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On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations. (English) Zbl 1262.35206
To start with, a brief survey of applications of local and global Carleman estimates for elliptic and parabolic equations is given. Then a number of new estimates are proved with the help of semi-classical microlocal techniques. Firstly, the authors introduce Carleman estimates for elliptic operators. They are shown to be local in the sense that they apply to functions (in the right-hand side of the equation) whose support is localized in a closed region strictly included in $$\overline{\Omega}$$. With these estimations at hand, the authors derive interpolation inequalities that satisfy some boundary conditions and derive a spectral inequality for finite linear combinations of eigenfunctions of the Laplace operator in $$\Omega$$ with homogeneous Dirichlet boundary conditions. Secondly, the authors prove Carleman estimates for parabolic equations. These estimates are characterized by an observation term. Such an estimate readily yields a so-called observability inequality, which is equivalent to the null controllability of the linear system.

##### MSC:
 35Q93 PDEs in connection with control and optimization 35B60 Continuation and prolongation of solutions to PDEs 35J15 Second-order elliptic equations 35K05 Heat equation 93B05 Controllability 93B07 Observability 35B45 A priori estimates in context of PDEs 35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
##### Keywords:
semiclassical analysis; observability inequality
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