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On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations

Jérôme Le Rousseau, Gilles Lebeau (2012)

ESAIM: Control, Optimisation and Calculus of Variations

Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation...

On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations∗∗∗

Jérôme Le Rousseau, Gilles Lebeau (2012)

ESAIM: Control, Optimisation and Calculus of Variations

Local and global Carleman estimates play a central role in the study of some partial differential equations regarding questions such as unique continuation and controllability. We survey and prove such estimates in the case of elliptic and parabolic operators by means of semi-classical microlocal techniques. Optimality results for these estimates and some of their consequences are presented. We point out the connexion of these optimality results to the local phase-space geometry after conjugation...

On Chemotaxis Models with Cell Population Interactions

Z. A. Wang (2010)

Mathematical Modelling of Natural Phenomena

This paper extends the volume filling chemotaxis model [18, 26] by taking into account the cell population interactions. The extended chemotaxis models have nonlinear diffusion and chemotactic sensitivity depending on cell population density, which is a modification of the classical Keller-Segel model in which the diffusion and chemotactic sensitivity are constants (linear). The existence and boundedness of global solutions of these models are discussed and...

On convergence of gradient-dependent integrands

Martin Kružík (2007)

Applications of Mathematics

We study convergence properties of { v ( u k ) } k if v C ( m × n ) , | v ( s ) | C ( 1 + | s | p ) , 1 < p < + , has a finite quasiconvex envelope, u k u weakly in W 1 , p ( Ω ; m ) and for some g C ( Ω ) it holds that Ω g ( x ) v ( u k ( x ) ) d x Ω g ( x ) Q v ( u ( x ) ) d x as k . In particular, we give necessary and sufficient conditions for L 1 -weak convergence of { det u k } k to det u if m = n = p .

On elliptic systems pertaining to the Schrödinger equation

J. Chabrowski, E. Tonkes (2003)

Annales Polonici Mathematici

We discuss the existence of solutions for a system of elliptic equations involving a coupling nonlinearity containing a critical and subcritical Sobolev exponent. We establish the existence of ground state solutions. The concentration of solutions is also established as a parameter λ becomes large.

On ergodic problem for Hamilton-Jacobi-Isaacs equations

Piernicola Bettiol (2005)

ESAIM: Control, Optimisation and Calculus of Variations

We study the asymptotic behavior of λ v λ as λ 0 + , where v λ is the viscosity solution of the following Hamilton-Jacobi-Isaacs equation (infinite horizon case) λ v λ + H ( x , D v λ ) = 0 , with H ( x , p ) : = min b B max a A { - f ( x , a , b ) · p - l ( x , a , b ) } . We discuss the cases in which the state of the system is required to stay in an n -dimensional torus, called periodic boundary conditions, or in the closure of a bounded connected domain Ω n with sufficiently smooth boundary. As far as the latter is concerned, we treat both the case of the Neumann boundary conditions (reflection on the boundary)...

On ergodic problem for Hamilton-Jacobi-Isaacs equations

Piernicola Bettiol (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We study the asymptotic behavior of λ v λ as λ 0 + , where v λ is the viscosity solution of the following Hamilton-Jacobi-Isaacs equation (infinite horizon case) λ v λ + H ( x , D v λ ) = 0 , with H ( x , p ) : = min b B max a A { - f ( x , a , b ) · p - l ( x , a , b ) } . We discuss the cases in which the state of the system is required to stay in an n-dimensional torus, called periodic boundary conditions, or in the closure of a bounded connected domain Ω n with sufficiently smooth boundary. As far as the latter is concerned, we treat both the case of the Neumann boundary conditions (reflection on the...

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