Elliptic and parabolic second-order pdes with growing coefficients

N. V. Krylov; E. Priola

doi:10.1080/03605300903424700

Elliptic and parabolic second-order pdes with growing coefficients

N. V. Krylov, E. Priola

Mathematics

Research output: Contribution to journal › Article › peer-review

39 Scopus citations

Abstract

We consider a second-order parabolic equation in ℝ^d+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 language	English (US)
Pages (from-to)	1-22
Number of pages	22
Journal	Communications in Partial Differential Equations
Volume	35
Issue number	1
DOIs	https://doi.org/10.1080/03605300903424700
State	Published - 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”.

Keywords

Schauder estimates
Second order elliptic and parabolic equations
Unbounded coefficients

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abstract = "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 H{\"o}lder 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.",

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AU - Priola, E.

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N2 - 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.

AB - 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.

KW - Schauder estimates

KW - Second order elliptic and parabolic equations

KW - Unbounded coefficients

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Elliptic and parabolic second-order pdes with growing coefficients

Abstract

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Keywords

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