One of the most important applications of the celebrated De Giorgi-Nash result concerns quasi-linear elliptic equations. For quasi-linear parabolic equations one might see theorems on maximal regularity as the counterpart of the De Giorgi-Nash result. This property of maximal regularity has been investigated intensely during the last decade. We will consider mainly the case of evolution equations governed by forms.
For the applications to quasi-linear parabolic equations one needs to consider non-autonomous forms and equations. This leads to considerable difficulties since the needed form of maximal regularity is yet not known to be true (Lions’ Problem). Still, in virtue of recent results joint with Dier, Laasri and Ouhabaz we can solve a variant of Lions’ problem and are able to treat at least equation with isotropic coefficients.
Reference: W.Arendt, D. Dier, H. Laasri, E.M. Ouhabaz: Maximal regularity for evolution equations governed by non-autonomous forms, arXiv:1303.1166 (2013).