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I am interested in Holder regularity for equations of the form $$u_t - div A(x,t) \nabla u = 0$$ where $A(x,t)$ is bounded, measurable and elliptic.

This was proved in the seminal paper of John Nash and later on by J. Moser.

I am looking for references where Ennio De Giorgi's methods are implemented to obtain the same regularity estimates.

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    $\begingroup$ You may enjoy reading De Giorgi's original paper (which only deals with the elliptic case, though). To get an English translation of his paper just google "Ennio De Giorgi : selected papers"... $\endgroup$
    – YangMills
    Commented May 31, 2018 at 20:46

2 Answers 2

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The proof of Harnack's inequality using De Giorgi method has a great flexibility and it can be exteneded even to doubling metric measure spaces that support Poincaré inequalities.

The elliptic case has been treated in:

J. Kinnunen, N. Shanmugalingam, Regularity of quasi-minimizers on metric spaces. Manuscripta Math. 105 (2001), 401–423. (MathScNet review.)

The parabolic case has been treated in:

J. Kinnunen, N. Marola, M. Miranda, Jr., F. Paronetto, Harnack's inequality for parabolic De Giorgi classes in metric spaces. Adv. Differential Equations 17 (2012), 801–832. (MathScNet review.)

There are many other related results. Just search references and citations of these two paper in MathSciNet.

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You can see the book of Emmanuele Dibenedetto, which deals with the operators (nonlinear) with divergence form. This book contains also a large reference, maybe you need it.

Degenerate Parabolic Equations

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