Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Is there a good resource from where I can learn about parabolic Hölder spaces? I see quite a few different definitions of this space in different papers. I am clueless about why, for example, one may want your initial condition (or solution) to some PDE to be in the space $C^{k+2, \alpha}$. This is more restrictive than the space $C^{k+2}$ and I don't know what the payoff is.

How do I know which function spaces to use?

share|cite|improve this question
Considering how much of PDE theory concerns finding suitable function spaces to use for a given equation, I doubt that there's any easy answer to "how do I know which function spaces to use" besides (a) experience (someone else used it and it worked) and (b) extreme cleverness (sadly, nothing that I am qualified to explain), plus a small dose of scaling considerations. – Willie Wong Jun 26 '12 at 15:38
One simple reason why you get a payoff is the following "gain of derivatives" (let's consider the elliptic case, the parabolic case is similar): if $\Delta u=f$ in some domain and $f\in C^k$ it does not follow that $u\in C^{k+2}_{loc}$. However, if $f\in C^{k,\alpha}$ and $0<\alpha<1$, then it does follow that $u\in C^{k+2,\alpha}_{loc}$ by Schauder estimates. – YangMills Jun 26 '12 at 16:10
A widely used reference for parabolic PDE is Friedman's "PDE of Parabolic Type" which might be of some help. – Otis Chodosh Jun 26 '12 at 16:41
Thanks for the responses. I'll look for that book in the library. – user24394 Jun 27 '12 at 8:36
@quentinknight a more recent book on Holder space theory in PDE is N. V. Krylov - Lectures on Elliptic and Parabolic Equations in Hölder Spaces… – Andrew Jul 7 '12 at 16:23

1 Answer 1

The book by N. Krylov is a great reference, but there are some other online references that may help with understanding:

  • Lectures on Elliptic and Parabolic Equations in Hölder spaces by N. Krylov

  • Generalized Lagrangian Mean Curvature Flow in Almost Calabi-Yau Manifolds doctoral thesis by T. Behrndt

  • Local Existence, Uniqueness and Smooth Dependence for Nonsmooth Quasilinear Parabolic Problems by J. Griepentrog and L. Recke


  • A Generalization of a Logarithmic Sobolev Inequality to the Hölder Class by H. Ibrahim


  • Schauder and $L^p$ Estimates for Parabolic Systems via Campanato Spaces by W. Schlag

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.