I've been reading Jost's lecture notes "Nonlinear Methods in Riemannian and Kählerian Geometry". In section 2.2 he gives a regular results about Elliptic and parabolic equations, but he doesn't prove it. And he says we can find elliptic results in "Gilbarg, D., and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Grundlehren 224, Berlin, Heidelberg, New York, 1977" A classic book on elliptic equations. the theorem But I did not find a proof of this theorem in this book. So how should I prove this regularity result? Or does anyone know where this theorem comes from?
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1$\begingroup$ The interior Holder estimates are Theorem 6.2 in Gilbarg-Trudinger wuth complete proof. I am looking at the 1983 edition but the authors say that they have not changed chapters 1-8. They have added a new chapter 9. $\endgroup$– Liviu NicolaescuCommented Mar 17 at 15:09
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$\begingroup$ @LiviuNicolaescu I noticed it early， but the domain seems different $\endgroup$– luyaoCommented Mar 17 at 15:26
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1$\begingroup$ To get the result in Jost take the domain $\Omega$ in Thm 6.2. Gilbarg-Trudinger to be the ball $B(0,r)$ in Jost Sec..2.2. You get a Holder estimate on $B(0,r)$ and thus a Holder estimate on the smaller region $B(0,r/2)$. $\endgroup$– Liviu NicolaescuCommented Mar 17 at 16:19
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