Related papers: Eigenfunctions and the Dirichlet problem for the C…
We investigate the homogeneous Dirichlet problem in uniformly convex domains for a large class of degenerate elliptic equations with singular zero order term. In particular we establish sharp existence and uniqueness results of positive…
A complete family of solutions for the one-dimensional reaction-diffusion equation \[ u_{xx}(x,t)-q(x)u(x,t) = u_t(x,t) \] with a coefficient $q$ depending on $x$ is constructed. The solutions represent the images of the heat polynomials…
In the present paper, motivated by point interaction, we propose a new and explicit approach to inverse Sturm-Liouville eigenvalue problems under Dirichlet boundary. More precisely, when a given Sturm-Liouville eigenvalue problem with the…
We present a novel integral-equation algorithm for evaluation of Zaremba eigenvalues and eigenfunctions}, that is, eigenvalues and eigenfunctions of the Laplace operator with mixed Dirichlet-Neumann boundary conditions; of course, (slight…
We study semilinear problems in bounded $C^{1,1}$ domains for non-local operators with a boundary condition. The operators cover and extend the case of the spectral fractional Laplacian. We also study harmonic functions with respect to the…
We consider positive solutions, possibly unbounded, to the semilinear equation $-\Delta u=f(u)$ on continuous epigraphs bounded from below. Under the homogeneous Dirichlet boundary condition, we prove new monotonicity results for $u$, when…
In this paper, usual Sturm-Liouville problems are extended for symmetric functions so that the corresponding solutions preserve the orthogonality property. Two basic examples, which are special cases of a generalized Sturm-Liouville…
In this paper we study the behaviour of the Neumann data of Dirichlet eigenfunctions on simplices. We prove that the $L^2$ norm of the (semi-classical) Neumann data on each face is equal to $2/n$ times the $(n-1)$-dimensional volume of the…
We present an existence and uniqueness result for weak solutions of Dirichlet boundary value problems governed by a nonlocal operator in divergence form and in the presence of a datum which is assumed to belong only to $L^1(\Omega)$ and to…
We consider the Poisson equation with homogeneous Dirichlet conditions in a family of domains in $R^{n}$ indexed by a small parameter $\epsilon$. The domains depend on $\epsilon$ only within a ball of radius proportional to $\epsilon$ and,…
We define and study the properties of Darboux-type transformations between Sturm--Liouville problems with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter (including the Dirichlet boundary…
We present quantitative results for the homogenization of uniformly convex integral functionals with random coefficients under independence assumptions. The main result is an error estimate for the Dirichlet problem which is algebraic (but…
Let $D \subset {\mathbb R}^d,\: d \geqslant 2,$ be the union of a finite collection of pairwise disjoint strictly convex compact obstacles. Let $\mu_j \in {\mathbb C},\: {\rm Im}\: \mu_j > 0,$ be the resonances of the Laplacian in the…
This note supplements the work of Gomez-Ullate, Kamran and Milson on the X_(1)-Laguerre polynomials which are orthogonal in a weighted Hilbert function space on the positive half-line of the real line. These polynomials are generated by a…
For a finite sequence of positive integers $A=\{a_j\}_{j=1}^{k},$ we prove a recursion for divisor function $\sigma_{x}^{(A)}(n)=\sum_{d|n,\enskip d\in A}d^x.$ As a corollary, we give an affirmative solution of the problem posed in 1969 by…
This note describes the solution of the Helmholtz equation inside a nanotorus with uniform Dirichlet boundary conditions. The eigenfunction symmetry is discussed and the lower-order eigenvalues and eigenfunctions are shown. The similarity…
In the paper, we investigate the uniqueness problem of entire functions concerning their linear differential polynomial in shift and obtain three results which improve and generalize the recent result due to Qi (Ann. Polon. Math., 102…
In the article we study the Dirichlet $(p,q)$-eigenvalue problem for subelliptic non-commutative operators on nilpotent Lie groups. We prove solvability of this eigenvalue problem and existence of the minimizer of the corresponding…
We study the $q$-hypergeometric difference operator $L$ on a particular Hilbert space. In this setting $L$ can be considered as an extension of the Jacobi operator for $q^{-1}$-Al-Salam--Chihara polynomials. Spectral analysis leads to…
We study diffusions, variational principles and associated boundary value problems on directed graphs with natural weightings. Using random walks and exit times, we associate to certain subgraphs (domains) a pair of sequences, each of which…