Arkiv för Matematik

Unique continuation for parabolic operators

Luis Escauriaza and Francisco Javier Fernández

Full-text: Open access


It is shown that if a function u satisfies a backward parabolic inequality in an open set Ω∉Rn+1 and vanishes to infinite order at a point (x0·t0) in Ω, then u(x, t0)=0 for all x in the connected component of x0 in Ω⌢(Rn×{t0}).

Article information

Ark. Mat., Volume 41, Number 1 (2003), 35-60.

Received: 26 November 2001
First available in Project Euclid: 31 January 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

2003 © Institut Mittag-Leffler


Escauriaza, Luis; Fernández, Francisco Javier. Unique continuation for parabolic operators. Ark. Mat. 41 (2003), no. 1, 35--60. doi:10.1007/BF02384566.

Export citation


  • Adolfson, V. and Escauriaza, L., C1·α domains and unique continuation at the boundary. Comm. Pure Appl. Math. 50 (1997), 935–969.
  • Ahlfors, L. V., Complex Analysis, 3rd ed., McGraw-Hill, New York, 1966.
  • Aronszajn, N., Krzywicki, A. and Szarski, J., A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. Mat. 4 (1962), 417–453.
  • Chen, X. Y., A strong unique continuation theorem for parabolic equations, Math. Ann. 311 (1996), 603–630.
  • Escauriaza, L., Carleman inequalities and the heat operator, Duke Math. J. 104 (2000), 113–127.
  • Escauriaza, L. and Vega, L., Carleman inequalities and the heat operator II, Indiana Univ. Math. J. 50 (2001), 1149–1169.
  • Evans, L. G., Partial Differential Equations, Amer. Math. Soc., Providence, R. I., 1998.
  • Grisvard, P., Elliptic Problems in Nonsmooth Domains, Pitman, Boston. Mass., 1985.
  • Hörmander, L., Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations 8 (1983), 21–64.
  • Kenig, C. E. and Wang, W., A note on boundary unique continuation for harmonic functions in non-smooth domains, Potential Anal. 8 (1998), 143–147.
  • Ladyzhenskaya, O. A., Solonnikov, V. A. and Uraltseva, N. N., Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1968 (Russian). English transl.: Translations of Math. Monographs 23, Amer. Math. Soc., Providence, R. I., 1968.
  • Landis, E. M. and Oleinik, O. A., Generalized analyticity and some related properties of solutions of elliptic and parabolic equations, Uspekhi Mat. Nauk 29(176):2 (1974), 190–206 (Russian). English transl.: Russian Math. Surveys 29 (1974), 195–212.
  • Lin, F. H., A uniqueness theorem for parabolic equations, Comm. Pure Appl. Math. 42 (1988), 125–136.
  • Miller, K., Non-unique continuation for certain ode’s in Hilbert space and for uniformly parabolic and elliptic equations in self-adjoint divergence form, Arch. Rational Mech. Anal. 54 (1963), 105–117.
  • Plis, A., On non-uniqueness in Cauchy problems for an elliptic second order differential equation, Bull. Acad. Polon. Sci. Math. Astronom. Phys. 11 (1963), 95–100.
  • Poon, C. C., Unique continuation for parabolic equations, Comm. Partial Differential Equations 21 (1996), 521–539.
  • Saut, J. C. and Scheurer, E., Unique continuation for evolution equations, J. Differential Equations 66 (1987). 118–137.
  • Sogge, C. D., A unique continuation theorem for second order parabolic differential operators, Ark. Mat. 28 (1990), 159–182.