is assumed.

Wave front sets, fourier integrals and propagation of singularities

Pseudo-differential operators have been developed as a tool for the study of elliptic differential equations. Suitably extended versions are also applicable to hypoelliptic equations but their value is rather limited in genuinely non-elliptic problems. Many operators arising in the solution of differential equations are not pseudolocal. For instance, if L is a hyperbolic operator, say the wave operator, c~ 2 c~ 2 L = Z ~t 2 ~3X~

High frequency dispersive estimates for the Schrödinger equation in high dimensions

We prove optimal dispersive estimates at high frequency for the Schrodinger group with real-valued potentials

Global Solvability and Hypoellipticity of Abstract Complexes and Equations

V(x)=O(|x|^{-\delta})

Uniform estimates of the resolvent of the Laplace-Beltrami operator on infinite volume Riemannian manifolds. II

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