I have heard in one of the lectures I attended that subsolutions cannot touch even tangentially since both the strong maximum principle and the weak maximum principle says that subsolution doesn't intersect each other. However I don't get to see this geometry behind the weak solution. Any help is very much appreciated.
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$\begingroup$ The two questions should be asked separately, since your question one is about hyperbolic PDEs and your second one is about elliptic PDEs. $\endgroup$– Willie WongCommented Oct 29, 2022 at 14:47
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1$\begingroup$ Yes sure. Editing this. Thank you Professor. $\endgroup$– User1723Commented Oct 29, 2022 at 14:52
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2$\begingroup$ You also have to be more careful in your second question: as stated it is not true. On the interval $[-1,1]$, both $x^4$ and $x^2$ satisfy $y'' \geq 0$ so are both subsolutions, but they touch tangentially at 0. The comparison principles requires comparing the source terms too. $\endgroup$– Willie WongCommented Oct 29, 2022 at 14:53
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$\begingroup$ For general stuff about the maximum principle, the book "The Maximum Principle" by Serrin and Pucci is a standard reference. But probably the notion you are looking for is that of the viscosity sub/supersolution. There are lots of resources you can find about it if you just do a search. $\endgroup$– Willie WongCommented Oct 29, 2022 at 20:49
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