34 research outputs found
Duality between Ahlfors-Liouville and Khas'minskii properties for nonlinear equations
In recent years, the study of the interplay between (fully) non-linear
potential theory and geometry received important new impulse. The purpose of
this work is to move a step further in this direction by investigating
appropriate versions of parabolicity and maximum principles at infinity for
large classes of non-linear (sub)equations on manifolds. The main goal is
to show a unifying duality between such properties and the existence of
suitable -subharmonic exhaustions, called Khas'minskii potentials, which is
new even for most of the "standard" operators arising from geometry, and
improves on partial results in the literature. Applications include new
characterizations of the classical maximum principles at infinity (Ekeland,
Omori-Yau and their weak versions by Pigola-Rigoli-Setti) and of conservation
properties for stochastic processes (martingale completeness). Applications to
the theory of submanifolds and Riemannian submersions are also discussed.Comment: 67 pages. Final versio
Dirichlet parabolicity and -Liouville property under localized geometric conditions
We shed a new light on the -Liouville property for positive,
superharmonic functions by providing many evidences that its validity relies on
geometric conditions localized on large enough portions of the space. We also
present examples in any dimension showing that the -Liouville property is
strictly weaker than the stochastic completeness of the manifold. The main tool
in our investigations is represented by the potential theory of a manifold with
boundary subject to Dirichlet boundary conditions. The paper incorporates,
under a unifying viewpoint, some old and new aspects of the theory, with a
special emphasis on global maximum principles and on the role of the Dirichlet
Green's kernel
Maximum principle for semi-elliptic trace operators and geometric applications
Based on ideas of L. Al\'ias, D. Impera and M. Rigoli developed in
"Hypersurfaces of constant higher order mean curvature in warped products", we
develope a fairly general weak/Omori-Yau maximum principle for trace operators.
We apply this version of maximum principle to generalize several higher order
mean curvature estimates and to give an extension of Alias-Impera-Rigoli Slice
Theore
Detecting the completeness of a Finsler manifold via potential theory for its infinity Laplacian
In this paper, we study some potential theoretic aspects of the eikonal and
infinity Laplace operator on a Finsler manifold . Our main result shows that
the forward completeness of can be detected in terms of Liouville
properties and maximum principles at infinity for subsolutions of suitable
inequalities, including . Also, an
-capacity criterion and a viscosity version of Ekeland principle are
proved to be equivalent to the forward completeness of . Part of the proof
hinges on a new boundary-to-interior Lipschitz estimate for solutions of
on relatively compact sets, that implies a uniform
Lipschitz estimate for certain entire, bounded solutions without requiring the
completeness of
Mean exit times from submanifolds with bounded mean curvature
We show that submanifolds with infinite mean exit time can not be
isometrically and minimally immersed into cylinders, horocylinders, cones, and
wedges of some product spaces. Our approach is not based on the weak maximum
principle at infinity, and thus it permits us to generalize previous results
concerning non-immersibility of stochastically complete submanifolds. We also
produce estimates for the complete tower of moments for submanifolds with small
mean curvature immersed into cylinders.Comment: We fixed one misprint in Remark 2.
Dimethyl fumarate in patients admitted to hospital with COVID-19 (RECOVERY): a randomised, controlled, open-label, platform trial
Dimethyl fumarate (DMF) inhibits inflammasome-mediated inflammation and has been proposed as a treatment for patients hospitalised with COVID-19. This randomised, controlled, open-label platform trial (Randomised Evaluation of COVID-19 Therapy [RECOVERY]), is assessing multiple treatments in patients hospitalised for COVID-19 (NCT04381936, ISRCTN50189673). In this assessment of DMF performed at 27 UK hospitals, adults were randomly allocated (1:1) to either usual standard of care alone or usual standard of care plus DMF. The primary outcome was clinical status on day 5 measured on a seven-point ordinal scale. Secondary outcomes were time to sustained improvement in clinical status, time to discharge, day 5 peripheral blood oxygenation, day 5 C-reactive protein, and improvement in day 10 clinical status. Between 2 March 2021 and 18 November 2021, 713 patients were enroled in the DMF evaluation, of whom 356 were randomly allocated to receive usual care plus DMF, and 357 to usual care alone. 95% of patients received corticosteroids as part of routine care. There was no evidence of a beneficial effect of DMF on clinical status at day 5 (common odds ratio of unfavourable outcome 1.12; 95% CI 0.86-1.47; p = 0.40). There was no significant effect of DMF on any secondary outcome