Related papers: A characteristic approach to the quasi-normal mode…
We consider an overdetermined problem for a two phase elliptic operator in divergence form with piecewise constant coefficients. We look for domains such that the solution $u$ of a Dirichlet boundary value problem also satisfies the…
As the fingerprints of black holes, quasinormal modes are closely associated with many properties of black holes. Especially, the ringdown phase of gravitational waveforms from the merger of compact binary components can be described by…
Linear perturbations of spherically symmetric spacetimes in general relativity are described by radial wave equations, with potentials that depend on the spin of the perturbing field. In previous work we studied the quasinormal mode…
The null-timelike initial-boundary value problem for a hyperbolic system of equations consists of the evolution of data given on an initial characteristic surface and on a timelike worldtube to produce a solution in the exterior of the…
The resonant mode spectrum of the Kerr-Newman spacetime is presently unknown. These modes, called the quasinormal modes, play a central role in determining the stability of Kerr-Newman black holes and their response to perturbations. We…
Consider the Laplacian in a bounded domain in R^d with general (mixed) homogeneous boundary conditions. We prove that its eigenfunctions are `quasi-orthogonal' on the boundary with respect to a certain norm. Boundary orthogonality is proved…
We describe a general multiplier method to obtain boundary stabilization of the wave equation by means of a (linear or quasi-linear) Neumann feedback. This also enables us to get Dirichlet boundary control of the wave equation. This method…
In this work we revisit axial perturbations of spherically symmetric and non-rotating neutron stars. Although it has been object of many studies, it still offers new insights that are of potential interest for more realistic scenarios or in…
Quasinormal modes are excited during the ringdown phase of black holes after merger. Determination of quasinormal modes of rapidly rotating black holes in alternative theories of gravity has remained a challenge for a long time. Here we…
By using the Cowling approximation, quasi-radial modes of rotating general relativistic stars are computed along equilibrium sequences from non-rotating to maximally rotating models. The eigenfrequencies of these modes are decreasing…
In this paper, we consider the problem of nonlinear (in particular, saturated) stabilization of the high-dimensional wave equation with Dirichlet boundary conditions. The wave dynamics are subject to a dissipative nonlinear velocity…
We prove well-posedness results for the solution to an initial and boundary-value problem for an Allen-Cahn type equation describing the phenomenon of phase transitions for a material contained in a bounded and regular domain. The dynamic…
We cast the quantum chemistry problem of computing bound states as that of solving a set of auxiliary eigenvalue problems for a family of parameterized compact integral operators. The compactness of operators assures that their spectrum is…
We propose a new method to deal with the essential boundary conditions encountered in the deep learning-based numerical solvers for partial differential equations. The trial functions representing by deep neural networks are…
We prove global second-order regularity for a class of quasilinear elliptic equations, both with homogeneous Dirichlet and Neumann boundary conditions. A condition on the integrability of the second fundamental form on the boundary of the…
We study the Dirichlet problem for the semi--linear partial differential equations ${\rm div}\,(A\nabla u)=f(u)$ in simply connected domains $D$ of the complex plane $\mathbb C$ with continuous boundary data. We prove the existence of the…
Optimization problems with composite functions consist of an objective function which is the sum of a smooth and a (convex) nonsmooth term. This particular structure is exploited by the class of proximal gradient methods and some of their…
Initial-boundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of so-called unified transform techniques. The main obstruction to applying these methods…
We consider the initial-boundary value problems on $\mathbb{R}^{+}\times \mathbb{R}^{+}$ for one-dimension systems of quasilinear wave equations with null conditions. We show that for homogeneous Dirichlet boundary values and sufficiently…
We prove global existence of solutions to multiple speed, Dirichlet-wave equations with quadratic nonlinearities satisfying the null condition in the exterior of compact obstacles. This extends the result of our previous paper by allowing…