122,199 research outputs found
Boundary regularity for minimizing biharmonic maps
We prove full boundary regularity for minimizing biharmonic maps with smooth
Dirichlet boundary conditions. Our result, similarly as in the case of harmonic
maps, is based on the nonexistence of nonconstant boundary tangent maps. With
the help of recently derivated boundary monotonicity formula for minimizing
biharmonic maps by Altuntas we prove compactness at the boundary following
Scheven's interior argument. Then we combine those results with the conditional
partial boundary regularity result for stationary biharmonic maps by
Gong--Lamm--Wang.Comment: 30 pages, 1 figur
Representation of Markov chains by random maps: existence and regularity conditions
We systematically investigate the problem of representing Markov chains by
families of random maps, and which regularity of these maps can be achieved
depending on the properties of the probability measures. Our key idea is to use
techniques from optimal transport to select optimal such maps. Optimal
transport theory also tells us how convexity properties of the supports of the
measures translate into regularity properties of the maps via Legendre
transforms. Thus, from this scheme, we cannot only deduce the representation by
measurable random maps, but we can also obtain conditions for the
representation by continuous random maps. Finally, we present conditions for
the representation of Markov chain by random diffeomorphisms.Comment: 22 pages, several changes from the previous version including
extended discussion of many detail
Obstructions to regularity in the classical Monge problem
We provide counterexamples to regularity of optimal maps in the classical
Monge problem under various assumptions on the initial data. Our construction
is based on a variant of the counterexample in \cite{LSW} to Lipschitz
regularity of the monotone optimal map between smooth densities supported on
convex domains
Function spaces and contractive extensions in Approach Theory: The role of regularity
Two classical results characterizing regularity of a convergence space in
terms of continuous extensions of maps on one hand, and in terms of continuity
of limits for the continuous convergence on the other, are extended to
convergence-approach spaces. Characterizations are obtained for two alternative
extensions of regularity to convergence-approach spaces: regularity and strong
regularity. The results improve upon what is known even in the convergence
case. On the way, a new notion of strictness for convergence-approach spaces is
introduced.Comment: previous version had an error, fixed here with a new definition of
strictnes
Epsilon-regularity for p-harmonic maps at a free boundary on a sphere
We prove an -regularity theorem for vector-valued p-harmonic maps,
which are critical with respect to a partially free boundary condition, namely
that they map the boundary into a round sphere.
This does not seem to follow from the reflection method that Scheven used for
harmonic maps with free boundary (i.e., the case ): the reflected equation
can be interpreted as a -harmonic map equation into a manifold, but the
regularity theory for such equations is only known for round targets.
Instead, we follow the spirit of the last-named author's recent work on free
boundary harmonic maps and choose a good frame directly at the free boundary.
This leads to growth estimates, which, in the critical regime , imply
H\"older regularity of solutions. In the supercritical regime, , we
combine the growth estimate with the geometric reflection argument: the
reflected equation is super-critical, but, under the assumption of growth
estimates, solutions are regular.
In the case , for stationary -harmonic maps with free boundary, as a
consequence of a monotonicity formula we obtain partial regularity up to the
boundary away from a set of -dimensional Hausdorff measure.Comment: Minor corrections, accepted to APD
Regularity of optimal transport maps on multiple products of spheres
This article addresses regularity of optimal transport maps for cost="squared
distance" on Riemannian manifolds that are products of arbitrarily many round
spheres with arbitrary sizes and dimensions. Such manifolds are known to be
non-negatively cross-curved [KM2]. Under boundedness and non-vanishing
assumptions on the transfered source and target densities we show that optimal
maps stay away from the cut-locus (where the cost exhibits singularity), and
obtain injectivity and continuity of optimal maps. This together with the
result of Liu, Trudinger and Wang [LTW] also implies higher regularity
(C^{1,\alpha}/C^\infty) of optimal maps for more smooth (C^\alpha /C^\infty))
densities. These are the first global regularity results which we are aware of
concerning optimal maps on non-flat Riemannian manifolds which possess some
vanishing sectional curvatures. Moreover, such product manifolds have potential
relevance in statistics (see [S]) and in statistical mechanics (where the state
of a system consisting of many spins is classically modeled by a point in the
phase space obtained by taking many products of spheres). For the proof we
apply and extend the method developed in [FKM1], where we showed injectivity
and continuity of optimal maps on domains in R^n for smooth non-negatively
cross-curved cost. The major obstacle in the present paper is to deal with the
non-trivial cut-locus and the presence of flat directions.Comment: 35 pages, 4 figure
Fractional div-curl quantities and applications to nonlocal geometric equations
We investigate a fractional notion of gradient and divergence operator. We
generalize the div-curl estimate by Coifman-Lions-Meyer-Semmes to fractional
div-curl quantities, obtaining, in particular, a nonlocal version of Wente's
lemma.
We demonstrate how these quantities appear naturally in nonlocal geometric
equations, which can be used to obtain a theory for fractional harmonic maps
analogous to the local theory. Firstly, regarding fractional harmonic maps into
spheres, we obtain a conservation law analogous to Shatah's conservation law
and give a new regularity proof analogous to H\'elein's for harmonic maps into
spheres.
Secondly, we prove regularity for solutions to critical systems with nonlocal
antisymmetric potentials on the right-hand side. Since the half-harmonic map
equation into general target manifolds has this form, as a corollary, we obtain
a new proof of the regularity of half-harmonic maps into general target
manifolds following closely Rivi\`{e}re's celebrated argument in the local
case.
Lastly, the fractional div-curl quantities provide also a new, simpler, proof
for H\"older continuity of -harmonic maps into spheres and we extend
this to an argument for -harmonic maps into homogeneous targets.
This is an analogue of Strzelecki's and Toro-Wang's proof for -harmonic maps
into spheres and homogeneous target manifolds, respectively
Wave maps on (1+2)-dimensional curved spacetimes
In this article we initiate the study of 1+ 2 dimensional wave maps on a
curved spacetime in the low regularity setting. Our main result asserts that in
this context the wave maps equation is locally well-posed at almost critical
regularity.
As a key part of the proof of this result, we generalize the classical
optimal bilinear L^2 estimates for the wave equation to variable coefficients,
by means of wave packet decompositions and characteristic energy estimates.
This allows us to iterate in a curved X^{s,b} space.Comment: 100 pages, 1 figur
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