|University of Bielefeld
Green's representation formula for solutions to Poisson's equation helps us to yield that the gradient of the solution is pointwise bounded by a Riesz potential of the right-hand side of the equation. In [Mingione, JEMS '11], the similar estimates (so-called gradient potential estimates) were obtained for the nonlinear generalization of Poisson's equation even though Green's representation for such a nonlinear equation is absent.
For the case of the nonlocal equation, the gradient potential estimates were only known for the linear equation with coefficients, see [Kuusi, Nowak, Sire, Arxiv '22]. However, the nonlinear case was not known. One of the main obstacles is that the gradient Hölder regularity is unknown for nonlinear nonlocal elliptic equations, and it is connected to the recent open question of whether the gradient Hölder regularity holds or not for any low order nonlocal equation between zero and two.
In this talk, we answer this open question affirmatively, and show that gradient potential estimates for nonlinear nonlocal elliptic equations actually hold like those available for nonlinear local equations and linear nonlocal equations. Moreover, we establish the higher differentiability estimates up to the maximal range which is naturally predicted from the order of the equation. We also obtain the singular set estimates and Lipschitz criteria for the solutions.
This is the joint work with Lars Diening (Bielefeld), Kyeongbae Kim (SNU), and Simon Nowak (Bielefeld).