Parabolic systems with polynomial growth and regularity / Frank Duzaar, Giuseppe Mingione, Klaus Steffen.

Format:

Book

Author/Creator:

Duzaar, Frank, 1957- author.

Mingione, Giuseppe, 1972- author.

Steffen, Klaus, 1945- author.

Series:

Memoirs of the American Mathematical Society ; Volume 214, Number 1005.

Memoirs of the American Mathematical Society, 0065-9266 ; Volume 214, Number 1005

Language:

English

Subjects (All):

Differential equations, Parabolic.

Polynomials.

Physical Description:

1 online resource (118 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2011.

Language Note:

English

Summary:

The authors establish a series of optimal regularity results for solutions to general non-linear parabolic systems $ u_t- \mathrm{div} \ a(x,t,u,Du)+H=0,$ under the main assumption of polynomial growth at rate $p$ i.e. $ a(x,t,u,Du) \leq L(1+ Du ^{p-1}), p \geq 2.$ They give a unified treatment of various interconnected aspects of the regularity theory: optimal partial regularity results for the spatial gradient of solutions, the first estimates on the (parabolic) Hausdorff dimension of the related singular set, and the first Calderon-Zygmund estimates for non-homogeneous problems are achieved here.

Contents:

""Contents""; ""Acknowledgments""; ""Introduction""; ""Chapter 1. Results""; ""1.1. Partial regularity""; ""1.2. Singular sets estimates""; ""1.3. Extended Calderón-Zygmund theory""; ""1.4. Outline of the paper""; ""Chapter 2. Basic material, assumptions""; ""2.1. Notation, parabolic cylinders""; ""2.2. Basic assumptions, especially for partial regularity""; ""2.3. General technical results""; ""2.4. Compactness in parabolic spaces""; ""2.5. Function spaces, preliminaries""; ""2.6. Parabolic Hausdorff dimension""; ""Chapter 3. The A-caloric approximation lemma""

""8.5. Proof of the a priori estimate""""8.6. Exit times""; ""8.7. Construction of comparison maps""; ""8.8. Estimates on cylinders""; ""8.9. Estimates for super-level sets""; ""8.10. Estimate 1.20 and proof of Theorem 1.6 concluded""; ""8.11. Proof of Theorem 1.5""; ""8.12. Proof of Theorems 1.7 and 1.9""; ""8.13. Interpolative nature of estimate 1.20""; ""Bibliography""

Notes:

"Volume 214, Number 1005 (first of 5 numbers )."

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-0622-5