51 research outputs found
Method of scaling spheres: Liouville theorems in inner or outer (unbounded or bounded) generalized radially convex domains, blowing-up analysis on domains with not necessarily boundary and other applications
In this paper, we aim to introduce the method of scaling spheres (MSS) as a
unified approach to Liouville theorems on general domains in , and
apply it to establish Liouville theorems on arbitrary unbounded or bounded MSS
applicable domains for general (-th order) PDEs and integral equations
without translation invariance or with singularities. The set of MSS applicable
domains includes any unbounded or bounded generalized radially convex domains
and any complementary sets of their closures, which is invariant under Kelvin
transforms and is the maximal collection of simply connected domains such that
the MSS works. For instance, , , balls, cone-like
domains, convex domains, star-shaped domains and all the complements of their
closures are MSS applicable domains. MSS applicable domains is to the MSS what
convex domains is to the method of moving planes. As applications, we derive a
priori estimates and existence of solutions from the boundary H\"{o}lder
estimates for Dirichlet or Navier problems of Lane-Emden equations by applying
the blowing-up argument on domains with blowing-up cone boundary (BCB domains
for short). After the blowing-up procedure, the BCB domains allow the limiting
shape of the domain to be a cone (half space is a cone). While the classical
blowing-up techniques in previous papers work on -smooth domains, we are
able to apply blowing-up analysis on more general BCB domains on which the
boundary H\"{o}lder estimates hold (can be guaranteed by uniform exterior cone
property etc). Since there are no smoothness conditions on the boundary of MSS
applicable domain and BCB domain, our results clearly reveal that the existence
and nonexistence of solutions mainly rely on topology (not smoothness) of the
domain.Comment: In this v3, we corrected a few typos and emphasized that the empty
set is a g-radially convex domain with arbitrary point as its radially
convex cente
Maximum principles and the method of moving planes for the uniformly elliptic nonlocal Bellman operator and applications
In this paper, we establish various maximum principles and develop the method
of moving planes for equations involving the uniformly elliptic nonlocal
Bellman operator. As a consequence, we derive multiple applications of these
maximum principles and the moving planes method. For instance, we prove
symmetry, monotonicity and uniqueness results and asymptotic properties for
solutions to various equations involving the uniformly elliptic nonlocal
Bellman operator in bounded domains, unbounded domains, epigraph or
. In particular, the uniformly elliptic nonlocal
Monge-Amp\`{e}re operator introduced by Caffarelli and Charro in \cite{CC} is a
typical example of the uniformly elliptic nonlocal Bellman operator.Comment: arXiv admin note: substantial text overlap with arXiv:2002.0992
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