Related papers: A Sharp Liouville Theorem for Elliptic Operators

We establish the Borg-Levinson theorem for elliptic operators of higher order with constant coefficients. The case of incomplete spectral data is also considered.

We prove Liouville theorem for the equation $\Delta v + N v^p + M |\nabla v|^{q}= 0$ in $\mathbb R^n$, with $M, N > 0, q = \frac{2p}{p + 1}$ in the critical and subcritical case. The proof is based on a differential identity and Young…

We use maximum principle to prove the Liouville theorem of the equation $\Delta U + b\cdot \nabla U + h U^{\alpha} = 0, U \geq 0, 0 < \alpha < \frac{n + 2}{n - 2}$ on the complete Riemannian manifold with non-negative Ricci tensor, which…

A fundamental theorem of Liouville asserts that positive entire harmonic functions in Euclidean spaces must be constant. A remarkable Liouville-type theorem of Caffarelli-Gidas-Spruck states that positive entire solutions of $-\Delta u=u^{…

We establish new Liouville-type theorems for the stationary Navier-Stokes equations in $\mathbb{R}^3$. A central open problem in this context is whether the classical $L^{9/2}(\mathbb{R}^3)$ condition of G.Galdi can be relaxed. In this note…

In this paper we study Liouville properties of smooth steady axially symmetric solutions of the Navier-Stokes equations. First, we provide another version of the Liouville theorem of \cite{kpr15} in the case of zero swirl, where we replaced…

In this paper, we obtain a necessary and sufficient condition for $L^{\infty}$-uniqueness of Sturm-Liouville operator $a(x)\frac{d^2}{dx^2} + b(x) \frac d{dx} -V$ on an open interval of $\rr$, which is equivalent to the $L^1$-uniqueness of…

We generalize A. Borb\'ely's condition for the conclusion of the Omori-Yau maximum principle for the Laplace operator on a complete Riemannian manifold to a second-order linear semi-elliptic operator $L$ with bounded coefficients and no…

We establish a Dahlberg-type perturbation theorem for second order divergence form elliptic operators with complex coefficients. In our previous paper, we showed the following result: If ${\mathcal L}_0=\mbox{div}…

We prove that the algebraic condition $|p-2| |< {\mathscr Im}{\mathscr A}\xi,\xi>| \leq 2 \sqrt{p-1} < {\mathscr Re}{\mathscr A}\xi,\xi>$ (for any $\xi\in\mathbb{R}^{n}$) is necessary and sufficient for the $L^{p}$-dissipativity of the…

The main aim of this article is to establish an $L_p$-theory for elliptic operators on manifolds with singularities. The particular class of differential operators discussed herein may exhibit degenerate or singular behavior near the…

We prove some $L^p$-Liouville theorems for hypoelliptic second order Partial Differential Operators left translation invariant with respect to a Lie group composition law in $\mathbb{R}^n$. Results for both solutions and subsolutions are…

The present paper commences the study of higher order differential equations in composition form. Specifically, we consider the equation Lu=\Div B^*\nabla(a\Div A\nabla u)=0, where A and B are elliptic matrices with complex-valued bounded…

A linear different operator L is called weakly hypoelliptic if any local solution u of Lu=0 is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important…

We consider the elliptic equation $-\Delta u = u^q|\nabla u|^p$ in $\mathbb R^n$ for any $p\ge 2$ and $q>0$. We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on…

In this paper we study solutions, possibly unbounded and sign-changing, of the following problem: -\D_{\lambda} u=|x|_{\lambda}^a |u|^{p-1}u, in R^n,\;n\geq 1,\; p>1, and a \geq 0, where \D_{\lambda} is a strongly degenerate elliptic…

We establish a Liouville type result for a backward global solution to the Navier-Stokes equations in the half plane with the no-slip boundary condition. No assumptions on spatial decay for the vorticity nor the velocity field are imposed.…

In this paper we prove three different Liouville type theorems for the steady Navier-Stokes equations in $\Bbb R^3$. In the first theorem we improve logarithmically the well-known $L^{\frac92} (\Bbb R^3)$ result. In the second theorem we…

We prove, with a purely analytic technique, a one-side Liouville theorem for a class of Ornstein--Uhlenbeck operators ${\mathcal L_0}$ in $\mathbb{R}^N$, as a consequence of a Liouville theorem at "$t=- \infty$" for the corresponding…

Suppose that ${\cal L}$ is a divergence form differential operator of the form ${\cal L}f:=(1/2) e^{U}\nabla_x\cdot\big[e^{-U}(I+H)\nabla_x f\big]$, where $U$ is scalar valued, $I$ identity matrix and $H$ an anti-symmetric matrix valued…