# Maximum principle for weak solutions

Hello,

maximum principles for parabolic PDEs seem to be well-known, if the solution is a priori C^2 (cf. Protter, Weinberger: Maximum principles in differential equations). However, what about weak solutions? To be specific, are there any maximum principles on the nonnegativity of solutions $u\in W^{1,p}(0,T;L^p(\Omega))\cap L^p(0,T;W^{2,p}(\Omega))$, $p\in(1,\infty)$, where $\Omega\subset R^n$ is a bounded domain? For given nonnegative initial data, does the solution remain positive, as long as it exists?

I assume yes, since there are numerous authors that use the results from Weinberger/Protter just for weak solutions. I would appreciate any hints on this topic.

Best regards, Marc

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Solutions of parabolic equations like heat equation are smooth away from the boundary of the space-time domain. Therefore they obey the maximum principle exactly as in P.-W. –  Denis Serre Apr 8 '11 at 11:45
Thank you for your reply. Actually im working on a heat equation with lower order perturbations whose coefficient functions are only continuous. Can one still expect smoothness of solutions in this situation? –  Marc Apr 8 '11 at 13:25
Yes. See the references I gave, or any pde book. –  András Bátkai Apr 8 '11 at 15:28

Yes. If you use operaor semigroups to represent the solutions, you can infer the positivity of the mild solutions (which are the same as the weak solutions) immediately.

There is an extensive treatment of positive semigroups in R. nagel (ed.): One-parameter semigroups of positive operators, Springer, 1986.

You can find a nice introduction, with a short discussion of this topic in Engel-Nagel: A short course on operator semigroups, Springer, 2006. ChapterVI.

Of course, there are less functional analytic arguments as well, but this is what I am familiar with

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You might want to distinguish between maximum principles (which assert typically things like "the max of the solution is attained on the boundary / parabolic boundary of the set") and positivity, which assert things like "if the data are non-negative on the (parabolic) boundary, then so is the solution in the entire domain". The latter often can be shown with functional analytic techniques (see previous post).

As to maximum principles for generalized solutions, there is work by Jensen on viscosity solutions of fully nonlinear elliptic problems. And there is work in the 70s that extends maximum principles for elliptic equations to solutions in $W^{2,n}$ where $n$ is the spatial dimension (if I remember correctly).

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