Related papers: On the Dirichlet and Neumann evolution operators i…

We prove a Hardy inequality for uniformly elliptic operators subject to Dirichlet or mixed boundary conditions on domains $\Omega$ with piecewiese smooth boundary in arbitrary Riemannian Manifolds (M, g). Employing an approach of E.B.…

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)…

The aim of this paper is to study the Dirichlet-to-Neumann operators in the context of Dirichlet forms and especially to figure out their probabilistic counterparts. Regarding irreducible Dirichlet forms, we will show that the…

We consider the elliptic estimates for Dirichlet-Neumann operator related to the water-wave problem on a two-dimensional corner domain in this paper. Due to the singularity of the boundary, there will be singular parts in the solution of…

We consider a bounded connected open set $\Omega \subset {\rm R}^d$ whose boundary $\Gamma$ has a finite $(d-1)$-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator $D_0$ on $L_2(\Gamma)$ by form methods. The…

It is shown that the theory of real symmetric second-order elliptic operators in divergence form on $\Ri^d$ can be formulated in terms of a regular strongly local Dirichlet form irregardless of the order of degeneracy. The behaviour of the…

We prove Poisson upper bounds for the heat kernel of the Dirichlet-to-Neumann operator with variable H{\"o}lder coefficients when the underlying domain is bounded and has a C 1+$\kappa$-boundary for some $\kappa$ > 0. We also prove a number…

This work considers the Neumann eigenvalue problem for the weighted Laplacian on a Riemannian manifold $(M,g,\partial M)$ under the singular perturbation. This perturbation involves the imposition of vanishing Dirichlet boundary conditions…

In this paper, we consider a fully-discrete approximation of an abstract evolution equation deploying a non-conforming spatial approximation and finite differences in time (Rothe-Galerkin method). The main result is the convergence of the…

The problem of characterizing sequences of real numbers that arise as spectra of Dirichlet-to-Neumann (DtN) maps for elliptic operators has attracted considerable attention over the past fifty years. In this article, we address this…

In this paper, the elastic Dirichlet-to-Neumann map $\Xi_g$ is studied for the stationary elasticity system in a compact Riemannian manifold $(\Omega,g)$ with smooth boundary $\partial \Omega$. By overcoming methodological difficulties, we…

We consider the nonautonomous Ornstein-Uhlenbeck operator in some weighted spaces of continuous functions in $\R^N$. We prove sharp uniform estimates for the spatial derivatives of the associated evolution operator $\OU$, which we use to…

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…

Recently, several works have been carried out in attempt to develop a theory for linear or sublinear elliptic equations involving a general class of nonlocal operators characterized by mild assumptions on the associated Green kernel. In…

The large time $t$ asymptotics for scalar, constant coefficient,linear, third order, dispersive equations are obtained for asymptotically time-periodic Dirichlet boundary data and zero initial data on the half-line modeling a wavemaker…

We determine sufficient conditions for the occurrence of a pointwise gradient estimate for the evolution operators associated to nonautonomous second order parabolic operators with (possibly) unbounded coefficients. Moreover we exhibit a…

We characterize the domain of the realization of the linear parabolic operator Gu := u_t + L(t)u (where, for each real t, L(t) is an Ornstein-Uhlenbeck operator), in L^2 spaces with respect to a suitable measure, that is invariant for the…

We introduce the notion of uniform gamma-radonification of a family of operators, which unifies the notions of R-boundedness of a family of operators and gamma-radonification of an individual operator. We study the the properties of…

Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: $$\dot{u}=A(t)u+F(t,u)+b(t), \quad t\ge 0; \quad u(0)=u_0. \qquad (*)$$ Here $\dot{u}:=\frac {du}{dt}$, $u=u(t)\in H$,…

The nonlinear semigroup generated by the subdifferential of a convex lower semicontinuous function $\varphi$ has a smoothing effect, discovered by H. Br\'ezis, which implies maximal regularity for the evolution equation. We use this and…