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Sharp L p Carleman estimates and unique continuation

David Dos Santos Ferreira (2003)

Journées équations aux dérivées partielles

We will present a unique continuation result for solutions of second order differential equations of real principal type P ( x , D ) u + V ( x ) u = 0 with critical potential V in L n / 2 (where n is the number of variables) across non-characteristic pseudo-convex hypersurfaces. To obtain unique continuation we prove L p Carleman estimates, this is achieved by constructing a parametrix for the operator conjugated by the Carleman exponential weight and investigating its L p - L p ' boundedness properties.

Spectre négatif d'un opérateur elliptique avec des conditions au bord de Robin.

Yuri V. Egorov, Mohammed El Aidi (2001)

Publicacions Matemàtiques

In this article we discuss some estimates of the number of the negative eigenvalues and their moments of energy for an elliptic operator L = L0 - V(x) defined in Hm(R+n) with the Robin boundary conditions containing a potential W(x), in terms of some integrals of V and W.

Stabilization of the Kawahara equation with localized damping

Carlos F. Vasconcellos, Patricia N. da Silva (2011)

ESAIM: Control, Optimisation and Calculus of Variations

We study the stabilization of global solutions of the Kawahara (K) equation in a bounded interval, under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. The proof requires of a unique continuation theorem and the smoothing effect of the (K) equation on the real line, which are proved in this work.

Stabilization of the Kawahara equation with localized damping

Carlos F. Vasconcellos, Patricia N. da Silva (2011)

ESAIM: Control, Optimisation and Calculus of Variations

We study the stabilization of global solutions of the Kawahara (K) equation in a bounded interval, under the effect of a localized damping mechanism. The Kawahara equation is a model for small amplitude long waves. Using multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model. The proof requires of a unique continuation theorem and the smoothing effect of the (K) equation on the real line, which are proved in this work.

Strong unique continuation for the Lamé system with Lipschitz coefficients in three dimensions

Hang Yu (2011)

ESAIM: Control, Optimisation and Calculus of Variations

This paper studies the strong unique continuation property for the Lamé system of elasticity with variable Lamé coefficients λ, µin three dimensions, div ( μ ( ∇ u + ∇ u t ) ) + ∇ ( λ div u ) + V u = 0 whereλ and μ are Lipschitz continuous and V∈L∞. The method is based on the Carleman estimate with polynomial weights for the Lamé operator.

Strong unique continuation for the Lamé system with Lipschitz coefficients in three dimensions*

Hang Yu (2011)

ESAIM: Control, Optimisation and Calculus of Variations

This paper studies the strong unique continuation property for the Lamé system of elasticity with variable Lamé coefficients λ, µ in three dimensions, div ( μ ( ∇ u + ∇ u t ) ) + ∇ ( λ div u ) + V u = 0 where λ and μ are Lipschitz continuous and V∈L∞. The method is based on the Carleman estimate with polynomial weights for the Lamé operator.

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