Related papers: The Landis Conjecture for variable coefficient sec…
In this article, we study the order of vanishing and a quantitative form of Landis' conjecture in the plane for solutions to second-order elliptic equations with variable coefficients and singular lower order terms. Precisely, we let $A$ be…
In this article, we investigate the quantitative unique continuation properties of real-valued solutions to elliptic equations in the plane. Under a general set of assumptions on the operator, we establish quantitative forms of Landis'…
In this paper we prove a quantitative form of Landis' conjecture in the plane. Precisely, let $W(z)$ be a measurable real vector-valued function and $V(z)\ge 0$ be a real measurable scalar function, satisfying $\|W\|_{L^{\infty}({\mathbf…
In this article, we study a quantitative form of the Landis conjecture on exponential decay for real-valued solutions to second order elliptic equations with variable coefficients in the plane. In particular, we prove the following…
In this article, we continue our investigation into the unique continuation properties of real-valued solutions to elliptic equations in the plane. More precisely, we make another step towards proving a quantitative version of Landis'…
We give partial affirmative answers to Landis conjecture in all dimensions for two different types of linear, second order, elliptic operators in a domain $\Omega\subset \mathbb{R}^N$. In particular, we provide a sharp decay criterion that…
We investigate the quantitative unique continuation properties of real-valued solutions to Schr\"odinger equations in the plane with potentials that exhibit growth at infinity. More precisely, for equations of the form $\Delta u - V u = 0$…
We investigate the quantitative unique continuation properties of real-valued solutions to planar Schr\"odinger equations with potential functions that exhibit pointwise decay at infinity. That is, for equations of the form $-\Delta u + V u…
Consider a solution $u$ to $\Delta u +Vu=0$ on $\mathbb{R}^2$, where $V$ is real-valued, measurable and $|V|\leq 1$. If $|u(x)| \leq \exp(-C |x| \log^{1/2}|x|)$, $|x|>2$, where $C$ is a sufficiently large absolute constant, then $u\equiv…
The so called Landis conjecture states that if a solution of the equation $$\Delta u+V(x)u=0$$ in an exterior domain decays faster than $e^{-\kappa|x|}$, for some $\kappa>\sqrt{\sup |V|}$, then it must be identically equal to $0$. This…
In this paper we derive quantitative uniqueness estimates at infinity for solutions to an elliptic equation with unbounded drift in the plane. More precisely, let $u$ be a real solution to $\Delta u+W\cdot\nabla u=0$ in ${\mathbf R}^2$,…
We obtain a unique continuation result at infinity for fully nonlinear elliptic integro-differential operators of order 2s which satisfy the maximum and minimum principles in bounded subdomains, under the decay assumption $o(|x|^{-(N+2s)})$…
The solvability in $W^{2}_{p}(\bR^{d})$ spaces is proved for second-order elliptic equations with coefficients which are measurable in one direction and VMO in the orthogonal directions in each small ball with the direction depending on the…
We present certain Liouville properties of eigenfunctions of second-order elliptic operators with real coefficients, via an approach that is based on stochastic representations of positive solutions, and criticality theory of second-order…
Consider positive solutions to second order elliptic equations with measurable coefficients in a bounded domain, which vanish on a portion of the boundary. We give simple necessary and sufficient geometric conditions on the domain, which…
The equation $- \Delta u + V u = 0$ in the cylinder $\mathbb{R} \times (0,2\pi)^d$ with periodic boundary conditions is considered. The potential $V$ is assumed to be bounded, and both functions $u$ and $V$ are assumed to be real-valued. It…
We consider second-order elliptic equations in a half space with leading coefficients measurable in a tangential direction. We prove the $W^2_p$-estimate and solvability for the Dirichlet problem when $p\in (1,2]$, and for the Neumann…
The asymptotic behavior of solutions to the second order elliptic equations in exterior domains is studied. In particular, under the assumption that the solution belongs to the Lorentz space $L^{p,q}$ or the weak Lebesgue space…
In this paper, we study the existence of positive solution for the following class of fractional elliptic equation $$ \epsilon^{2s} (-\Delta)^{s}{u}+V(z)u=\lambda |u|^{q-2}u+|u|^{2^{*}_{s}-2}u\,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, $$ where…
We derive a priori second order estimates for fully nonlinear elliptic equations which depend on the gradients of solutions in critical ways on Hermitian manifolds. The global estimates we obtained apply to an equation arising from a…