Quadratic expansions and partial regularity for fully nonlinear uniformly parabolic equations

Abstract

Abstract. For a parabolic equation associated to a uniformly elliptic operator, we obtain a W 3,ε estimate, which provides a lower bound on the Lebesgue measure of the set on which a viscosity solution has a quadratic expansion. The argument combines parabolic W 2,ε estimates with a comparison principle argument. As an application, we show, assuming the operator is C 1 , that a viscosity solution is C 2,α on the complement of a closed set of Hausdorff dimension ε less than that of the ambient space, where the constant ε > 0 depends only on the dimension and the ellipticity.

Global $$W^{2,\delta }$$ W 2 , δ estimates for a type of singular fully nonlinear elliptic equations

“…This idea has originated in the work of X. Cabré [Cab97] and continued in the work of O. Savin [Sav07] (see also [IS13]). Following the same idea, J.-P. Daniel [Dan15] proved an estimate equivalent to local W 2,δ estimate for uniformly parabolic equation. For singular fully nonlinear elliptic equations, intuitively, once we have a universal control of Du L ∞ , for instance C 1,α estimate (see [BD10]), the W 2,δ estimate will then be a natural corollary of the traditional results of [Caf89], [CC95] and [Win09].…”

GlobalW2,δestimates for singular fully nonlinear elliptic equations withL

“…The sliding paraboloid argument we mentioned above has originated in the work of X. Cabré [4] and continued in the work of O. Savin [15], see also [12], [7] and [8]. We now give some other historical remarks concerning the W 2,δ estimates and the singular elliptic equations.…”

Error estimates for approximations of nonlinear uniformly parabolic equations

“…His proof also uses methods similar to those of Caffarelli and Souganidis in [8, Theorem A]. Theorem 3.2 is the only overlap between this paper and [12].…”

“…While this paper was in preparation, we learned of the preprint of Daniel [12]. It contains a result [12, Theorem 1.2] that is similar to our Theorem 3.2.…”