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A Harnack inequality approach to the interior regularity of parabolic equations. (English) Zbl 0811.35046
From author’s introduction: We prove $$C^{1, \alpha}$$ regularity for the evolution $$S(t)$$ of surfaces whose normal velocity $$S_ t$$ equals to a function of its curvature tensor $$II$$, normal vector $$\nu$$, position vector $$X$$ and time $$t$$: $$S_ t - F(II, \nu, X,t) = R(\nu, X,t)$$, where the curvature tensor $$II$$ is the second fundamental form in the space direction only and $$\nu$$ its normal in the space direction. By the compactness arguments $$\dots$$ we see that the $$C^{1,\alpha}$$ regularity for this equation actually reduces to $$C^{1, \alpha}$$ regularity for much simpler equations of the type $$S_ t - F(II, \nu) = 0$$.
Reviewer: O.John (Praha)

##### MSC:
 35K55 Nonlinear parabolic equations 35D10 Regularity of generalized solutions of PDE (MSC2000)
##### Keywords:
evolution of surfaces; Harnack inequality; regularity
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