برآوردهای کارلمن، نابرابری‌های قابل مشاهده و کنترل‌پذیری صفر برای معادلات پارابولیک غیرهموار و داخلی

The authors consider a parabolic problem with degeneracy in the interior of the spatial domain, and they focus on observability results through Carleman estimates for the associated adjoint problem. The novelties of the present paper are two. First, the coefficient of the leading operator only belongs to a Sobolev space. Second, the degeneracy point is allowed to lie even in the interior of the control region, so that no previous result can be adapted to this situation; however, different cases can be handled, and new controllability results are established as a consequence. Cover 1 Title page 2 Chapter 1. Introduction 8 Chapter 2. Mathematical tools and preliminary results 14 2.1. Well-posedness in the divergence case 14 2.2. Well-posedness in the non divergence case 16 Chapter 3. Carleman estimate for non degenerate parabolic problems with non smooth coefficient 22 3.1. Preliminaries 22 3.2. The divergence case. 23 3.3. The non divergence case 33 Chapter 4. Carleman estimate for degenerate non smooth parabolic problems 40 4.1. Carleman estimate for the problem in divergence form 40 4.2. Carleman estimate for the problem in non divergence form 45 Chapter 5. Observability inequalities and application to null controllability 52 5.1. The divergence case 52 5.2. The non divergence case 64 Chapter 6. Linear and Semilinear Extensions 74 Chapter 7. Final Comments 80 Appendix A. Rigorous derivation of Lemma 3.5 84 Acknowledgement 87 Bibliography 88 Back Cover 96