Elliptic and parabolic second-order pdes with growing coefficients

N. V. Krylov, E. Priola

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40 Scopus citations


We consider a second-order parabolic equation in ℝd+1 with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally Hölder continuous in the space variables. We show that global Schauder estimates hold even in this case. The proof introduces a new localization procedure. Our results show that the constant appearing in the classical Schauder estimates is in fact independent of the L-norms of the lower order coefficients. We also give a proof of uniqueness which is of independent interest even in the case of bounded coefficients.

Original languageEnglish (US)
Pages (from-to)1-22
Number of pages22
JournalCommunications in Partial Differential Equations
Issue number1
StatePublished - 2009

Bibliographical note

Funding Information:
The first author was partially supported by NSF Grant DMS-0653121. The second author gratefully acknowledges the support by the M.I.U.R. research projects Prin 2004 and 2006 “Kolmogorov Equations”.


  • Schauder estimates
  • Second order elliptic and parabolic equations
  • Unbounded coefficients


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