Carleman estimates and unique continuation for second order parabolic equations with nonsmooth coefficients

Communications in Partial Differential Equations

Volumn 34, Issue 4, 2009, Pages 305-366

  Koch, Herbert ^a   Tataru, Daniel ^b  

^a UNIVERSITY OF BONN   (Germany)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Carleman estimates; Parabolic equations; Strong unique continuation

Indexed keywords

EID: 68749093942     PISSN: 03605302     EISSN: 15324133     Source Type: Journal    
DOI: 10.1080/03605300902740395     Document Type: Article

References (33)

1. Alessandrini, G., Vessella, S. (1988). Local behaviour of solutions to parabolic equations. Comm. Partial Differential Equations 13:1041-1058.
2. Alessandrini, G., Vessella, S. (2004). Remark on the strong unique continuation property for parabolic operators. Proc. Amer. Math. Soc. 132:499-501.
3. Aronszajn, N. (1957). A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pure Appl. 36:235-249.
4. Aronszajn, N., Krzywicki, A., Szarski, J. (1962). A unique continuation theorem for exterior differential forms on Riemannian manifolds. Ark. Mat. 4:417-453.
5. Carleman, T. (1939). Sur un problème d'unicité pur les systèmes d'équations aux dérivées partielles à deux variables indépendantes [A uniqueness question for systems of partial differential equations in two space dimensions]. Ark. Mat. Astr. Fys. 26:9.
6. Chen, X.-Y. (1998). A strong unique continuation theorem for parabolic equations. Math. Ann. 311:603-630.
7. Escauriaza, L. (2000). Carleman inequalities and the heat operator. Duke Math. J. 104:113-127.
8. Escauriaza, L., Fernández, F. J. (2003). Unique continuation for parabolic operators. Ark. Mat. 41:35-60.
9. Escauriaza, L., Vega, L. (2001). Carleman inequalities and the heat operator. II. Indiana Univ. Math. J. 50:1149-1169.
10. Escauriaza, L., Fernández, F. J., Vessella, S. (2006). Doubling properties of caloric functions. Appl. Anal. 85:205-223.
11. Fernández, F. J. (2003). Unique continuation for parabolic operators. II. Comm. Partial Differential Equations 28:1597-1604.
12. p weights and unique continuation. Indiana Univ. Math. J. 35:245-268.
13. Hörmander, L. (1983). Uniqueness theorems for second order elliptic differential equations. Comm. Partial Differential Equations 8:21-64.
14. Hörmander, L. (1994). The Analysis of Linear Partial Differential Operators. IV, Volume 275 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin: Springer-Verlag. Corrected reprint of the 1985 original.
15. Jerison, D. (1986). Carleman inequalities for the Dirac and Laplace operators and unique continuation. Adv. Math. 62:118-134.
16. Jerison, D., Kenig, C. E. (1985). Unique continuation and absence of positive eigenvalues for Schrödinger operators. Ann. Math. 121:463-494.
17. p spaces. C. R. Acad. Bulgare Sci. 47:5-8.
18. Koch, H., Tataru, D. (2001). Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients. Comm. Pure Appl. Math. 54:339-360.
19. Koch, H., Tataru, D. (2005). Dispersive estimates for principally normal pseudodifferential operators. Comm. Pure Appl. Math. 58:217-284.
20. p eigenfunction bounds for the Hermite operator. Duke Math. J. 128:369-392.
21. Landis, E. M. (1963). Some problems of the qualitative theory of second order ellitpic equations (case of several independent variables). Russian Math. Surv. 18:1-62.
22. Landis, E. M., Oleǐnik, O. A. (1974). Generalized analyticity and some related properties of solutions of elliptic and parabolic equations. Russian Math. Surv. 29:195-212.
23. Lin, F.-H. (1990). A uniqueness theorem for parabolic equations. Comm. Pure Appl. Math. 43:127-136.
24. Mizohata, S. (1958). Unicité du prolongement des solutions pour quelques opérateurs différentiels paraboliques [Unique continuation for solutions to some parabolic equations]. Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math. 31:219-239.
25. Poon, C.-C. (1996). Unique continuation for parabolic equations. Comm. Partial Differential Equations 21:521-539.
26. Saut, J.-C., Scheurer, B. (1987). Unique continuation for some evolution equations. J. Differential Equations 66:118-139.
27. p norm of spectral clusters for secondorder elliptic operators on compact manifolds. J. Funct. Anal. 77:123-138.
28. Sogge, C. D. (1989). Oscillatory integrals and unique continuation for second order elliptic differential equations. J. Amer. Math. Soc. 2:491-515.
29. Sogge, C. D. (1990). A unique continuation theorem for second order parabolic differential operators. Ark. Mat. 28:159-182.
30. Sogge, C. D. (1990). Strong uniqueness theorems for second order elliptic differential equations. Amer. J. Math. 112:943-984.
31. Thangavelu, S. (1989). Summability of Hermite expansions. I, II. Trans. Amer. Math. Soc. 314:119-142; 143-170.
32. N and an application to unique continuation. Geom. Funct. Anal. 2:225-284.
33. Yamabe, H. (1959). A unique continuation theorem of a diffusion equation. Ann. Math. 69:462-466.