Strong unique continuation for higher order elliptic equations with Gevrey coefficients

We address the strong unique continuation problem for higher order elliptic partial differential equations in 2D with Gevrey coefficients. We provide a quantitative estimate of unique continuation (observability estimate) and prove that the solutions satisfy the strong unique continuation property for ranges of the Gevrey exponent strictly including non-analytic Gevrey classes. As an application, we obtain a new upper bound on the Hausdorff length of the nodal sets of solutions with a polynomial dependence on the coefficients.     

凸区域上椭圆方程弱解的边界唯一延拓性和B_p权特性

本文研究非光滑凸区域上的散度型二阶椭圆方程 ｉ（ａｉｊ（Ｘ）　ｊｕ（Ｘ））＝０的非零弱解的近边无穷次消失性和双倍性质，刻划弱解梯度在区域边界上的Ｂｐ权特性，建立弱解和弱解的梯度在凸区域边界的任意开子集上不可能同时消失的边界唯一延拓性定理．     

Quantitative uniqueness for elliptic equations with singular lower order terms

We use a Carleman type inequality of Koch and Tataru to obtain quantitative estimates of unique continuation for solutions of second-order elliptic equations with singular lower order terms. First we prove a three sphere inequality and then describe two methods of propagation of smallness from sets of positive measure.     

Strong unique continuation for general elliptic equations in 2D

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

Smooth Compactly Supported Solutions of Some Underdetermined Elliptic PDE,with Gluing Applications

We give sufficient conditions for some underdetermined elliptic PDE of any order to construct smooth compactly supported solutions. In particular we show that two smooth elements in the kernel of certain underdetermined linear elliptic operators P can be glued in a chosen region in order to obtain a new smooth solution. This new solution is exactly equal to the initial elements outside the gluing region. This result completely contrasts with the usual unique continuation for determined or overdetermined elliptic operators. As a corollary we obtain compactly supported solutions in the kernel of P and also solutions vanishing in a chosen relatively compact open region. We apply the result for natural geometric and physics contexts such as divergence free fields or TT-tensors.     

Unique Continuation Property and Local Asymptotics of Solutions to Fractional Elliptic Equations

Asymptotics of solutions to fractional elliptic equations with Hardy type potentials is studied in this paper. By using an Almgren type monotonicity formula, separation of variables, and blow-up arguments, we describe the exact behavior near the singularity of solutions to linear and semilinear fractional elliptic equations with a homogeneous singular potential related to the fractional Hardy inequality. As a consequence we obtain unique continuation properties for fractional elliptic equations.     

Strong unique continuation property for a class of fourth order elliptic equations with strongly singular potentials

In this paper we prove the strong unique continuation property for a class of fourth order elliptic equations involving strongly singular potentials.Our argument is to establish some Hardy-Rellich type inequalities with boundary terms and introduce an Almgren’s type frequency function to show some doubling conditions for the solutions to the above-mentioned equations.     

Unique continuation for positive solutions of degenerate elliptic equations

We establish the strong unique continuation property for positive weak solutions to degenerate quasilinear elliptic equations. The degeneracy is given by a suitable power of a strong A_∞ weight (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Sharp three sphere inequality for perturbations of a product of two second order elliptic operators and stability for the Cauchy problem for the anisotropic plate equation

We prove a sharp three sphere inequality for solutions to third order perturbations of a product of two second order elliptic operators with real coefficients. Then we derive various kinds of quantitative estimates of unique continuation for the anisotropic plate equation. Among these, we prove a stability estimate for the Cauchy problem for such an equation and we illustrate some applications to the size estimates of an unknown inclusion made of different material that might be present in the plate. The paper is self-contained and the Carleman estimate, from which the sharp three sphere inequality is derived, is proved in an elementary and direct way based on standard integration by parts.     

Asymptotic behavior of the Korteweg-de Vries equation posed in a quarter plane

The purpose of this work is to study the exponential stabilization of the Korteweg-de Vries equation in the right half-line under the effect of a localized damping term. We follow the methods in [G.P. Menzala, C.F. Vasconcellos, E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping, Quart. Appl. Math. LX (1) (2002) 111-129] which combine multiplier techniques and compactness arguments and reduce the problem to prove the unique continuation property of weak solutions. Here, the unique continuation is obtained in two steps: we first prove that solutions vanishing on the support of the damping function are necessarily smooth and then we apply the unique continuation results proved in [J.C. Saut, B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations 66 (1987) 118-139]. In particular, we show that the exponential rate of decay is uniform in bounded sets of initial data.     

Nonexistence of positive solutions of nonlinear elliptic systems with potentials vanishing at infinity

We study the nonexistence of positive solutions for nonlinear elliptic systems with potentials vanishing at infinity, and establish the optimal vanishing order of potentials for the nonexistence of positive supersolutions in exterior domains.     

Quantitative uniqueness of solutions to second order elliptic equations with singular potentials in two dimensions

In this article, we study the vanishing order of solutions to second order elliptic equations with singular lower order terms in the plane. In particular, we derive lower bounds for solutions on arbitrarily small balls in terms of the Lebesgue norms of the lower order terms for all admissible exponents. Then we show that a scaling argument allows us to pass from these vanishing order estimates to estimates for the rate of decay of solutions at infinity. Our proofs rely on a new \(L^p - L^q\) Carleman estimate for the Laplacian in \(\mathbb {R}^2\).     

On unique continuation principles for some elliptic systems

In this paper we prove unique continuation principles for some systems of elliptic partial differential equations satisfying a suitable superlinearity condition. As an application, we obtain nonexistence of nontrivial (not necessarily positive) radial solutions for the Lane-Emden system posed in a ball, in the critical and supercritical regimes. Some of our results also apply to general fully nonlinear operators, such as Pucci's extremal operators, being new even for scalar equations.     

Polynomial asymptotics near zero points of solutions of general elliptic equations

We extend a classical result of Lipman Bers concerning the local behavior of solutions to a wide class of elliptic equations, systems and inequalities with singular coefficients. The main theorem states that near any zero point of finite vanishing order, the solution is asymptotic to a homogeneous polynomial solution of the “osculating” equation, under mild hypotheses on coefficients. The analysis invloves homothety blow-up arguments with the aid of some elements of Lyapunov exponents in the theory of dynamical systems.     

Unique continuation for the Schrödinger equation with gradient vector potentials

We obtain unique continuation results for Schrödinger equations with time dependent gradient vector potentials. This result with an appropriate modification also yields unique continuation properties for solutions of certain nonlinear Schrödinger equations.

     

Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities

We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension. We also prove the existence of Fowler solutions to the global equations, and establish a sup ? inf type Harnack inequality of Schoen for integral equations.     

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumptions of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma developed recently in Polá?ik et al. (2007) [20], we improve several Liouville-type theorems in higher order elliptic equations, some semilinear equations and elliptic systems. More specifically, we remove the boundedness assumption of the solutions which is required in the proofs of the corresponding Liouville-type theorems in the recent literature. Moreover, we also investigate the singularity and decay estimates of higher order elliptic equations.     

Solutions of singular elliptic equations via the oblique derivative problem

We study a class of second elliptic equations whose highest order coefficients vanish everywhere on the boundary. Under suitable conditions on the lower order coefficients, Langlais proved in 1985 that such equations have unique smooth solutions up to the boundary provided the data are smooth enough. Our goal here is to prove some Schauder estimates for these equations and to obtain results even in Lipschitz domains. In addition, we show that bounded solutions of such problems are as smooth as the data allow. A key step is to observe that smooth solutions must satisfy an oblique derivative boundary condition.     

On the first eigenpair of singularly perturbed operators with oscillating coefficients

The paper deals with a Dirichlet spectral problem for a singularly perturbed second order elliptic operator with rapidly oscillating locally periodic coefficients. We study the limit behavior of the first eigenpair (ground state) of this problem. The main tool in deriving the limit \mbox(effective) problem is the viscosity solutions technique for Hamilton-Jacobi equations. The effective problem need not have a unique solution. We study the non-uniqueness issue in a particular case of zero potential and construct the higher order term of the ground state asymptotics.     

The mixed problem for degenerate elliptic equations of second order

The present paper deals with the mixed boundary value problem for elliptic equations with degenerate rank 0. We first give the formulation of the problem and estimates of solutions of the problem, and then prove the existence of solutions of the above problem for elliptic equations by the above estimates and the method of parameter extension. We use the complex method, namely first discuss the corresponding problem for degenerate elliptic complex equations of first order, afterwards discuss the above problem for degenerate elliptic equations of second order.