Related papers: Partial balayage for the Helmholtz equation
In this paper, we introduce quadrature domains for the Helmholtz equation. We show existence results for such domains and implement the so-called partial balayage procedure. We also give an application to inverse scattering problems, and…
The primary objective of this paper is to explore the multi-phase variant of quadrature domains associated with the Helmholtz equation, commonly referred to as $k$-quadrature domains. Our investigation employs both the minimization problem…
The theory of quadrature domains for harmonic functions and the Hele-Shaw problem of the fluid dynamics are related subjects of the complex variables and mathematical physics. We present results generalizing the above subjects for elliptic…
A general theory of partial balayage on Riemannian manifolds is developed, with emphasis on compact manifolds. Partial balayage is an operation of sweeping measures, or charge distributions, to a prescribed density, and it is closely…
In this note, we discuss the quantum Hele-Shaw flow, a random measure process in the complex plane introduced by the physicists P.Wiegmann, A. Zabrodin, et al. This process arises in the theory of electronic droplets confined to a plane…
We prove that the Cauchy problem is well-posed in a strong sense and in a general setting. Our main result is the construction of an abstract semi-flow for the Hele-Shaw problem within general fluid domains (enabling, for instance, changes…
We consider the Hele-Shaw flow that arises from injection of two-dimensional fluid into a point of a curved surface. The resulting fluid domains have and are more or less determined implicitly by a mean value property for harmonic…
We present a computational framework to address the flow of two immiscible viscous liquids which co-flow into a shallow rectangular container at one side, and flow out into a holding container at the opposite side. Assumptions based on the…
Method of the quantum hydrodynamics has been applied in quantum plasmas studies. As the first step in our consideration, derivation of classical semi-relativistic (i. e. described by the Darwin Lagrangian on microscopic level)…
We are interested in the Helmholtz equation in a junction of two periodic half-spaces. When the overall medium is periodic in the direction of the interface, Fliss and Joly (2019) proposed a method which consists in applying a partial…
The objective of this paper is to derive analytical solutions of fractional order Laplace, Poisson and Helmholtz equations in two variables derived from the corresponding standard equations in two dimensions by replacing the integer order…
We propose a method of construction of exact solutions of free boundary problems corresponding to Hele-Shaw flows in presence of an external field. Such a field may arise, in particular, due to electrokinetic phenomena. Both a general…
New numerical algorithms based on rational functions are introduced that can solve certain Laplace and Helmholtz problems on two-dimensional domains with corners faster and more accurately than the standard methods of finite elements and…
We introduce and investigate a generalization of the Hele-Shaw flow with injection where several droplets compete for space as they try to expand due to internal pressure while still preserving their topology. Droplets are described by…
In this paper we study some boundary value problems for a fractional analogue of second order elliptic equation with an involution perturbation in a rectangular domain. Theorems on existence and uniqueness of a solution of the considered…
We consider the numerical solution of high-frequency scattering problems modeled by the Helmholtz equation with a bounded obstacle. Although the analysis of this problem dates back at least 50 years, over the past decade or so, tools and…
We study a minimization problem with free boundary, resulting in hybrid quadrature domains for the Helmholtz equation, as well as some application to inverse scattering problem.
A correspondence between scalar field fluctuations and generalized fluctuations in a hydrodynamic approximation of fields is obtained. The results presented here are of interest to field-fluid correspondences and form part of theoretical…
We make several improvements on the results of M.-T. Wang in [8] and his joint paper with M.-P. Tsui [7] concerning the long time existence and convergence for solutions of mean curvature flow in higher co-dimension. Both the curvature…
In this paper, we study the stochastic Hamiltonian flow in Wasserstein manifold, the probability density space equipped with $L^2$-Wasserstein metric tensor, via the Wong--Zakai approximation. We begin our investigation by showing that the…