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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.

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
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