# Maximum principle for weak solutions

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

• 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. Apr 8, 2011 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, 2011 at 13:25
• Yes. See the references I gave, or any pde book. Apr 8, 2011 at 15:28
• @DenisSerre: I guess this question has nontheless merit in the spirit of showing positivity via a quick argument (not having to bother with a demonstration of the Weyl lemma, which always strikes me as difficult?), for an occasion when you are not interested in the regularity that usually comes as the byproduct of the usual arguments. Feb 18, 2020 at 1:05

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

• Your references don't delve into the question when precisely the resolvents are positive (they shouldn't: because that's Lax-Milgram theory, not semi-group theory). As far as I know (?), the Lax-Milgram apparatus is not very efficient in demonstrating positivity of resolvents directly and a long and bothersome detour (establishing local regularity) must be made to show that the resolvent maps to strong solutions which are then shown to be positive using classical maximum principle. This is the account I see in Qian, Qian, Jiang's book "Mathematical theory of nonequilibrium steady states". Feb 18, 2020 at 0:58
• Let me know if I have overlooked something. To be clear: my point is that you're not addressing the OP's question. Feb 18, 2020 at 1:00
• @ThibautDemaerel: what I wanted to say (badly) ist that 1) Maximum (and) minimum princiles imply positivity of the semigroup and 2) Weak solutions are mild solutions.I did not want to say anything about Lax-Milgram or resolvent positivity. Feb 18, 2020 at 11:52
• @ThibautDemaerel: I thought this would adresses the question (Maximum princilpe implies positivity of weak solution). See also my more recent book on the subject. Feb 18, 2020 at 11:54
• @ThibautDemaerel: There are at least two ways to show positivity of a semigroup without extensive use of regularity theory: (i) If you use form methods (which works on $L^2$-spaces and comes essentially down to the Lax-Milgram lemma that you mentioned) you can use the Beurling-Deny criterion (and its generalization by Ouhabaz) to prove positivity. In fact, this is one of the most efficient ways that I know to obtain positivity of a semigroup. (ii) On other spaces, it is known that dispersive operators generate positive semigroups - and this assumption can also be checked in some applications. Feb 20, 2020 at 19:27

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).