Titlebook: Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume I; Dirichlet Boundary C Jér?me Le Rousseau,Gilles

GUISE

Carleman Estimate for a Second-Order Elliptic OperatorConsider a general second-order elliptic operator . with a principal part of the form . where . with all derivatives bounded and such that ..?=?.., 1?≤?., .?≤?.. We recall that .?=??.. The elliptic operator under consideration is then . where ., 1?≤?.?≤?..

AMITY

Postulate

珠寶

Elliptic Operator with Dirichlet Data and Associated SemigroupOn a smooth bounded open set . of ., we consider elliptic second-order operator .. given by . where . with furthermore ..?=?.., 1?≤?., .?≤?.. In addition we shall impose Dirichlet boundary conditions, that is, the trace of the solution at the boundary ..

現(xiàn)代

Some Elements of Functional AnalysisHere, . and .? will denote Banach spaces with their norms denoted by ∥.∥., ∥.∥., or simply ∥.∥when there is no ambiguity.

幾何學家

CARE

Book 2022ons, including the stabilization property of the damped wave equation and the null-controllability of the heat equation.? All analysis is performed in the case of open sets in the Euclidean space; a second volume will extend this treatment to Riemannian manifolds..The first three chapters illustrate

藝術

Introductionan estimates has gone beyond the original domain; they are also used in the study of stabilization and controllability properties of partial differential equations, two applications we shall consider in this book. Inverse problems are also a field of applications for Carleman estimate; we shall however not touch upon that subject.

HEED

fodlder

Controllability of Parabolic Equations, in the time interval (0, .), with homogeneous Dirichlet boundary conditions, and for an initial condition .. in ..(.), is given by . The function . is the control. The goal is to drive the solution . to a prescribe state at time .?>?0, yet only acting in the sub-domain .. We shall make precise what can actually be achieved below.