8/11/2019 Strong maximum principle

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Hari Rau-Murthy

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The proof given here is a similar, but heavily simplified version of what is givenin A. Friedmans journal article Friedman Remarks on the maximum principlefor parabolic PDEs, 1952. I also borrow from Evans proof of Hopf boundarypoint lemma. No weird functions needed. All I need is path connectedness anda way to lift the path in in such that implicit function theorem holds. I dontneed any big inequalities like Harnacks inequality. I thought it was a goodwarmup for what we are doing in class today.

1.1 Simplified proof of the strong (parabolic)maximum

principle

1.1.1

If a closed ballEin the interior ofD = [0, T] containsP = (x0, t0) Es.t.u(P)> u(x, t) in EPand (t)u 0 in D, thenx0= xcwhere (xc, tc) is thecenter of the ellipse : If not x0 =xc. Make r small enough that u(P)> u(x, t)in Br(P) P and Br(P) D0. Br(P) is compact and P / Br(P) somaxB

r(p) u < u(P). But by W.M.P. maxBr(p) u u(P) = maxBr(p) = u(P) .contradiction.

1.1.2

Interior max at P u((), t0) constant: If not x1 s.t. u(x1, t0) < u(P).Take injective path : [0, 1] {0}, (0) = (x1, t0), (1) = P. In a nbhdof (x1, t0), u < u(P), so 0 < inf()=u(P) = 0 1. Since is smooth,

take 1 < 0 s.t. B|(1)(0)|=R((1)) D0. Since > 0 s.t. u < u(P)

in B((1)), we have 0 < r = inf{r|P Br ((1)) {t0} s.t. u(P) u((t), t) which contradicts section 1.1.2, andthereforeu(x, t) < u(P) in (t1, t0). But by the weak maximum principleon the problem on (t1, t0), U(P) = max[0,t] u max[t1,t0]t1u.But max[t1,t0]t1u < u(P) by the previous remark. Contradiction.

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