7,351 research outputs found

    Hilbert transforms and the Cauchy integral in euclidean space

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    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

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    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

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    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 A\mathbb{A}-Variation

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    In this paper, we consider the space BVA(Ω)\mathrm{BV}^{\mathbb A}(\Omega) of functions of bounded A\mathbb A-variation. For a given first order linear homogeneous differential operator with constant coefficients A\mathbb A, this is the space of L1L^1--functions u:Ω→RNu:\Omega\rightarrow\mathbb R^N such that the distributional differential expression Au\mathbb A u is a finite (vectorial) Radon measure. We show that for Lipschitz domains Ω⊂Rn\Omega\subset\mathbb R^{n}, BVA(Ω)\mathrm{BV}^{\mathbb A}(\Omega)-functions have an L1(∂Ω)L^1(\partial\Omega)-trace if and only if A\mathbb A is C\mathbb C-elliptic (or, equivalently, if the kernel of A\mathbb A is finite dimensional). The existence of an L1(∂Ω)L^1(\partial\Omega)-trace was previously only known for the special cases that Au\mathbb A u coincides either with the full or the symmetric gradient of the function uu (and hence covered the special cases BV\mathrm{BV} or BD\mathrm{BD}). 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 BV\mathrm{BV}- and BD\mathrm{BD}-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 Au\mathbb A u

    Co-dimension one stable blowup for the supercritical cubic wave equation

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    For the focusing cubic wave equation, we find an explicit, non-trivial self-similar blowup solution uT∗u^*_T, which is defined on the whole space and exists in all supercritical dimensions d≥5d \geq 5. For d=7d=7, 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 uT∗u^*_T (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 uT∗u^*_T 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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