[HG_AP] Self-improving properties for nonlocal elliptic equations
ABSTRACT
Self-improving property is one of the fundamental regularity result in the theory of linear and nonlinear elliptic equations. For nonlocal elliptic equations, however, establishing such a property is significantly more complex than in the local setting. In the first part of this talk, we will discuss self improving property for fractional p-Laplacian via nonlinear commutator method developed by Armin Schikorra.
In the second part, we will address the optimality of such properties. Classically, this optimality was established by Meyers in 1963 through the construction of a counterexample. We will present a nonlocal analogue of Meyers' example, the construction is based on Fourier transform techniques for distributional convolutions. One of the key features of our example is its robustness: it remains valid as the order of the nonlocal operator converges to 2, the order of a classical second-order elliptic operator.
