Related papers: Explicit $\infty$-harmonic functions in high dimen…
We present a rigorous convergence analysis for cylindrical approximations of nonlinear functionals, functional derivatives, and functional differential equations (FDEs). The purpose of this analysis is twofold: first, we prove that…
We introduce suitable coordinate systems for pipes and their variants that allow us to transform partial differential equations (PDEs) on the pipe surfaces or in the solid pipes into computational domains with fixed limits/ranges. Such a…
Below, the explicit solution to a certain finite-difference equation is given and the required steps for derivation of these results are outlined. Everything is included as Mathematica formulae, so the notebook itself can be used for…
A numerical method to solve linear integro-differential equations is presented. This method has been used to solve the QCD Altarelli-Parisi evolution equations within the H1 Collaboration at DESY-Hamburg. Mathematical aspects and numerical…
We show existence and uniqueness for the solutions of the regularity and the Neumann problems for harmonic functions on Lipschitz domains with data in the Hardy spaces H^p, p>2/3, where This in turn implies that solutions to the Dirichlet…
We study existence, uniqueness and regularity of solutions for linear equations in infinitely many derivatives. We develop a natural framework based on Laplace transform as a correspondence between appropriate $L^p$ and Hardy spaces: this…
The aim of this paper is to establish properties of the solutions to the $\alpha$-harmonic equations: $\Delta_{\alpha}(f(z))=\partial{z}[(1-{|{z}|}^{2})^{-\alpha} \overline{\partial}{z}f](z)=g(z)$, where…
This paper investigates the existence, nonexistence, and qualitative properties of p-harmonic functions in the upper half-space $\mathbb{R}^N_+ \, (N \geq 3)$ satisfying nonlinear boundary conditions for $1<p<N$. Moreover, the symmetry of…
We study the existence and nonexistence of positive (super) solutions to the nonlinear $p$-Laplace equation $$-\Delta_p u-\frac{\mu}{|x|^p}u^{p-1}=\frac{C}{|x|^{\sigma}}u^q$$ in exterior domains of ${\R}^N$ ($N\ge 2$). Here…
In this paper, we solve Laplace equation analytically by using differential transform method. For this purpose, we consider four models with two Dirichlet and two Neumann boundary conditions and obtain the corresponding exact solutions. The…
Hardy spaces in the complex plane and in higher dimensions have natural finite-dimensional subspaces formed by polynomials or by linear maps. We use the restriction of Hardy norms to such subspaces to describe the set of possible…
In this work we use Lie group theoretic methods and the theory of prolonged group actions to study two fully nonlinear partial differential equations (PDEs). First we consider a third order PDE in two spatial dimensions that arises as the…
Using properties of harmonic functions in multidimensional space, we transform the Hartree-Fock eigenvalue problem into a more tractable eigenvalue problem in which the Laplacian is eliminated. This new formulation may facilitate the…
Quantum superintegrable systems are solvable eigenvalue problems. Their solvability is due to symmetry, but the symmetry is often "hidden". The symmetry generators of 2nd order superintegrable systems in 2 dimensions close under commutation…
Let $0<\alpha<2$, $p\geq 1$, $m\in\mathbb{N}_+$. Consider $u$ to be the positive solution of the PDE \begin{equation}\label{abstract PDE} (-\Delta)^{\frac{\alpha}{2}+m} u(x)=u^p(x) \quad\text{in }\mathbb{R}^n. \end{equation} Cao, Dai and…
Classical harmonic analysis says that the spaces of homogeneous harmonic polynomials (solutions of Laplace equation) are irreducible modules of the corresponding orthogonal Lie group (algebra) and the whole polynomial algebra is a free…
This paper establishes existence of solutions for a partial differential equation in which a differential operator involving variable exponent growth conditions is present. This operator represents a generalization of the $p(\cdot)$-Laplace…
We consider certain solutions of the Infinity-Laplace Equation in planar convex rings. Their ascending streamlines are unique while the descending ones may bifurcate. We prove that bifurcation occurs in the generic situation and as a…
It is shown that eigenvalues of Laplace-Beltrami operators on compact Riemannian manifolds can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polyharmonic functions with singularities. In…
We develop a method to obtain the general solution of the Laplace equation in $d$-dimension in ultraspherical coordinates.