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A Liouville theorem for minimizers with finite potential energy for the vectorial Allen-Cahn equation
Get PDFWe prove that if a globally minimizing solution to the vectorial Allen-Cahn equation has finite potential energy, then it is a constant
On a selection principle for multivalued semiclassical flows
Get PDFWe study the semiclassical behaviour of solutions of a Schr ̈odinger equation with a scalar po- tential displaying a conical singularity. When a pure state interacts strongly with the singularity of the flow, there are several possible classical evolutions, and it is not known whether the semiclassical limit cor- responds to one of them. Based on recent results, we propose that one of the classical evolutions captures the semiclassical dynamics; moreover, we propose a selection principle for the straightforward calculation of the regularized semiclassical asymptotics. We proceed to investigate numerically the validity of the proposed scheme, by employing a solver based on a posteriori error control for the Schr ̈odinger equation. Thus, for the problems we study, we generate rigorous upper bounds for the error in our asymptotic approximation. For 1-dimensional problems without interference, we obtain compelling agreement between the regularized asymptotics and the full solution. In problems with interference, there is a quantum effect that seems to survive in the classical limit. We discuss the scope of applicability of the proposed regularization approach, and formulate a precise conjecture
Asymptotic Solutions of the Phase Space Schrodinger Equation: Anisotropic Gaussian Approximation
Get PDFWe consider the singular semiclassical initial value problem for the phase space Schrodinger equation. We approximate semiclassical quantum evolution in phase space by analyzing initial states as superpositions of Gaussian wave packets and applying individually semiclassical anisotropic Gaussian wave packet dynamics, which is based on the the nearby orbit
approximation; we accordingly construct a semiclassical approximation of the phase space propagator, semiclassical wave packet propagator, which admits WKBM semiclassical states as initial data. By the semiclassical propagator we
construct asymptotic solutions of the phase space Schrodinger equation, noting the connection of this construction to the initial value repsresentations for the
Schrodinger equation
The heteroclinic connection problem for general double-well potentials
Get PDFBy variational methods, we provide a simple proof of existence of a heteroclinic orbit to a second order Hamiltonian ODE that connects the two global minima of a double-well potential. Moreover, we consider several inhomogeneous extensions
Regularity of weak solutions to rate-independent systems in one-dimension
Get PDFWe show that under some appropriate assumptions, every weak solution (e.g.
energetic solution) to a given rate-independent system is of class SBV, or has fi�nite jumps, or is even piecewise C1. Our assumption is essentially imposed on the energy functional, but not convexity is required
Asymptotic Solutions of the Phase Space Schrodinger Equation: Anisotropic Gaussian Approximation
Get PDFWe consider the singular semiclassical initial value problem for the phase space Schrodinger equation. We approximate semiclassical quantum evolution in phase space by analyzing initial states as superpositions of Gaussian wave packets and applying individually semiclassical anisotropic Gaussian wave packet dynamics, which is based on the the nearby orbit
approximation; we accordingly construct a semiclassical approximation of the phase space propagator, semiclassical wave packet propagator, which admits WKBM semiclassical states as initial data. By the semiclassical propagator we
construct asymptotic solutions of the phase space Schrodinger equation, noting the connection of this construction to the initial value repsresentations for the
Schrodinger equation
Optimal energy growth lower bounds for a class of solutions to the vectorial Allen-Cahn equation
Get PDFWe prove optimal lower bounds for the growth of the energy over balls of minimizers to the vectorial Allen-Cahn energy in two spatial dimensions, as the radius tends to infinity. In the case of radially symmetric solutions, we can prove a stronger result in all dimensions
On periodic orbits in a slow-fast system with normally elliptic slow manifold
Get PDFIn this note we consider the bifurcation of a singular homoclinic orbit to periodic ones in a 4-dimensional slow-fast system of ordinary differential equations, having a 2-dimensional normally elliptic slow manifold, originally studied by Feckan and Rothos. Assuming an extra degree of differentiability on the system, we can refine their perturbation scheme, in particular the choice of approximate
solution, and obtain improved estimates
Signal to noise ratio analysis in virtual source array imaging
Get PDFWe consider correlation-based imaging of a reflector located on one side of a passive array where
the medium is homogeneous. On the other side of the array the illumination by remote impulsive sources
goes through a strongly scattering medium. It has been shown in [J. Garnier and G. Papanicolaou, Inverse Problems 28 (2012), 075002] that
migrating the cross correlations of the passive array gives an image whose resolution is as good as if
the array was active and the array response matrix was that of a homogeneous medium.
In this paper we study the signal to noise ratio of the image as a function of statistical properties of the
strongly scattering medium, the signal bandwidth and the source and passive receiver array characteristics.
Using a Kronecker model for the strongly scattering medium we show that image resolution is as
expected and that the signal to noise ratio can be computed in an essentially explicit way. We
show with direct numerical simulations using full wave propagation solvers in random media that
the theoretical predictions based on the Kronecker model are accurate
Relative entropy in diffusive relaxation
Get PDFWe establish convergence in the diffusive limit from entropy weak solutions of
the equations of compressible gas dynamics with friction to the porous media equation away from vacuum.
The result is based on a Lyapunov type of functional provided by a calculation of the relative entropy.
The relative entropy method is also employed to establish convergence from entropic weak solutions
of viscoelasticity with memory to the system of viscoelasticity of the rate-type