Jeffrey Rauch

Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary

For the observation or control of solutions of second-order hyperbolic equation in

Potential and scattering theory on wildly perturbed domains

\mathbb{R}_t \times \Omega

Symmetric positive systems with boundary characteristic of constant multiplicity

, Ralston’s construction of localized states [Comm. Pure Appl. Math., 22 (1969), pp. ...

Perturbation theory for eigenvalues and resonances of Schrodinger hamiltonians

Abstract We study the potential, scattering, and spectral theory associated with boundary value problems for the Laplacian on domains which are perturbed in very irregular fashions. Of particular interest are problems in which a “thin” set is deleted and the behavior of the Laplace operator changes very little, and problems where many tiny domains are deleted. In the latter case the “clouds” of tiny obstacles may tend to disappear, to solidify, or to produce an intermediate effect, depending on the relative numbers and sizes of the tiny domains. These phenomena vary according to the specific boundary value problem and in many cases their behavior is contrary to crude intuitive guesses.

Focusing at a point and absorption of nonlinear oscillations

The theory of maximal positive boundary value problems for symmetric positive systems is developed assuming that the boundary is characteristic of constant multiplicity. No such hypothesis is needed on a neighborhood of the boundary. Both regularity theorems and mixed initial boundary value problems are discussed. Many classical ideas are sharpened in the process.

Jump discontinuities of semilinear, strictly hyperbolic systems in two variables: Creation and propagation

Suppose that e2ϵ|x|V ∈ ReLP(R3) for some p > 2 and for g ∈ R, H(g) = − Δ + g V, H(g) = −Δ + gV. The main result, Theorem 3, uses Puiseaux expansions of the eigenvalues and resonances of H(g) to study the behavior of eigenvalues λ(g) as they are absorbed by the continuous spectrum, that is λ(g) ↗6 0 as g ↘5 g0 > 0. We find a series expansion in powers of (g − g0)12, λ(g) = ∑n = 2∞ an(g − g0)n2 whose values for g < g0 correspond to resonances near the origin. These resonances can be viewed as the traces left by the just absorbed eigenvalues.

BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one

Several recent papers give rigorous justifications of weakly nonlinear geometric optics. All of them consider oscillating wave trains on domains where focusing phenomena do not exist, either because the space dimension is equal to one, or thanks to a coherence assumption on the phases. This paper is devoted to a study of some nonlinear effects of focusing. In a previous paper, the authors have given a variety of examples which show how focusing in nonlinear equations can spoil even local existence in the sense that the domain of existence shrinks to zero as the wavelength decreases to zero. On the other hand, there are many problems for which global existence is known and in those cases it is natural to ask what happens to oscillations as they pass through a focus. The main goal of this paper is to present such a study for some strongly dissipative semilinear wave equations and spherical wavefronts which focus at the origin. We show that the strongly nonlinear phenomenon which is produced is that oscillations are killed by the simultaneous action of focusing and dissipation. Our study relies on the analysis of Young measures and two-scale Young measures associated to sequences of solutions. The main step is to prove that these measures satisfy appropriate transport equations. Then, their variances are shown to satisfy differential inequalities which imply a propagation result for their support.

Global Solutions to Maxwell Equations in a Ferromagnetic Medium

AbstractThe creation and propagation of jump discontinuities in the solutions of semilinear strictly hyperbolic systems is studied in the case where the initial data has a discrete set, {xi}i=1n, of jump discontinuities. LetS be the smallest closed set which satisfies:(i)S is a union of forward characteristics.(ii)S contains all the forward characteristics from the points {xi}i=1n.(iii)if two forward characteristics inS intersect, then all forward characteristics from the point of intersection lie inS. We prove that the singular support of the solution lies inS. We derive a sum law which gives a lower bound on the smoothness of the solution across forward characteristics from an intersection point. We prove a sufficient condition which guarantees that in many cases the lower bound is also an upper bound.

Hyperbolic partial differential equations and geometric optics

AbstractWe show that for most non-scalar systems of conservation laws in dimension greater than one, one does not have BV estimates of the form