Maximum Principle fails when u∉C²(Ω)? Can't find example. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T15:03:08Z http://mathoverflow.net/feeds/question/39317 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/39317/maximum-principle-fails-when-uc-cant-find-example Maximum Principle fails when u∉C²(Ω)? Can't find example. Dorian 2010-09-19T17:28:20Z 2010-09-19T20:54:09Z <p>I would like an example where the maximum principle fails in a bounded smooth domain $\Omega$ where one has a solution which is not $C^2(\Omega)$ to $Lu=0$ where $L$ is elliptic and linear. This obviously must rely on the coefficients being discontiuous for the elliptic operator since otherwise one can do interior regularity estimates. All of the examples I have tried to come up with turn out to not actually be weak solutions so I'm stuck on this. Perhaps maximum principles extend to the non-smooth setting? Any insight on this would be appreciated.</p> http://mathoverflow.net/questions/39317/maximum-principle-fails-when-uc-cant-find-example/39327#39327 Answer by Denis Serre for Maximum Principle fails when u∉C²(Ω)? Can't find example. Denis Serre 2010-09-19T19:24:39Z 2010-09-19T19:24:39Z <p>Yes indeed, the maximum principle extends to the non-smooth setting. I am not sure that there is a complete theory, because so many situations can occur. But at least let me mention the following situations.</p> <ol> <li>In Hopf's maximum principle, one only needs that $u$ admits second order derivatives (this is less than ${\mathcal C}^2$) and the coefficients of $L$ be bounded.</li> <li>In the variational case, that if $L$ is self-adjoint, the $H^1$ solution is unique and minimizes a functional $\int (Lu,u)dx$. Using the fact that if $v$ is in the Sobolev space $H^1$, then the absolute value $|v|$ is in this space too, one again proves a maximum principle.</li> <li>The previous item covers the one-dimensional case, because then second order operator can always be multiplied by an appropriate weight in such a way that they become self-adjoint. This is the reason why the spectrum of a one-D second-order operator is real (Liouville's theory).</li> <li>If $L$ is elliptic with smooth coefficients, and if $u$ is a distribution such that $Lu\in{\mathcal C}^\infty$, then $u\in{\mathcal C}^\infty$. Therefore the maximum principle is valid.</li> <li>The modern theory of second-order elliptic (not necessarily linear) equations is based upon the maximum principle. One declares that $u\in{\mathcal C}^0$ is a solution if, given an arbitrary point, there exists a super-solution (resp. a sub-solution) $\phi_\pm$ such that $\phi_\pm(x)=u(x)$, and $\phi_+\le u\le \phi_-$ otherwise.</li> </ol> http://mathoverflow.net/questions/39317/maximum-principle-fails-when-uc-cant-find-example/39337#39337 Answer by Hung Tran for Maximum Principle fails when u∉C²(Ω)? Can't find example. Hung Tran 2010-09-19T20:54:09Z 2010-09-19T20:54:09Z <p>Maximum principle, in general, can be applied in the viscosity solutions setting (in this case, the viscosity solution is only continuous). You can have a look at the excellent paper by Crandall-Ishii-Lions as following: <a href="http://arxiv.org/abs/math/9207212" rel="nofollow">http://arxiv.org/abs/math/9207212</a></p>