regularity of solution of linear elliptic PDE - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T00:10:24Z http://mathoverflow.net/feeds/question/71203 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71203/regularity-of-solution-of-linear-elliptic-pde regularity of solution of linear elliptic PDE craig 2011-07-25T09:12:41Z 2012-02-17T01:22:12Z <p>I am interested in the boundary regularity of solutions of $L(u) = f(x) \ge 0$ in $\Omega$ with zero Dirichlet boundary conditions, here $L(u) = (-\Delta)^\frac{\alpha}{2}$ where $0 &lt; \alpha &lt;2$. </p> <p>I have found results like:<br> - if $f$ bounded and $\alpha&lt;1$ then $u \in C^{t}$ (to the boundary) for all $t &lt; \alpha$. </p> <ul> <li>if $\alpha=1$ and $f$ is smooth with $f=0$ on the boundary then $u$ is $C^{2,\gamma}$ (to the boundary) for some $\gamma >0$. </li> </ul> <p>My question is related to the following calculation which seems to contradict the above results. We let $G(x,y)$ denote the Greens function associated to $L$. One can show that </p> <p>$G(x,y)= \frac{ \delta(x)^\frac{\alpha}{2} \delta(y)^\frac{\alpha}{2} }{ |x-y|^{N-\alpha} \left( \max{ |x-y|^2, \delta(x)\delta(y) } \right)^\frac{\alpha}{2} }$ </p> <p>or at least is bounded above and below by constant multiples of this and where $\delta(x)$ is the distance from $x$ to the boundary of $\Omega$. Here $N$ is the space dimension of $\Omega$. </p> <p>So using the integral representation and taking $f(x) \ge 0$ smooth and zero in a neighborhood of the boundary of $\Omega$ i seem to be able to show that $u(x)$ cannot be Holder continuous of order $> \frac{\alpha}{2}$ at the boundary. To do this let $x_m$ be a sequence that converges to $x_0$ which lies on the boundary and assume that $x_m$ approaches the boundary at right angles. Use the above representation to write out </p> <p>$u(x_m)$ and note that one can calculate the maximum for big enough $m$ since $f$ is identially zero near the boundary. Then one uses this to get a lower bound on the Holder quotient of $u$ at $x_m$ and $x_0$ and arrives at a contradiction. (I will add more details of the exact calculation if this would help). </p> <p>In anycase I cannot spot the error in my logic. </p> <p>Any comments would be apprecaited. thanks </p> http://mathoverflow.net/questions/71203/regularity-of-solution-of-linear-elliptic-pde/79313#79313 Answer by Ray Yang for regularity of solution of linear elliptic PDE Ray Yang 2011-10-27T20:55:31Z 2011-10-27T20:55:31Z <p>I don't quite have the points to make this a comment like it should be (moderators, feel free to move), but there are multiple definitions of the fractional Laplacian, which might (as in, I don't know, not they definitely can) have different regularity properties at the boundary. </p> <p>Which definition are you using? Can you give a reference for the Green's function or expound your calculations in more detail? </p>