Basic questions about parabolic Holder space - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T18:01:28Z http://mathoverflow.net/feeds/question/104035 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104035/basic-questions-about-parabolic-holder-space Basic questions about parabolic Holder space tagwoh 2012-08-05T18:15:15Z 2012-08-24T22:04:15Z <p>Hi, I am interested in learning a bit more about this space. I have exhausted all the books available at my disposal, and none of them explain much of the basics for me. Here's a definition of this space.</p> <p>The seminorm is $$[u] = \sup_{(x,t), (y,s) \in Q} \frac{|u(x,t) - u(y,s)|}{(|x-y|^2 + |t-s|)^{\frac{\alpha}{2}}},$$ and norm $$\lVert{u}\rVert_{{C}^{k, \alpha}(\overline{Q})} = \sum_{i+2j \leq k} \lVert{\frac{\partial^{i+j}u}{\partial x^i \partial t^j}}\rVert_{C(\overline{Q})} + \sum_{i+2j = k} \bigg[\frac{\partial^{i+j}u}{\partial x^i \partial t^j}\bigg].$$</p> <p>Would someone please explain to me why the parabolic Holder space norm is chosen in the way that it is? For example, why aren't we interested in the quantity $u_{xt}$? Because it doesn't pop up in PDEs very often? Why only take the highest order seminorms in the norm?</p> <p>Also, in the denominator of the expression for seminorms, usually we have the spatial $|x-y|$ term to a power higher than the $|t-s|$ term (eg. $|x-y|^2 + |t-s|$). Why is this?</p> <p>Also, there are a number of different definitions for the norm of these spaces. Since these are norms we equip these spaces, are they somewhat equivalent? Does it really matter which one we use? </p> http://mathoverflow.net/questions/104035/basic-questions-about-parabolic-holder-space/105427#105427 Answer by Hassan Jolany for Basic questions about parabolic Holder space Hassan Jolany 2012-08-24T22:04:15Z 2012-08-24T22:04:15Z <p>In this PhD thesis you can find all necessary information about parabolic Holder Space <a href="http://www.google.fr/url?sa=t&amp;rct=j&amp;q=&amp;esrc=s&amp;source=web&amp;cd=4&amp;cad=rja&amp;ved=0CEgQFjAD&amp;url=http%3A%2F%2Fpeople.maths.ox.ac.uk%2Fjoyce%2Ftheses%2FBehrndtDPhil.pdf&amp;ei=UPk3UMG4Iue70QWh0oDIBQ&amp;usg=AFQjCNEWZ_LHmQEZsgNr47KxPWmDD8lk-g&amp;sig2=owLhC_qXR-ooOKo3DUnpYA" rel="nofollow">http://www.google.fr/url?sa=t&amp;rct=j&amp;q=&amp;esrc=s&amp;source=web&amp;cd=4&amp;cad=rja&amp;ved=0CEgQFjAD&amp;url=http%3A%2F%2Fpeople.maths.ox.ac.uk%2Fjoyce%2Ftheses%2FBehrndtDPhil.pdf&amp;ei=UPk3UMG4Iue70QWh0oDIBQ&amp;usg=AFQjCNEWZ_LHmQEZsgNr47KxPWmDD8lk-g&amp;sig2=owLhC_qXR-ooOKo3DUnpYA</a> </p>