Related papers: Generalized Q-functions and Dirichlet-to-Neumann m…

A large class of initial-boundary value problems of linear evolution partial differential equations formulated on the half-line is analyzed via the unified transform method. In particular, explicit formulae are presented for the generalized…

In this article we focus on inverse problems for a semilinear elliptic equation. We show that a potential $q$ in $L^{n/2+\varepsilon}$, $\varepsilon>0$, can be determined from the full and partial Dirichlet-to-Neumann map. This extends the…

The technique of Caffarelli and Silvestre, characterizing the fractional Laplacian as the Dirichlet-to-Neumann map for a function U satisfying an elliptic equation in the upper half space with one extra spatial dimension, is shown to hold…

Jacobi's elliptic integrals and elliptic functions arise naturally from the Schwarz-Christoffel conformal transformation of the upper half plane onto a rectangle. In this paper we study generalized elliptic integrals which arise from the…

In this note we describe how the Neumann homogenization of fully nonlinear elliptic equations can be recast as the study of nonlocal (integro-differential) equations involving elliptic integro-differential operators on the boundary. This is…

We consider second order uniformly elliptic operators of divergence form in $\R^{d+1}$ whose coefficients are independent of one variable. For such a class of operators we establish a factorization into a product of first order operators…

In this paper, authors study the generalized complete $(p,q)$-elliptic integrals of the first and the second kind as an application of generalized trigonometric functions with two parameters, and establish the Tur\'an type inequalities of…

We aim to introduce a new extension of Mittag-Leffler function via q-analogue and obtained their significant properties including integral representation, q-differentiation, q-Laplace transform, image formula under q-derivative operators.…

The Dirichlet-to-Neumann maps connect boundary values of harmonic functions. It is an amazing fact that the square of the non-local Dirichlet-to-Neumann map for the uniform conductivity 1 on the unit disc equals minus the local(!) Laplace…

In this work, we review and extend some well known results for the eigenvalues of the Dirichlet $p-$Laplace operator to a more general class of monotone quasilinear elliptic operators. As an application we obtain some homogenization results…

We consider the Dirichlet-to-Neumann operator ${\cal N}$ associated with a general elliptic operator \[ {\cal A} u = - \sum_{k,l=1}^d \partial_k (c_{kl}\, \partial_l u) + \sum_{k=1}^d \Big( c_k\, \partial_k u - \partial_k (b_k\, u) \Big)…

We numerically investigate the generalized Steklov problem for the modified Helmholtz equation and focus on the relation between its spectrum and the geometric structure of the domain. We address three distinct aspects: (i) the asymptotic…

We show that the knowledge of the Dirichlet-to-Neumann map on an arbitrary open portion of the boundary of a domain in $\mathbb{R}^n$, $n\ge 2$, for a class of semilinear elliptic equations, determines the nonlinearity uniquely.

Operator-valued $Q$-functions for special pairs of nonnegative selfadjoint extensions of nonnegative not necessarily densely defined operators are defined and their analytical properties are studied. It is shown that the Kre\u\i…

Let $\Omega\subset{\mathbb R}^n$ be bounded with a smooth boundary $\Gamma$ and let $S$ be the symmetric operator in $L^2(\Omega)$ given by the minimal realization of a second order elliptic differential operator. We give a complete…

We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fr\'echet…

We introduce and systematically develop two classes of discrete integrable operators: those with $2\times 2$ matrix kernels and those possessing general differential kernels, thereby generalizing the discrete analogue previously studied. A…

We provide the necessary and sufficient conditions for a generalized Nevanlinna function $Q$ ($Q\in N_{\kappa }\left( \mathcal{H} \right)$) to be a Weyl function (also known as a Weyl-Titchmarch function). We also investigate an important…

We extend the theory of distributional kernel operators to a framework of generalized functions, in which they are replaced by integral kernel operators. Moreover, in contrast to the distributional case, we show that these generalized…

In this paper we study nonlocal equations driven by the fractional powers of hypoelliptic operators in the form $$\mathscr K u = \mathscr A u - \partial_t u \overset{def}{=} \operatorname{tr}(Q \nabla^2 u) + <BX,\nabla u> - \partial_t u,$$…