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Nonlocal self-improving properties

Tuomo Kuusi, Giuseppe Mingione and Yannick Sire

Vol. 8 (2015), No. 1, 57–114

Solutions to nonlocal equations with measurable coefficients are higher differentiable.

Specifically, we consider nonlocal integrodifferential equations with measurable coefficients whose model is given by

nn[u(x)u(y)][η(x)η(y)]K(x,y)dxdy =nfηdxforallη Cc(n),

where the kernel K( ) is a measurable function and satisfies the bounds

1 Λ|x y|n+2α K(x,y) Λ |x y|n+2α

with 0 < α < 1, Λ > 1, while f Llocq(n) for some q > 2n(n + 2α). The main result states that there exists a positive, universal exponent δ δ(n,α,Λ,q) such that for every weak solution u the self-improving property

u Wα,2(n)u W locα+δ,2+δ(n)

holds. This differentiability improvement is a genuinely nonlocal phenomenon and does not appear in the local case, where solutions to linear equations in divergence form with measurable coefficients are known to be higher integrable but are not, in general, higher differentiable.

The result is achieved by proving a new version of the Gehring lemma involving certain families of lifted reverse Hölder-type inequalities in 2n and which is implied by delicate covering and exit-time arguments. In turn, such reverse Hölder inequalities are based on the concept of dual pairs, that is, pairs (μ,U) of measures and functions in 2n which are canonically associated to solutions. We also allow for more general equations involving as a source term an integrodifferential operator whose kernel does not necessarily have to be of order α.

elliptic equations, fractional differentiability, nonlocal operators
Mathematical Subject Classification 2010
Primary: 35D10
Secondary: 35R11
Received: 19 February 2014
Accepted: 22 October 2014
Published: 15 April 2015
Tuomo Kuusi
Institute of Mathematics
Aalto University
P.O. Box 11100
FI-00076 Aalto
Giuseppe Mingione
Dipartimento di Matematica e Informatica
Universitá di Parma
Parco Area delle Scienze 53/a, Campus
I-43100 Parma
Yannick Sire
Institut de Mathématique de Marseille
Université Aix-Marseille
9 rue F. Joliot Curie
13453 Marseille Cedex 13