Related papers: Kato's inequality when $\Delta u$ is a measure

In this paper we study the equation $-\Delta u+\rho^{-(\alpha+2)}h(\rho^{\alpha}u)=0$ in a smooth bounded domain $\Omega$ where $\rho(x)=\textrm{dist}\,(x,\partial \Omega)$, $\alpha>0$ and $h$ is a non-decreasing function which satisfies…

This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -\Delta u + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in…

In the following paper, one studies, given a bounded, connected open set $\Omega$ $\subseteq$ R n , $\kappa$ > 0, a positive Radon measure $\mu$ 0 in $\Omega$ and a (signed) Radon measure $\mu$ on $\Omega$ satisfying $\mu$($\Omega$) = 0 and…

Equation $(-\Delta+k^2)u+f(u)=0$ in $D$, $u\mid_{\partial D}=0$, where $k=\const>0$ and $D\subset\R^3$ is a bounded domain, has a solution if $f:\R\to\R$ is a continuous function in the region $|u|\geq a$, piecewise-continuous in the region…

We classify the several classes of the set of smooth measures from the perspective of the denseness and the locality, and consider their relationships, in particular, that of the Kato class and Radon measures of finite energy integrals. We…

In this paper, we consider a class of important nonlinear elliptic equations $$\Delta u + a(x)u\log u + b(x)u = 0$$ on a collapsed complete Riemannian manifold and its parabolic counterpart under integral curvature conditions, where $a(x)$…

We study nonnegative classical solutions $u$ of the polyharmonic inequality $-\Delta^m u > 0$ in a punctured neighborhood of the origin in $R^n$. We give necessary and sufficient conditions on integers $n\ge 2$ and $m\ge 1$ such that these…

We consider Rado numbers of the regular equations $\mathcal{E}(b)$ of the form \[ c_1x_1+c_2x_2+\dots+ c_{k-1}x_{k-1} = x_k + b, \] where $b \in \mathbb{Z}$ and $c_i \in \mathbb{Z}^{+}$ for all $i$. We give the upper bounds and the…

We present several operator extensions of the Chebyshev inequality for Hilbert space operators. The main version deals with the synchronous Hadamard property for Hilbert space operators. Among other inequalities, it is shown that if…

We study the following nonlinear scalar field equation $$ -\Delta u=f(u)-\mu u, \quad u \in H^1(\mathbb{R}^N) \quad \text{with} \quad \|u\|^2_{L^2(\mathbb{R}^N)}=m. $$ Here $f\in C(\mathbb{R},\mathbb{R})$, $m>0$ is a given constant and…

We establish uniform bounds for the solutions $e^{it\Delta}u$ of the Schr\"{o}dinger equation on arithmetic flat tori, generalising earlier results by J. Bourgain. We also study the regularity properties of weak-* limits of sequences of…

We show the estimates \inf_T \int_T^{T+\delta} |\zeta(1+it)|^{-1} dt =e^{-\gamma}/4 \delta^2+ O(\delta^4) and \inf_T \int_T^{T+\delta} |\zeta(1+it)| dt =e^{-\gamma} \pi^2/24 \delta^2+ O(\delta^4) as well as corresponding results for…

In this work we study the existence and regularity of solutions to the following equation: $$-\Delta_p u + g(x) u = \frac{\lambda}{|x|^{p}} |u|^{p-2}u + f,$$ where $1< p < N$ and $f\in L^m$, where $m\ge 1$.

We study the existence and non-existence of classical solutions for inequalities of type $$ \pm \Delta^m u \geq \big(\Psi(|x|)*u^p\big)u^q \quad\mbox{ in } {\mathbb R}^N (N\geq 1). $$ Here, $\Delta^m$ $(m\geq 1)$ is the polyharmonic…

We study semilinear elliptic equations \begin{equation*} \begin{cases} -\Delta u = f(u) & \text{in } \Omega, \\ \partial_\nu u = 0 & \text{on } \partial\Omega, \end{cases} \end{equation*} with homogeneous Neumann boundary conditions in…

We discuss the existence of positive superharmonic functions $u$ in $\mathbb{R}^N_+=\mathbb{R}^{N-1}\times (0, \infty)$, $N\geq 3$, in the sense $-\Delta u=\mu$ for some Radon measure $\mu$, so that $u$ satisfies the nonlocal boundary…

We identify a set of sufficient local conditions under which a significant portion of a Radon measure $\mu$ on $\mathbb{R}^{n+1}$ with compact support can be covered by an $n$-uniformly rectifiable set at the level of a ball $B\subset…

We study the Kato problem for degenerate divergence form operators. This was begun by Cruz-Uribe and Rios who proved that given an operator $L_w=-w^{-1}{\rm div}(A\nabla)$, where $w\in A_2$ and $A$ is a $w$-degenerate elliptic measure (i.e,…

We give bilateral pointwise estimates for positive solutions $u$ to the sublinear integral equation \[ u = \mathbf{G}(\sigma u^q) + f \quad \textrm{in} \,\, \Omega,\] for $0 < q < 1$, where $\sigma\ge 0$ is a measurable function, or a Radon…

We study the inequality $ -\Delta u - \frac{\mu}{|x|^2} u \geq (|x|^{-\alpha} * u^p)u^q$ in an unbounded cone $\mathcal{C}_\Omega^\rho\subset \mathbb{R}^N$ ($N\geq 2$) generated by a subdomain $\Omega$ of the unit sphere $S^{N-1}\subset…