Related papers: Kato's inequalities up to the boundary for a quasi…

We solve the Kato square root problem for parabolic operators of arbitrary order $2m$ whose coefficients are allowed to depend on both space and time in a merely measurable way and possess boundedness and ellipticity controlled by a…

This paper consists of two parts. In the first part, which is of more abstract nature, the notion of quasi boundary triples and associated Weyl functions is developed further in such a way that it can be applied to elliptic boundary value…

We study Rellich inequalities associated to higher-order elliptic operators in the Euclidean space. The inequalities are expressed in terms of an associated Finsler metric. In the case of half-spaces we obtain the sharp constant while for a…

We consider a partially overdetermined problem for the $p$-Laplace equation in a convex cone $\mathcal{C}$ intersected with the exterior of a smooth bounded domain $\overline{\Omega}$ in $\mathbb{R}^n$($n\geq2$). First, we establish the…

In this paper we establish a Hardy inequality for Laplace operators with Robin boundary conditions. For convex domains, in particular, we show explicitly how the corresponding Hardy weight depends on the coefficient of the Robin boundary…

The squared Laplace operator acting on symmetric rank-two tensor fields is studied on a (flat) Riemannian manifold with smooth boundary. Symmetry of this fourth-order elliptic operator is obtained provided that such tensor fields and their…

We prove the Kato conjecture for elliptic operators, $L=-\nabla\cdot\left((\mathbf A+\mathbf D)\nabla\ \right)$, with $\mathbf A$ a complex measurable bounded coercive matrix and $\mathbf D$ a measurable real-valued skew-symmetric matrix in…

Assume that $p > 1$ and $p - 1 \le \alpha \le p$ are real numbers and $\Omega$ is a non-empty open subset of ${\mathbb R}^n$, $n \ge 2$. We consider the inequality $$ {\rm div} \, A (x, D u) + b (x) |D u|^\alpha \ge 0, $$ where $D =…

We prove the Kato conjecture for degenerate elliptic operators in R^n. More precisely, we consider the divergence form operator L_w = -1/w div (wA) grad, where w is a Muckenhoupt A_2 weight and A is a complex valued n x n matrix which is…

This article focuses on Lp-estimates for the square root of elliptic systems of second order in divergence form on a bounded domain. We treat complex bounded measurable coefficients and allow for mixed Dirichlet/Neumann boundary conditions…

We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with $L^2$ boundary data. We…

We analyze a non-linear elliptic boundary value problem, that involves $(p, q)$ Laplace operator, for the existence of its positive solution in an arbitrary smooth bounded domain. The non-linearity here is driven by a continuous function in…

In this paper, a discrete form of the Kato inequality for discrete magnetic Laplacians on graphs is used to study asymptotic properties of the spectrum of discrete magnetic Schrodinger operators. We use the existence of a ground state with…

In this paper we consider Sobolev inequalities associated with singular problems for the fractional $p$-Laplacian operator in a bounded domain of $\mathbb{R}^{N}$, $N\geq 2$.

We establish refinements of the classical Kato inequality for sections of a vector bundle which lie in the kernel of a natural injectively elliptic first-order linear differential operator. Our main result is a general expression which…

We extend the classical Kato's inequality in order to allow functions $u \in L^1_\mathrm{loc}$ such that $\Delta u$ is a Radon measure. This inequality has been applied by Brezis, Marcus, and Ponce to study the existence of solutions of the…

Let L(t) = --div (A(x, t)$\nabla$ x) for t $\in$ (0, $\tau$) be a uniformly elliptic operator with boundary conditions on a domain $\Omega$ of R d and $\partial$ = $\partial$ $\partial$t. Define the parabolic operator L = $\partial$ + L on…

In this article we develop a simplistic approach to revisit the classical Kato-Ponce inequality, which is also known as 'fractional Leibniz rule.' As a consequence, we derive the validity of this inequality even in quasi-Banach spaces $L^p$…

Kato's inequality is shown for the magnetic relativistic Schr\"odinger operator $H_{A,m}$ defined as the operator theoretical {\it square root} of the selfadjoint, magnetic nonrelativistic Schr\"odinger operator $(-i\nabla-A(x))^2+m^2$ with…

We study a nonlinear elliptic boundary value problem defined on a smooth bounded domain involving the fractional Laplace operator, a concave-convex powers term together with mixed Dirichlet-Neumann boundary conditions.