相关论文: Properties of some elliptic Hill's potentials
We establish pointwise estimates expressed in terms of a nonlinear potential of a generalized Wolff type for $A$-superharmonic functions with nonlinear operator $A:\Omega\times\mathbb{R}^n\to\mathbb{R}^n$ having measurable dependence on the…
Several local elliptic coordinates are used to build a new polyelliptic coordinate system which is orthogonal and admits the separation of variables. Such coordinate systems can give the exact solutions of some unsolved problems in quantum…
The paper concerns the problem for the ultrahyperbolic equation in the Euclidean space with data on a characteristic hyperplane. Smoothness and asymptotics of the solution along characteristic lines transversal to the initial hyperplane are…
The equations of motion of planar elliptic restricted three body problem are transformed to four decoupled Hill's equations. By using the Floquet theorem analytic solution to the oscillator equations with time dependent periodic…
We investigate the existence and multiplicity of positive solutions to the following problem driven by the superposition of the Laplacian and the fractional Laplacian with Hardy potential \begin{equation*} \left\{ \begin{aligned} -\Delta u…
This paper reports a study of the nonlinear partial differential equation that arises in the local potential approximation to the Wetterich formulation of the functional renormalization group equation. A cut-off-dependent shift of the…
An elliptic equation of order $2m$ with general nonlocal boundary-value conditions, in a plane bounded domain $G$ with piecewise smooth boundary, is considered. Generalized solutions belonging to the Sobolev space $W_2^m(G)$ are studied.…
In this paper, we study the asymptotic behavior as $x_1\to+\infty$ of solutions of semilinear elliptic equations in quarter- or half-spaces, for which the value at $x_1=0$ is given. We prove the uniqueness and characterize the…
A nonlocal contact problem for two-dimensional linear elliptic equations is stated and investigated. The method of separation of variables is used to find the solution of a stated problem in case of Poisson's equation. Then the more general…
In this article we obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the self-adjoint operator generated by a system of Sturm-Liouville equations with summable coefficients and the quasiperiodic boundary conditions.…
We study the asymptotic limit of the Cahn-Hilliard equation on an evolving surface with prescribed velocity. The method of formally matched asymptotic expansions is extended to account for the movement of the domain. We consider various…
We study the asymptotic behaviour, as the small parameter $\varepsilon$ tends to zero, of the resolvents of uniformly elliptic second-order differential operators with locally periodic coefficients depending on the slow variable $x$ and the…
We study mixed local and nonlocal elliptic equation with a variable coefficient $\rho$. Under suitable assumptions on the behaviour at infinity of $\rho$, we obtain uniqueness of solutions belonging to certain weighted Lebsgue spaces, with…
This work is devoted to the asymptotic behavior of eigenvalues of an elliptic operator with rapidly oscillating random coefficients on a bounded domain with Dirichlet boundary conditions. A sharp convergence rate is obtained for isolated…
We propose a refined version of the existing conjectural asymptotic formula for the moments of the family of quadratic Dirichlet L-functions over rational function fields. Our prediction is motivated by two natural conjectures that provide…
The double-layer potential plays an important role in solving boundary value problems for elliptic equations, and in the study of which for a certain equation, the properties of the fundamental solutions of the given equation are used. All…
The fractional Laplacian $(-\Delta)^{\alpha/2}$ is the prototypical non-local elliptic operator. While analytical theory has been advanced and understood for some time, there remain many open problems in the numerical analysis of the…
We describe a numerical framework that uses random sampling to efficiently capture low-rank local solution spaces of multiscale PDE problems arising in domain decomposition. In contrast to existing techniques, our method does not rely on…
We consider Schr\"odinger operators on the line with potentials that are periodic with respect to the coordinate variable and real analytic with respect to the energy variable. We prove that if the imaginary part of the potential is bounded…
We present a constructive method to devise boundary conditions for solutions of second-order elliptic equations so that these solutions satisfy specific qualitative properties such as: (i) the norm of the gradient of one solution is bounded…