Integrability of superharmonic functions in a John domain

Abstract

Abstract. The integrability of positive superharmonic functions on a bounded fat John domain is established. No exterior conditions are assumed. For a general bounded John domain the L p -integrability is proved with the estimate of p in terms of the John constant.

Parabolic BMO and global integrability of supersolutions to doubly nonlinear parabolic equations

“…Remark 2.5. The general BMO space with lag mapping (in sense of Definition 2.3) satisfies the same inequalities but U ± from Definition 2.1 are replaced by 1 8 B ± from Definition 2.3. This difference is not essential, and all the following arguments will work also in that case.…”

“…that − log u ∈ BMO for positive supersolutions, Lindqvist composed the general result. For more about research related to global integrability, see also [24], [9] and [1].…”

Global integrability and boundary estimates for uniformly elliptic PDE in divergence form

“…A classical result by Armitage [5], [6], states that if u > 0 is superharmonic in Ω then u ∈ L p (Ω) for each p < n/(n − 1), and that bound is sharp. Extensions of Armitage's result to superharmonic functions in more general domains can be found in [1], [18], [27].…”

The integrability of negative powers of the soution to the Saint Venant problem

“…This result has been subsequently extended by Maeda and Suzuki in [30] to the class of bounded Lipschitz domains for a range of p's which depends on the Lipschitz constant of the domain in question, in such a way that p % n/(n 1) as the domain is progressively closer and closer to being of class C 1 (i.e., as the Lipschitz constant approaches zero). Further refinements of this result, in the class of John domains and Hölder domains (in which scenario p is typically small), have been studied, respectively by Aikawa in [6] and by Stegenga and Ullrich in [37].…”