Related papers: Quantitative unique continuation for Schr\"odinger…

In this paper we study the local behavior of a solution to the $l$th power of Laplacian with singular coefficients in lower order terms. We obtain a bound on the vanishing order of the nontrivial solution. Our proofs use Carleman estimates…

We prove sharpness of quantitative unique continuation results for solutions of $-\Delta u + W\cdot \nabla u + V u = \la u$, where $\la \in \C$ and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \lesssim <x>^{-N}$…

We prove that solutions to elliptic equations in two variables in divergence form, possibly non-selfadjoint and with lower order terms, satisfy the strong unique continuation property.

In this article we determine bounds on the maximal order of vanishing for eigenfunctions of a generalized Dirichlet-to-Neumann map (which is associated with fractional Schr\"odinger equations) on a compact, smooth Riemannian manifold,…

Given two continuous functions $V\left(r \right)\geq 0$ and $K\left(r\right)> 0$ ($r>0$), which may be singular or vanishing at zero as well as at infinity, we study the quasilinear elliptic equation \[ -\Delta w+ V\left( \left| x\right|…

We prove strong unique continuation property for the differential inequality $|(\partial_t +\Delta)u(x,t)|\le V(x,t)|u(x,t)|$ with $V$ contained in weak spaces. In particular, we establish the strong unique continuation property for $V\in…

We are concerned with the properties of weak solutions of the stationary Schr\"odinger equation $-\Delta u + Vu = f(u)$, $u\in H^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, where $V$ is H\"older continuous and $\inf V>0$. Assuming $f$ to…

In this paper we establish some new $L^{2}-L^{2}$ Carleman estimates for the Baouendi-Grushin operators $\mathscr{B}_\gamma$, in (1.1) below. We apply such estimates to obtain: (i) an extension of the Bourgain-Kenig quantitative unique…

We prove quantitative unique continuation results for solutions of $\Delta w - k^2 w = V w + W\cdot \nabla w$ in a neighborhood of infinity, where $k > 0$, and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \lesssim…

We establish the strong unique continuation property of fractional orders of linear elliptic equations with Lipschitz coefficients by establishing monotonicity of some Almgren-type frequency functional via an extension procedure.

We establish new, optimal gradient continuity estimates for solutions to a class of 2nd order partial differential equations, $\mathscr{L}(X, \nabla u, D^2 u) = f$, whose diffusion properties (ellipticity) degenerate along the \textit{a…

For general second order evolution equations, we prove an optimal condition on the degree of unboundedness of the damping, that rules out finite-time extinction. We show that control estimates give energy decay rates that explicitly depend…

In this paper we describe some recent works on quantitative unique continuation for elliptic, parabolic and dispersive equations. The elliptic results are joint work with J.Bourgain, while the remainder of the works discussed are joint…

We demonstrate a quantitative version of the usual properties related to unique continuation from an interior datum for the Schr\"odinger equation with bounded or unbounded potential. The inequalities we establish have constants that…

This work is devoted to prove the exponential decay for the energy of solutions of a higher order Korteweg -de Vries (KdV)--Benjamin-Bona-Mahony (BBM) equation on a periodic domain with a localized damping mechanism. Following the method in…

We establish the validity of a strong unique continuation property for weakly coupled elliptic systems, including competitive ones. Our proof exploits the system structure and uses Carleman estimates. We apply this result to obtain some…

In this article we present two mechanisms for deducing logarithmic quantitative unique continuation bounds for certain classes of integral operators. In our first method, expanding the corresponding integral kernels, we exploit the…

We consider the complement value problem for a class of second order elliptic integro-differential operators. Let $D$ be a bounded Lipschitz domain of $\mathbb{R}^d$. Under mild conditions, we show that there exists a unique bounded…

We prove the existence of positive solutions for the supercritical nonlinear fractional Schr\"odinger equation $(-\Delta)^s u+V(x)u-u^p=0$ in $\mathbb R^n$, with $u(x)\to 0$ as $|x|\to +\infty$, where $p>\frac{n+2s}{n-2s}$ for $s\in (0,1),…

Unique continuation principles are fundamental properties of elliptic partial differential equations, giving conditions that guarantee that the solution to an elliptic equation must be uniformly zero. Since finite-element discretizations…