7,351 research outputs found
Hilbert transforms and the Cauchy integral in euclidean space
We generalize the notion of harmonic conjugate functions and Hilbert
transforms to higher dimensional euclidean spaces, in the setting of
differential forms and the Hodge-Dirac system. These conjugate functions are in
general far from being unique, but under suitable boundary conditions we prove
existence and uniqueness of conjugates. The proof also yields invertibility
results for a new class of generalized double layer potential operators on
Lipschitz surfaces and boundedness of related Hilbert transforms.Comment: Some minor corrections mad
Inertial manifolds and finite-dimensional reduction for dissipative PDEs
These notes are devoted to the problem of finite-dimensional reduction for
parabolic PDEs. We give a detailed exposition of the classical theory of
inertial manifolds as well as various attempts to generalize it based on the
so-called Man\'e projection theorems. The recent counterexamples which show
that the underlying dynamics may be in a sense infinite-dimensional if the
spectral gap condition is violated as well as the discussion on the most
important open problems are also included.Comment: This manuscript is an extended version of the lecture notes taught by
the author as a part of the crash course in the Analysis of Nonlinear PDEs at
Maxwell Center for Analysis and Nonlinear PDEs (Edinburgh, November, 8-9,
2012
Inverse limit spaces satisfying a Poincare inequality
We give conditions on Gromov-Hausdorff convergent inverse systems of metric
measure graphs (and certain higher dimensional inverse systems of metric
measure spaces) which imply that the measured Gromov-Hausdorff limit
(equivalently, the inverse limit) is a PI space, i.e. it satisfies a doubling
condition and a Poincare inequality in the sense of Heinonen-Koskela. We also
give a systematic construction of examples for which our conditions are
satisfied. Included are known examples of PI spaces, such as Laakso spaces, and
a large class of new examples. Generically our graph examples have the property
that they do not bilipschitz embed in any Banach space with Radon-Nikodym
property, but they do embed in the Banach space L_1. For Laakso spaces, these
facts were discussed in our earlier papers
On the Trace Operator for Functions of Bounded -Variation
In this paper, we consider the space of
functions of bounded -variation. For a given first order linear
homogeneous differential operator with constant coefficients , this
is the space of --functions such that the
distributional differential expression is a finite (vectorial)
Radon measure. We show that for Lipschitz domains ,
-functions have an -trace
if and only if is -elliptic (or, equivalently, if the
kernel of is finite dimensional). The existence of an
-trace was previously only known for the special cases
that coincides either with the full or the symmetric gradient of
the function (and hence covered the special cases or
). As a main novelty, we do not use the fundamental theorem of
calculus to construct the trace operator (an approach which is only available
in the - and -setting) but rather compare projections
onto the nullspace as we approach the boundary. As a sample application, we
study the Dirichlet problem for quasiconvex variational functionals with linear
growth depending on
Co-dimension one stable blowup for the supercritical cubic wave equation
For the focusing cubic wave equation, we find an explicit, non-trivial
self-similar blowup solution , which is defined on the whole space and
exists in all supercritical dimensions . For , we analyze its
stability properties without any symmetry assumptions and prove the existence
of a co-dimension one Lipschitz manifold consisting of initial data whose
solutions blowup in finite time and converge asymptotically to (modulo
space-time shifts and Lorentz boosts) in the backward lightcone of the blowup
point. Furthermore, based on numerical simulations we perform in a separate
work, we conjecture that the stable manifold of is in fact a threshold
for blowup.Comment: 47 pages; v2 abstract and introduction adjusted to include the
reference to authors' separate numerical work with Maliborsk
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