Strong maximum principle for weak solutions

Question

Suppose I have a linear parabolic equation with solutions in the Bochner-Sobolev spaces (eg. $L^2(0,T;H^1) \cap H^1(0,T;H^{-1})$). Is it possible to obtain a strong maximum principle with a proof that does not rely on classical solutions or techniques (and with minimal semigroup theory as possible)?

By the strong maximum principle I mean that:

if the parabolic operator has some given sign and the maximum occurs in the interior then the solution must be a constant.

I read the thread Maximum principle for weak solutions but it is slightly different. Note I also posted this on MSE https://math.stackexchange.com/questions/910740/strong-maximum-principle-for-weak-solutions.

Answer 1

I'm not immediately sure how to formulate a strong maximum principle if you are trying to avoid "classical" techniques where you understand the solution as being regular enough to be well-defined pointwise. I'm also a little unclear as to which techniques you do or don't want to use.

However, I will mention that Grigor'yan and Hu obtained a maximum principle for weak solutions in the context of a metric measure space equipped with a symmetric Dirichlet form, which may include examples of interest or suggest a useful approach. See Section 4 of:

Grigorʹyan, Alexander; Hu, Jiaxin. Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces. Invent. Math. 174 (2008), no. 1, 81–126. MathSciNet Article

Laurent Saloff-Coste and I recorded a slightly different version (and included a few other details in the proof) in our paper "Widder's theorem for symmetric local Dirichlet spaces", which is on arXiv and will appear in Journal of Theoretical Probability.