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2 - Sobolev Space Basics
from Part I - The Sobolev Space Setting
Houman Owhadi, California Institute of Technology, Clint Scovel, California Institute of Technology
Book:

Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization

Published online:

10 October 2019

Print publication:

24 October 2019, pp 25-33
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Summary

This chapter introduces the Sobolev spaces and self-adjointellipticoperators on those spaces that will be used through the book. It also introduces basic concepts and tools such as Gelfand triples, the Sobolev embedding theorem, the equivalence between energy norm and the Sobolev space norm, the dual norm, the Green's function, and eigenfunctions.

Embeddings for the space $LD_\gamma ^{p}$ on sets of finite perimeter
Nikolai V. Chemetov, Anna L. Mazzucato
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 5 / October 2020

Published online by Cambridge University Press:

07 May 2019, pp. 2442-2461

Print publication:

October 2020
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Given an open set with finite perimeter $\Omega \subset {\open R}^n$, we consider the space $LD_\gamma ^{p}(\Omega )$, $1\les p<\infty $, of functions with pth-integrable deformation tensor on Ω and with pth-integrable trace value on the essential boundary of Ω. We establish the continuous embedding $LD_\gamma ^{p}(\Omega )\subset L^{pN/(N-1)}(\Omega )$. The space $LD_\gamma ^{p}(\Omega )$ and this embedding arise naturally in studying the motion of rigid bodies in a viscous, incompressible fluid.

Heat Kernel Estimates Under the Ricci–Harmonic Map Flow
Part of
- Parabolic equations and systems
- Global differential geometry
Mihai Băileşteanu, Hung Tran
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 60 / Issue 4 / November 2017

Published online by Cambridge University Press:

22 February 2017, pp. 831-857
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This paper considers the Ricci flow coupled with the harmonic map flow between two manifolds. We derive estimates for the fundamental solution of the corresponding conjugate heat equation and we prove an analogue of Perelman's differential Harnack inequality. As an application, we find a connection between the entropy functional and the best constant in the Sobolev embedding theorem in ℝn .

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