Unique continuation for the elasticity system and a counterexample for second order elliptic systems
نویسندگان
چکیده
In this paper we study the global unique continuation property for the elasticity system and the general second order elliptic system in two dimensions. For the isotropic and the anisotropic systems with measurable coefficients, under certain conditions on coefficients, we show that the global unique continuation property holds. On the other hand, for the anisotropic system, if the coefficients are Lipschitz, we can prove that the global unique continuation is satisfied for a more general class of media. In addition to the positive results, we also present counterexamples to unique continuation and strong unique continuation for general second elliptic systems.
منابع مشابه
Analytic Solution for Hypersonic Flow Past a Slender Elliptic Cone Using Second-Order Perturbation Approximations
An approximate analytical solution is obtained for hypersonic flow past a slender elliptic cone using second-order perturbation techniques in spherical coordinate systems. The analysis is based on perturbations of hypersonic flow past a circular cone aligned with the free stream, the perturbations stemming from the small cross-section eccentricity. By means of hypersonic approximations for the ...
متن کاملOscillatory Integrals and Unique Continuation for Second Order Elliptic Differential Equations
A well-known counterexample (see [11]) indicates that the condition V E L~2 is in the best possible nature. This counterexample, however, has to do with unique continuation from a point rather than from open sets as in Theorem 1.1. We hope to treat unique continuation from a point in a subsequent paper. Results of this type for constant coefficient operators were obtained by Sawyer [14] for low...
متن کاملStrong Unique Continuation for Products of Elliptic Operators of Second Order
We study strong unique continuation for products of elliptic operators. The main tools are Carleman inequalities for second order elliptic operators. We obtain strong unique continuation assuming either Gevrey regularity of the coefficients or some pointwise conditions on the coefficients.
متن کاملZero Sets of Solutions to Semilinear Elliptic Systems of First Order
Consider a nontrivial solution to a semilinear elliptic system of first order with smooth coefficients defined over an n-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution is contained in a countable union of smooth (n − 2)dimensional submanifolds. Hence it is countably (n − 2)-rectifiable and its Hausdorff dimension ...
متن کاملSmooth counterexamples to strong unique continuation for a Beltrami system in C
We construct an example of a smooth map C → C which vanishes to infinite order at the origin, and such that the ratio of the norm of the z̄ derivative to the norm of the z derivative also vanishes to infinite order. This gives a counterexample to strong unique continuation for a vector valued analogue of the Beltrami equation.
متن کامل