共查询到20条相似文献,搜索用时 15 毫秒
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Juha Kinnunen Riikka Korte Tuomo Kuusi Mikko Parviainen 《Mathematische Annalen》2013,355(4):1349-1381
We extend the theory of the thermal capacity for the heat equation to nonlinear parabolic equations of the $p$ -Laplacian type. We study definitions and properties of the nonlinear parabolic capacity and show that the capacity of a compact set can be represented via a capacitary potential. As an application, we show that polar sets of superparabolic functions are of zero capacity. The main technical tools used include estimates for equations with measure data and obstacle problems. 相似文献
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In this paper, we mainly investigate the dynamical properties of entire solutions of complex differential equations. With some conditions on coefficients, we prove that the set of common limiting directions of Julia sets of solutions, their derivatives and their primitives must have a definite range of measure. 相似文献
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Ricardo Mañé 《Bulletin of the Brazilian Mathematical Society》1975,6(2):145-153
In this note we considerC r semiflows on Banach spaces, roughly speakingC r flows defined only for positive values of time. Such semiflows arise as the “general solution” of a large class of partial differential equations that includes the Navier-Stokes equation. Our main result (Proposition B) is that under certain assumptions on the P.D.E. (satisfield by the Navier-Stokes equation) a hyperbolic set for the corresponding semiflow (hyperbolicity is defined following closely the finite dimensional case) is always ε-equivalent to a hyperbolic set for an ordinary differential equation that can be easily deduced from the P.D.E. As an example we consider the P.D.E. (0) $$\frac{{\partial u}}{{\partial t}} = - \Delta u + \varepsilon F(x,u,u')$$ where u:M → ? k andM is a closed smooth Riemannian manifold. Applying normal hyperbolicity techniques the phase portrait of (0) can be analyzed proving that every example of hyperbolic set for O.D.E. can appear as a hyperbolic set for the semiflow generated by (0). 相似文献
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In this note,it is shown that if a rational function fofdegree≥2 has a nonempty set of buried points ,then for a generic choice of the point z in the Julia set ,z is a buried point ,and if the Julia set is disconnected,it has uncountably many buried components. 相似文献
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John Garnett Stan Yoshinobu 《Proceedings of the American Mathematical Society》2001,129(12):3543-3548
We prove that certain Cantor sets with non-sigma-finite one- dimensional Hausdorff measure have zero analytic capacity.
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John Maginnis 《Journal of Combinatorial Theory, Series A》2010,117(7):872-883
We determine the nature of the fixed point sets of groups of order p, acting on complexes of distinguished p-subgroups (those p-subgroups containing p-central elements in their centers). The case when G has parabolic characteristic p is analyzed in detail. 相似文献
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Verena Bögelein 《Annali di Matematica Pura ed Applicata》2009,188(1):61-122
In the present paper we provide a broad survey of the regularity theory for non-differentiable higher order parabolic systems
of the type
Initially, we prove a partial regularity result with the method of A-polycaloric approximation, which is a parabolic analogue of the harmonic approximation lemma of De Giorgi. Moreover, we prove
better estimates for the maximal parabolic Hausdorff-dimension of the singular set of weak solutions, using fractional parabolic
Sobolev spaces. Thereby, we also consider different situations, which yield a better dimension reduction result, including
the low dimensional case and coefficients A(z, D
m
u), independent of the lower order derivatives of u.
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In this paper we obtain rather precise estimates for the analytic capacity of a big class of planar Cantors sets. In fact, we show that analytic capacity and positive analytic capacity are comparable for these sets. The main tool for the proof is an appropriate version of the -Theorem.
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Manfred Stoll 《Potential Analysis》1995,4(2):141-155
In the paper we investigate tangential boundary limits of invariant Green potentials on the unit ballB in ? n ,n≥1. LetG(z, w) denote the Green function for the Laplace-Beltrami operator onB, and let λ denote the invariant measure onB. If μ is a non-negative measure, orf is a non-negative measurable function onB,G μ andG f denote the Green potential of μ andf respectively. For ξ∈S=δB, τ≥1, andc>0, let $$\mathcal{T}_{\tau ,c} (\zeta ) = \{ z \in B:\left| {1 - \left\langle {z,\xi } \right\rangle } \right|^\tau< c(1 - \left| z \right|^2 )\} $$ . The main result of the paper is as follows: Letf be a non-negative measurable function onB satisfying $$\int_B {(1 - \left| w \right|^2 )^\beta f^p (w)d\lambda (w)< \infty } $$ for some β, 0<β<n, and somep>n. Then for each τ, 1≤τ<n/β, there exists a setE t ?S withH βτ (E t )=0, such that $$\mathop {\lim }\limits_{\mathop {z \to \zeta }\limits_{z \in \mathcal{T}_{\tau ,c} (\zeta )} } G_f (z) = 0,forall\zeta \in S \sim E_\tau $$ In the above, for 0<α≤n,H α denotes the non-isotropic α-dimensional Hausdorff capacity onS. We also prove that if {a k } is a sequence inB satisfying Σ(1?|a k |2) β <∞ for some β, 0 <β<n, and μ=Σδ ak , where δ a denotes point mass measure ata, then the same conclusion holds for the potentialG μ . 相似文献
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Frank Duzaar Giuseppe Mingione 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2005,22(6):705-751
We present a new, complete approach to the partial regularity of solutions to non-linear, second order parabolic systems of the form
ut−divA(x,t,u,Du)=0.