Quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients

We investigate the quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients. Quantitative unique continuation described by the vanishing order is a quantitative form of strong unique continuation property. We characterize the vanishing order of solutions for higher order elliptic equations in terms of the norms of coefficient functions in their respective Lebesgue spaces. New versions of quantitative Carleman estimates are established.     

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.     

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.     

Quantitative unique continuation for Neumann problems of elliptic equations with weight

Let P(ω, ?) be an elliptic operator with weight ω, and let u be a solution in some Lipschitz domains to ?P(ω, ?u)+W?u+Vu=0 with sharp singular potentials W and V. The weighted doubling estimates, the weighted three‐ball inequalities and the unique continuation at the boundary for solution u of the Neumann problem are established in this paper. Copyright © 2009 John Wiley & Sons, Ltd.     

Three sphere inequality for second order elliptic equations with coefficients with jump discontinuity

This is a short note to complete the paper appeared in Francini et al. (2016) [4], where a rough version of the classical well known Hadamard three-circle theorem for solution of an elliptic PDE in divergence form has been proved. Precisely, instead of circles, the authors obtain a similar inequality in a more complicated geometry. In this paper we clean the geometry and obtain a generalized version of the three-circle inequality for elliptic equation with coefficients with discontinuity of jump type.     

Global regularity for higher order divergence elliptic and parabolic equations

In this paper we obtain the global regularity estimates of the weak solutions in Sobolev spaces and Orlicz spaces for higher order elliptic and parabolic equations of divergence form with small BMO coefficients in the whole space. We only focus on the parabolic case while the corresponding result in the elliptic case can be obtained as a corollary.     

Gradient estimates for higher order elliptic equations on nonsmooth domains

We establish optimal gradient estimates in Orlicz space for a nonhomogeneous elliptic equation of higher order with discontinuous coefficients on a nonsmooth domain. Our assumption is that for each point and for each sufficiently small scale the coefficients have small mean oscillation and the boundary of the domain is sufficiently close to a hyperplane. As a consequence we prove the classical Wm,p, m=1,2,…, 1<p<∞, estimates for such a higher order equation. Our results easily extend to higher order elliptic and parabolic systems.     

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.     

The dirichlet problem in lipschitz domains for higher order elliptic systems with rough coefficients

We study the Dirichlet problem, in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order with bounded, complex-valued coefficients. A sharp corollary of our main solvability result is that the operator of this problem performs an isomorphism between weighted Sobolev spaces when its coefficients and the unit normal of the boundary belong to the space VMO.     

High-order methods for elliptic equations with variable coefficients

In this article, we give a simple method for developing finite difference schemes on a uniform square gird. We consider a general, two-dimensional, second-order, partial differential equation with variable coefficients. In the case of a nine-point scheme, we obtain the known results of Young and Dauwalder in a fairly elegant fashion. We show how this can be extended to obtain fourth-order schemes on thirteen points. We derive two such schemes which are attractive because they can be adapted quite easily bnto obtain formulas for gird points near the boundary. In addition to this, these formulas only require nine evaluations for the typical forcing function. Numerical examples are given to demonstrate the performance of one of the fourth-order schemes.     

On the Gevrey well posedness of the Cauchy problem for weakly hyperbolic equations of higher order

We shall consider the Cauchy problem for weakly hyperbolic equations of higher order with coefficients depending only on time. The regularities of the distinct characteristic roots and the multiple characteristic roots independently influence Gevrey well posedness of the Cauchy problem.