6,194 research outputs found
New formulas for the Laplacian of distance functions and applications
The goal of the paper is to prove an exact representation formula for the
Laplacian of the distance (and more generally for an arbitrary 1-Lipschitz
function) in the framework of metric measure spaces satisfying Ricci curvature
lower bounds in a synthetic sense (more precisely in essentially non-branching
MCP(K,N)-spaces). Such a representation formula makes apparent the classical
upper bounds and also some new lower bounds, together with a precise
description of the singular part. The exact representation formula for the
Laplacian of 1-Lipschitz functions (in particular for distance functions) holds
also (and seems new) in a general complete Riemannian manifold. We apply these
results to prove the equivalence of CD(K,N) and a dimensional Bochner
inequality on signed distance functions. Moreover we obtain a measure-theoretic
Splitting Theorem for infinitesimally Hilbertian essentially non-branching
spaces verifying MCP(0,N).Comment: Final version to appear in Analysis and PD
Steady nearly incompressible vector fields in 2D: chain rule and renormalization
Given bounded vector field , scalar field and a smooth function we study the characterization of the distribution
in terms of and . In the case of vector fields (and under some further assumptions)
such characterization was obtained by L. Ambrosio, C. De Lellis and J. Mal\'y,
up to an error term which is a measure concentrated on so-called
\emph{tangential set} of . We answer some questions posed in their paper
concerning the properties of this term. In particular we construct a nearly
incompressible vector field and a bounded function for which this
term is nonzero.
For steady nearly incompressible vector fields (and under some further
assumptions) in case when we provide complete characterization of
in terms of and . Our approach relies on the structure of level sets of Lipschitz functions
on obtained by G. Alberti, S. Bianchini and G. Crippa.
Extending our technique we obtain new sufficient conditions when any bounded
weak solution of is
\emph{renormalized}, i.e. also solves for any smooth function . As a
consequence we obtain new uniqueness result for this equation.Comment: 50 pages, 8 figure
On the differential structure of metric measure spaces and applications
The main goals of this paper are: i) To develop an abstract differential
calculus on metric measure spaces by investigating the duality relations
between differentials and gradients of Sobolev functions. This will be achieved
without calling into play any sort of analysis in charts, our assumptions
being: the metric space is complete and separable and the measure is Borel, non
negative and locally finite. ii) To employ these notions of calculus to
provide, via integration by parts, a general definition of distributional
Laplacian, thus giving a meaning to an expression like , where
is a function and is a measure. iii) To show that on spaces with
Ricci curvature bounded from below and dimension bounded from above, the
Laplacian of the distance function is always a measure and that this measure
has the standard sharp comparison properties. This result requires an
additional assumption on the space, which reduces to strict convexity of the
norm in the case of smooth Finsler structures and is always satisfied on spaces
with linear Laplacian, a situation which is analyzed in detail.Comment: Clarified the dependence on the Sobolev exponent of various
objects built in the paper. Updated bibliography. Corrected typo
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