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Thanks. The upper bound is nonetheless informative. I wasn't familiar with irrationality measure prior to this exchange so learning a bit about that alone has been interesting.
I'd still be quite interested to know if a larger lower bound (in terms of order of delta) could be established. I'm mostly asking out of curiosity given I attended a seminar in which this was a (relevant to me anyway) query which was unanswered.
In relation to your question, it seems that there would potentially be something helpful in Watson's Bessel functions book. I can't say for sure though.
I'm not certain but a continuously differentiable function, such that at each $p\in (0,1)$, the derivative of the function is Dini continuous, but not Holder continuous, seems like it will be enough (maybe further details are required). I should emphasise the first 3 words of my comment though.
This is very much loosely related to your question (in that it doesn't require the Hermitian part ... or the strict diagonal dominance ... actually it is not that related), but the underlying idea in the Hawkins-Simon condition might be somewhat helpful to some specific problems you might consider. See en.wikipedia.org/wiki/Hawkins%E2%80%93Simon_condition.
Instead of the book, you can see the following 2 references for similar results: J. Serrin, Local behavior of solutions of quasilinear elliptic equations", Acta Math., 111, (1964), 247-302. N.S.Trudinger On Harnack type inequalities and their applications to quasilinear elliptic equations.", Comm. Pure. Appl. Math., 20, (1967), 721-747. Alternatively, a somewhat more contemporary discussion of related Harnack inequalities appears in A.I.Nazarov and N.N.Ural'tseva foud at arxiv.org/abs/1011.1888. Hopefully that helps and sorry about the way the comment is rendered.
You need growth conditions on norms of $f$ for large $s$ to make such a conclusion. Just look at a proof of a Harnack inequality for general elliptic pdi and follow it through for your pdi and refine it as much as possible. See for example "The Maximum Principle" by Pucci and Serrin for a general self contained proof of an associated Harnack inequality.
The following paper by Lieberman may help answer this question (if you increase boundary regularity in the regularised distance function construction and prove a few more results): msp.org/pjm/1985/117-2/p08.xhtml
In terms of the original question, to establish that $g'$ and $g''$ exist and are continuous, since your $v$ solves an a priori bounded bistable reaction-diffusion equation, with symmetric initial data, a global classical solution exists, which is also symmetric. Additionally, since the lap number of $v$ doesn't increase, and is initially equal to 1, it follows that $g(t)=v(t,0)$. In addition, the nonlinear term is $C^\infty (\mathbb{R})$ and initial data is $C^\infty(\mathbb{R} )$, so $v$ is smooth. Finally it follows that both $g'$ and $g''$ exist and are continuous on [0,\infty )$.
