Higher order partial derivatives and global regularity. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T00:37:27Z http://mathoverflow.net/feeds/question/98807 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98807/higher-order-partial-derivatives-and-global-regularity Higher order partial derivatives and global regularity. Ainu 2012-06-04T20:16:14Z 2012-06-11T07:27:41Z <p>Let $f$ be a function of two variables $x$ and $y$. Assume that $f$ is $C^1$. Assume that $f_{xx}$ exists and continuous. </p> <ol> <li>Is it true that $f_{xy}$ exists and continuous?</li> <li>Is it true that $f_{yx}$ exists and continuous?</li> </ol> <p>I suspect that the answers are negative, so let me ask a more general question. </p> <p><b>Question</b> <i> If f is $C^k$ and $\partial f^n/\partial x^n$, $n>>k$, exists and continuous. Can one say anything about $f_x$ better than $C^{k-1}$?</i></p> http://mathoverflow.net/questions/98807/higher-order-partial-derivatives-and-global-regularity/98823#98823 Answer by timur for Higher order partial derivatives and global regularity. timur 2012-06-04T22:59:44Z 2012-06-05T03:46:09Z <p>This does not directly answer your question, but is a related phenomenon:</p> <p>Even if $f_{xx}+f_{yy}$ exists and continuous, it is possible that $f$ is not in $C^2$. This is a basic property of the solutions of $\Delta u = g$. One recovers $C^2$ if the right hand side is a bit more than merely continuous (e.g. Dini continuity is enough). In order to get the full regularity gain of 2, one has to use (at least) Hölder or Sobolev spaces. The situation is similar for the heat equation $u_{xx}-u_y=g$.</p> http://mathoverflow.net/questions/98807/higher-order-partial-derivatives-and-global-regularity/99274#99274 Answer by Ray Yang for Higher order partial derivatives and global regularity. Ray Yang 2012-06-11T07:27:41Z 2012-06-11T07:27:41Z <p>From Counterexamples in Analysis (9.10), consider the function $f(x) = xy\frac{x^2-y^2}{x^2 +y^2}$. A bit of calculation confirms that $f_{xx}$ exists and is continuous, while $f_{xy}$ is discontinuous at the origin. </p> <p>In general, consider, for example, $f(x,y) = \frac{x^k y^3}{x^2+y^2}$. You can take any number of $x$ derivatives, and the result will still be continuous, but if you take $k$ $x$-derivatives, and then a $y$-derivative, you will have a term of the form $\frac{y^2}{x^2+y^2}$ in your sum, which is problematic at the origin. So you can't get better than $C^k$ for this function. </p>