Related papers: Linearizable Initial-Boundary Value Problems for t…
We study qualitative properties of initial traces of nonnegative solutions to a semilinear heat equation in a smooth domain under the Dirichlet boundary condition. Furthermore, for the corresponding Cauchy--Dirichlet problem, we obtain…
In this paper, we analyze an eigenvalue problem for quasi-linear elliptic operators involving homogeneous Dirichlet boundary conditions in a open smooth bounded domain. We show that the eigenfunctions corresponding to the eigenvalues belong…
In this paper, we consider the Dirichlet boundary value problem for fully nonlinear Yamabe equations on Riemannian manifolds with boundary. Assuming the existence of a subsolution, we derive \emph{a priori} boundary second derivative…
In this paper we show how a second order scalar uniformly elliptic equation on divergence form with measurable coefficients and Dirichlet boundary conditions can be transformed into a first order elliptic system with half-Dirichlet boundary…
We consider the Hirota equation on the quarter plane with the initial and boundary values belonging to the Schwartz space. The goal of this paper is to study the long-time behavior of the solution of this initial-boundary value problem…
Based upon elements of the modern Pseudoanalytic Function Theory, we analyse a new method for numerically approaching the solution of the Dirichlet boundary value problem, corresponding to the two-dimensional Electrical Impedance Equation.…
We establish existence and regularity results for boundary value problems arising from the first variation of the Willmore energy in the graphical setting. Our focus lies on two-dimensional surfaces with fixed clamped boundary conditions,…
We provide explicit formulas for the Green function of an elliptic PDE in the infinite strip and the half-plane. They are expressed in elementary and special functions. Proofs of uniqueness and existence are also given.
We characterize the behavior of the solutions of linear evolution partial differential equations on the half line in the presence of discontinuous initial conditions or discontinuous boundary conditions, as well as the behavior of the…
We investigate the initial-boundary value problem for the general three-component nonlinear Schrodinger (gtc-NLS) equation with a 4x4 Lax pair on a finite interval by extending the Fokas unified approach. The solutions of the gtc-NLS…
Let $\mathcal{O}\subset \mathbb{R}^d$ be a bounded domain of class $C^{1,1}$. In $ L_2(\mathcal{O};\mathbb{C}^n)$, we consider a matrix elliptic second order differential operator $A_{D,\varepsilon}$ with the Dirichlet boundary condition.…
We consider the spectral structure of indefinite second order boundary-value problems on graphs. A variational formulation for such boundary-value problems on graphs is given and we obtain both full and half-range completeness results. This…
The boundary behaviour of solutions of stochastic PDEs with Dirichlet boundary conditions can be surprisingly - and in a sense, arbitrarily - bad: as shown by Krylov, for any $\alpha>0$ one can find a simple $1$-dimensional constant…
We show how to solve initial-boundary value problems for integrable nonlinear differential-difference equations on a finite set of integers. The method we employ is the discrete analogue of the unified transform (Fokas method). The…
We consider the nonlinear Schr\"odinger equation on the half-line with a given Dirichlet (Neumann) boundary datum which for large $t$ tends to the periodic function $g_0^b(t)$ ($g_1^b(t)$). Assuming that the unknown Neumann (Dirichlet)…
We derive new boundary conditions and implementation procedures for nonlinear initial boundary value problems (IBVPs) with non-zero boundary data that lead to bounded solutions. The new boundary procedure is applied to nonlinear IBVPs in…
Maximally dissipative boundary conditions are applied to the initial-boundary value problem for Einstein's equations in harmonic coordinates to show that it is well-posed for homogeneous boundary data and for boundary data that is small in…
We obtained estimates for first eigenvalues of the Dirichlet boundary value problem for elliptic operators in divergence form (i.e. for the principal frequency of non-homogeneous membranes) in bounded domains $\Omega \subset \mathbb C$…
The current paper is the second part of a series of two papers dedicated to 2D problem of diffraction of acoustic waves by a segment bearing impedance boundary conditions. In the first part some preliminary steps were made, namely, the…
The main objective of this paper is analysis of the initial-boundary value problems for the linear time-fractional diffusion equations with a uniformly elliptic spatial differential operator of the second order and the Caputo type…