An Existence Result for a Class of Coupled Polyharmonic Systems

This work deals with the existence of positive continuous solutions for a nonlinear coupled polyharmonic system. Our analysis is based on some potential theory tools, properties of functions in the Kato class Km, n and the Schauder fixed point theorem.

Navier–Stokes Cauchy Problem with |v0(x)|2 Lying in the Kato Class K3

We investigate the 3D Navier–Stokes Cauchy problem. We assume the initial datum v0 is weakly divergence free, supR3∫R3|v0(y)|2|x−y|dy<∞ and |v0(y)|2∈K3, where K3 denotes the Kato class. The existence is local for arbitrary data and global if supR3∫R3|v0(y)|2|x−y|dy is small. Regularity and uniqueness also hold.

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

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

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 $t&gt;0$, there exists an explicitly given constant $A(V,K,\alpha ,t)&lt;\infty$, 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.