25 research outputs found
Rigidity of the 1-Bakry-\'Emery inequality and sets of finite perimeter in RCD spaces
This note is dedicated to the study of the asymptotic behaviour of sets of
finite perimeter over RCD(K,N) metric measure spaces. Our main result asserts
existence of a Euclidean tangent half-space almost everywhere with respect to
the perimeter measure and it can be improved to an existence and uniqueness
statement when the ambient is non collapsed. As an intermediate tool, we
provide a complete characterization of the class of RCD(0,N) spaces for which
there exists a nontrivial function satisfying the equality in the
1-Bakry-\'Emery inequality. This result is of independent interest and it is
new, up to our knowledge, even in the smooth framework
Regularity of Lagrangian flows over spaces
The aim of this note is to provide regularity results for Regular Lagrangian
flows of Sobolev vector fields over compact metric measure spaces verifying the
Riemannian curvature dimension condition. We first prove, borrowing some ideas
already present in the literature, that flows generated by vector fields with
bounded symmetric derivative are Lipschitz, providing the natural extension of
the standard Cauchy-Lipschitz theorem to this setting. Then we prove a
Lusin-type regularity result in the Sobolev case (under the additional
assumption that the m.m.s. is Ahlfors regular) therefore extending the already
known Euclidean result
Boundary regularity and stability for spaces with Ricci bounded below
This paper studies the structure and stability of boundaries in noncollapsed
spaces, that is, metric-measure spaces
with lower Ricci curvature bounded below. Our
main structural result is that the boundary is homeomorphic to a
manifold away from a set of codimension 2, and is rectifiable. Along the
way we show effective measure bounds on the boundary and its tubular
neighborhoods. These results are new even for Gromov-Hausdorff limits
of smooth manifolds
with boundary, and require new techniques beyond those needed to prove the
analogous statements for the regular set, in particular when it comes to the
manifold structure of the boundary .
The key local result is an -regularity theorem, which tells us that
if a ball is sufficiently close to a half space
in the Gromov-Hausdorff sense, then
is biH\"older to an open set of . In particular,
is itself homeomorphic to near . Further, the boundary
is rectifiable and the boundary measure
is Ahlfors regular on with volume
close to the Euclidean volume.
Our second collection of results involve the stability of the boundary with
respect to noncollapsed mGH convergence . Specifically, we show a
boundary volume convergence which tells us that the Hausdorff measures on
the boundaries converge to the limit Hausdorff measure on .
We will see that a consequence of this is that if the are boundary free
then so is
Gluing non-unique Navier-Stokes solutions
We construct non-unique Leray solutions of the forced Navier-Stokes equations
in bounded domains via gluing methods. This demonstrates a certain locality and
robustness of the non-uniqueness discovered by the authors in [1]