Form-perturbation theory for higher-order elliptic operators and systems by singular potentials

2020 ◽  
Vol 378 (2185) ◽  
pp. 20190621
Author(s):  
Mustapha Mokhtar-Kharroubi
Keyword(s):  
Perturbation Theory ◽  
Elliptic Systems ◽  
Elliptic Operators ◽  
Gaussian Kernel ◽  
Singular Potentials ◽  
Singular Matrix ◽  
Kernel Estimates ◽  
Theme Issue ◽  
Hölder Continuous ◽  
Kato Class

We give a form-perturbation theory by singular potentials for scalar elliptic operators on L 2 ( R d ) of order 2 m with Hölder continuous coefficients. The form-bounds are obtained from an L 1 functional analytic approach which takes advantage of both the existence of m -gaussian kernel estimates and the holomorphy of the semigroup in L 1 ( R d ) . We also explore the (local) Kato class potentials in terms of (local) weak compactness properties. Finally, we extend the results to elliptic systems and singular matrix potentials. This article is part of the theme issue ‘Semigroup applications everywhere’.

RCD*(K,N) Spaces and the Geometry of Multi-Particle Schrödinger Semigroups

2020 ◽  
Author(s):  
Batu Güneysu
Keyword(s):  
Perturbation Theory ◽  
Schrödinger Operator ◽  
The Self ◽  
Schrodinger Operator ◽  
Hölder Continuous ◽  
Kato Class ◽  
Coupling Methods

Abstract Dedicated to the memory of Kazumasa Kuwada. Let $(X,\mathfrak{d},{\mathfrak{m}})$ be an $\textrm{RCD}^*(K,N)$ space for some $K\in{\mathbb{R}}$, $N\in [1,\infty )$, and let $H$ be the self-adjoint Laplacian induced by the underlying Cheeger form. Given $\alpha \in [0,1]$, we introduce the $\alpha$-Kato class of potentials on $(X,\mathfrak{d},{\mathfrak{m}})$, and given a potential $V:X\to{\mathbb{R}}$ in this class, we denote with $H_V$ the natural self-adjoint realization of the Schrödinger operator $H+V$ in $L^2(X,{\mathfrak{m}})$. We use Brownian coupling methods and perturbation theory to prove that for all $t>0$, there exists an explicitly given constant $A(V,K,\alpha ,t)<\infty$, such that for all $\Psi \in L^{\infty }(X,{\mathfrak{m}})$, $x,y\in X$ one has $$\begin{align*}\big|e^{-tH_V}\Psi(x)-e^{-tH_V}\Psi(y)\big|\leq A(V,K,\alpha,t) \|\Psi\|_{L^{\infty}}\mathfrak{d}(x,y)^{\alpha}.\end{align*}$$In particular, all $L^{\infty }$-eigenfunctions of $H_V$ are globally $\alpha$-Hölder continuous. This result applies to multi-particle Schrödinger semigroups and, by the explicitness of the Hölder constants, sheds some light into the geometry of such operators.

scholarly journals Kernel estimates for elliptic operators with unbounded diffusion, drift and potential terms

2019 ◽  
Vol 39 (2) ◽  
pp. 803-817 ◽  
Author(s):  
Sallah Eddine Boutiah ◽  
◽  
Abdelaziz Rhandi ◽  
Cristian Tacelli ◽  
Keyword(s):  
Elliptic Operators ◽  
Kernel Estimates

Positive solutions for some competitive elliptic systems

2014 ◽  
Vol 64 (1) ◽  
Author(s):  
Ramzi Alsaedi ◽  
Habib Mâagli ◽  
Noureddine Zeddini
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Elliptic System ◽  
Elliptic Systems ◽  
Global Behavior ◽  
Continuous Solutions ◽  
Kato Class ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic System ◽  
Compact Boundary

AbstractUsing some potential theory tools and the Schauder fixed point theorem, we prove the existence of positive bounded continuous solutions with a precise global behavior for the semilinear elliptic system Δu = p(x)u α ν r in domains D of ℝn, n ≥ 3, with compact boundary (bounded or unbounded) subject to some Dirichlet conditions, where α ≥ 1, β ≥ 1, r ≥ 0, s ≥ 0 and the potentials p, q are nonnegative and belong to the Kato class K(D).

scholarly journals Positive Solutions for Some Nonlinear Elliptic Systems in Exterior Domains of

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Ramzi Alsaedi
Keyword(s):  
Exterior Domain ◽  
Elliptic Systems ◽  
Global Behavior ◽  
Exterior Domains ◽  
Nonlinear Elliptic Systems ◽  
Continuous Solutions ◽  
Kato Class ◽  
Nonlinear Elliptic ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic System

Using some potential theory tools and the Schauder fixed point theorem, we prove the existence of positive continuous solutions with a precise global behavior for the competitive semilinear elliptic system , in an exterior domain of , subject to some Dirichlet conditions, where , , , and the potentials are nonnegative and satisfy some hypotheses related to the Kato class .

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