共查询到20条相似文献,搜索用时 15 毫秒
1.
We develop a large-scale regularity theory of higher order for divergence-form elliptic equations with heterogeneous coefficient fields a in the context of stochastic homogenization. The large-scale regularity of a-harmonic functions is encoded by Liouville principles: The space of a-harmonic functions that grow at most like a polynomial of degree k has the same dimension as in the constant-coefficient case. This result can be seen as the qualitative side of a large-scale Ck,α-regularity theory, which in the present work is developed in the form of a corresponding Ck,α-“excess decay” estimate: For a given a-harmonic function u on a ball BR, its energy distance on some ball Br to the above space of a-harmonic functions that grow at most like a polynomial of degree k has the natural decay in the radius r above some minimal radius r0.Though motivated by stochastic homogenization, the contribution of this paper is of purely deterministic nature: We work under the assumption that for the given realization a of the coefficient field, the couple (φ, σ) of scalar and vector potentials of the harmonic coordinates, where φ is the usual corrector, grows sublinearly in a mildly quantified way. We then construct “kth-order correctors” and thereby the space of a-harmonic functions that grow at most like a polynomial of degree k, establish the above excess decay, and then the corresponding Liouville principle. 相似文献
2.
Jacek Gulgowski 《Mathematische Nachrichten》2005,278(4):401-408
In this paper we deal with boundary value problems where l : C1([a, b], ?k) → ?k × ?k is continuous, μ ≤ 0 and φ is a Caratheodory map. We define the class S of maps l, for which a global bifurcation theorem holds for the problem (+), with φ(t, x, y, λ) = λ(|x1|, …, |xk|) + o(|x| + |y|). We show that the class S contains Sturm‐Liouville boundary conditions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
3.
Daniele Bartolucci 《Mathematische Nachrichten》2011,284(13):1639-1651
We generalize a Harnack‐type inequality (I. Shafrir, C. R. Acad. Sci. Paris, 315 (1992), 159–164), for solutions of Liouville equations to the case where the weight function may admit zeroes or singularities of power‐type |x|2α, with α ∈ (?1, 1). © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim 相似文献
4.
We prove that a solution of an elliptic operator with periodic coefficients behaves on large scales like an analytic function in the sense of approximation by polynomials with periodic corrections. Equivalently, the constants in the large-scale Ck, 1 estimate scale exponentially in k , just as for the classical estimate for harmonic functions, and the minimal scale grows at most linearly in k . As a consequence, we characterize entire solutions of periodic, uniformly elliptic equations that exhibit growth like O(exp(δ| x| )) for small δ > 0 . The large-scale analyticity also implies quantitative unique continuation results, namely a three-ball theorem with an optimal error term as well as a proof of the nonexistence of L2 eigenfunctions at the bottom of the spectrum. © 2020 Wiley Periodicals LLC. 相似文献
5.
Nils Svanstedt Niklas Wellander John Wyller 《Numerical Methods for Partial Differential Equations》1996,12(4):423-440
A numerical algorithm is constructed for the solution to a class of nonlinear parabolic operators in the case of homogenization. We consider parabolic operators of the form d/dt + Aϵ, where Aϵ is monotone. More precisely, we consider the case when Aϵu=−div (a(x/ϵ, e/ϵk) |Du|p−2Du), where p≥2 and k>0. © 1996 John Wiley & Sons, Inc. 相似文献
6.
In this paper, we mainly study the Rm (m>0) Riemann boundary value problems for functions with values in a Clifford algebra C?(V3, 3). We prove a generalized Liouville‐type theorem for harmonic functions and biharmonic functions by combining the growth behaviour estimates with the series expansions for k‐monogenic functions. We obtain the result under only one growth condition at infinity by using the integral representation formulas for harmonic functions and biharmonic functions. By using the Plemelj formula and the integral representation formulas, a more generalized Liouville theorem for harmonic functions and biharmonic functions are presented. Combining the Plemelj formula and the integral representation formulas with the above generalized Liouville theorem, we prove that the Rm (m>0) Riemann boundary value problems for monogenic functions, harmonic functions and biharmonic functions are solvable. Explicit representation formulas of the solutions are given. Copyright © 2009 John Wiley & Sons, Ltd. 相似文献
7.
Given a solution u to a linear, homogeneous, second‐order elliptic equation with Lipschitz coefficients, we introduce techniques for giving improved estimates of the critical set ??(u)u {x :|δu|(x) = 0}, as well as the first estimates on the effective critical set ??r(u), which roughly consists of points x such that the gradient of u is small somewhere on Br(x) compared to the nonconstancy of u. The results are new even for harmonic functions on ?n. Given such a u, the standard first‐order stratification {lk} of u separates points x based on the degrees of symmetry of the leading‐order polynomial of u‐u(x). In this paper we give a quantitative stratification of u, which separates points based on the number of almost symmetries of approximate leading‐order polynomials of u at various scales. We prove effective estimates on the volume of the tubular neighborhood of each , which lead directly to (n‐2 + ?)‐Minkowski type estimates for the critical set of u. With some additional regularity assumptions on the coefficients of the equation, we refine the estimate to give new proofs of the uniform (n‐2)‐Hausdorff measure estimate on the critical set and singular sets of u.© 2014 Wiley Periodicals, Inc. 相似文献
8.
Arjun Krishnan 《纯数学与应用数学通讯》2016,69(10):1984-2012
We consider first‐passage percolation with positive, stationary‐ergodic weights on the square lattice ?d. Let T(x) be the first‐passage time from the origin to a point x in ?d. The convergence of the scaled first‐passage time T([nx])/n to the time constant as n → ∞ can be viewed as a problem of homogenization for a discrete Hamilton‐Jacobi‐Bellman (HJB) equation. We derive an exact variational formula for the time constant and construct an explicit iteration that produces a minimizer of the variational formula (under a symmetry assumption). We explicitly identify when the iteration produces correctors.© 2016 Wiley Periodicals, Inc. 相似文献
9.
R. Dante DeBlassie 《Stochastics An International Journal of Probability and Stochastic Processes》2013,85(4):181-203
Consider a scale invariant diffusion whose state space is a closed cone in R d , minus the vertex. Then the process is either recurrent, transient to ∞ or transient to the vertex of the cone. In the latter case, the diffusion has finite lifetime (a.s.) and converges to the vertex at the lifetime. The Martin boundary consists of two points, and the corresponding minimal harmonic functions are of the form 1 and |x| α ψ(x/|x|). 相似文献
10.
We study the regularity and behavior at the origin of solutions to the two‐dimensional degenerate Monge‐Ampère equation det D2u = |x|α with α > ?2. We show that when α > 0, solutions admit only two possible behaviors near the origin, radial and nonradial, which in turn implies C2, δ‐regularity. We also show that the radial behavior is unstable. For α < 0 we prove that solutions admit only the radial behavior near the origin. © 2008 Wiley Periodicals, Inc. 相似文献
11.
M. Revers 《Archiv der Mathematik》2000,74(5):385-391
In this paper we present a generalized quantitative version of a result due to D. L. Berman concerning the exact convergence rate at zero of Lagrange interpolation polynomials to | x| a\left | x\right | ^{\alpha } based on equally spaced nodes in [-1, 1]. The estimates obtained turn out to be best possible. 相似文献
12.
Grace Piper 《Mathematical Logic Quarterly》2009,55(5):562-568
We develop the theory of Cκ, λi, a strongly normal filter over ??κ λ for Mahlo κ. We prove a minimality result, showing that any strongly normal filter containing {x ∈ ??κ λ: |x | = |x ∩ κ | and |x | is inaccessible} also contains Cκ, λi. We also show that functions can be used to obtain a basis for Cκ, λi (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
13.
We study the Cauchy problem for a class of quasilinear hyperbolic systems with coefficients depending on (t, x) ∈ [0, T ] × ?n and presenting a linear growth for |x | → ∞. We prove well‐posedness in the Schwartz space ?? (?n ). The result is obtained by deriving an energy estimate for the solution of the linearized problem in some weighted Sobolev spaces and applying a fixed point argument. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
14.
Robert Rettinger Xizhong Zheng Romain Gengler Burchard von Braunmühl 《Mathematical Logic Quarterly》2002,48(3):459-479
Area number x is called k‐monotonically computable (k‐mc), for constant k > 0, if there is a computable sequence (xn)n ∈ ℕ of rational numbers which converges to x such that the convergence is k‐monotonic in the sense that k · |x — xn| ≥ |x — xm| for any m > n and x is monotonically computable (mc) if it is k‐mc for some k > 0. x is weakly computable if there is a computable sequence (xs)s ∈ ℕ of rational numbers converging to x such that the sum $\sum _{s \in \mathbb{N}}$|xs — xs + 1| is finite. In this paper we show that a mc real numbers are weakly computable but the converse fails. Furthermore, we show also an infinite hierarchy of mc real numbers. 相似文献
15.
Consider a Poisson process X in R d with density 1. We connect each point of X to its k nearest neighbors by undirected edges. The number k is the parameter in this model. We show that, for k = 1, no percolation occurs in any dimension, while, for k = 2, percolation occurs when the dimension is sufficiently large. We also show that if percolation occurs, then there is exactly one infinite cluster. Another percolation model is obtained by putting balls of radius zero around each point of X and let the radii grow linearly in time until they hit another ball. We show that this model exists and that there is no percolation in the limiting configuration. Finally we discuss some general properties of percolation models where balls placed at Poisson points are not allowed to overlap (but are allowed to be tangent). © 1996 John Wiley & Sons, Inc. 相似文献
16.
For fixed k ≥ 3, let Ek(x) denote the error term of the sum
?n £ xrk(n)\sum_{n\le x}\rho_k(n)
, where
rk(n) = ?n=|m|k+|l|k, g.c.d.(m,l)=1\rho_k(n) = \sum_{n=|m|^k+|l|^k, g.c.d.(m,l)=1}
1. It is proved that if the Riemann hypothesis is true, then
E3(x) << x331/1254+eE_3(x)\ll x^{331/1254+\varepsilon}
,
E4(x) << x37/184+eE_4(x)\ll x^{37/184+\varepsilon}
. A short interval result is also obtained. 相似文献
17.
Lucio Boccardo 《Annali di Matematica Pura ed Applicata》2009,188(4):591-601
The method introduced by Ennio De Giorgi and Guido Stampacchia for the study of the regularity (L
p
, Marcinkiewicz or C
0,α
) of the weak solutions of Dirichlet problems hinges on the handle of inequalities concerning the integral of on the subsets where |u(x)| is greater than k. In this framework, here we give a contribution with the study of the Marcinkiewicz regularity of the gradient of infinite
energy solutions of Dirichlet problems with nonregular data.
Dedicated to Juan Luis Vazquez for his 60th birthday (“El verano del Patriarca”, see [19]). 相似文献
18.
Let p be a prime, χ denote the Dirichlet character modulo p, f (x) = a
0 + a
1
x + ... + a
k
x
k
is a k-degree polynomial with integral coefficients such that (p, a
0, a
1, ..., a
k
) = 1, for any integer m, we study the asymptotic property of
$
\sum\limits_{\chi \ne \chi _0 } {\left| {\sum\limits_{a = 1}^{p - 1} {\chi (a)e\left( {\frac{{f(a)}}
{p}} \right)} } \right|^2 \left| {L(1,\chi )} \right|^{2m} } ,
$
\sum\limits_{\chi \ne \chi _0 } {\left| {\sum\limits_{a = 1}^{p - 1} {\chi (a)e\left( {\frac{{f(a)}}
{p}} \right)} } \right|^2 \left| {L(1,\chi )} \right|^{2m} } ,
相似文献
19.
We consider the simple random walk on the (unique) infinite cluster of super-critical bond percolation in ℤ
d
with d≥2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that
of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the
infinite cluster on which the random walk becomes a square-integrable martingale. The size of the deformation, expressed by
the so called corrector, is estimated by means of ergodicity arguments. 相似文献
20.
Let k be fixed, 1 < k < 2. There exists an infinite word over a finite alphabet which contains no subword of the form xyz with |xyz | / | xy | ≥ k and where z is a permutation of x. 相似文献
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