共查询到20条相似文献,搜索用时 15 毫秒
1.
Christian Bär 《Inventiones Mathematicae》1999,138(1):183-202
Consider a nontrivial smooth solution to a semilinear elliptic system of first order with smooth coefficients defined over
an n-dimensional manifold. Assume the operator has the strong unique continuation property. We show that the zero set of the solution
is contained in a countable union of smooth (n−2)-dimensional submanifolds. Hence it is countably (n−2)-rectifiable and its Hausdorff dimension is at most n−2. Moreover, it has locally finite (n−2)-dimensional Hausdorff measure. We show by example that every real number between 0 and n−2 actually occurs as the Hausdorff dimension (for a suitable choice of operator). We also derive results for scalar elliptic
equations of second order.
Oblatum 22-V-1998 & 26-III-1999 / Published online: 10 June 1999 相似文献
2.
Connor Mooney 《纯数学与应用数学通讯》2015,68(6):1066-1084
We prove that solutions to the Monge‐Ampère inequality in ?n are strictly convex away from a singular set of Hausdorff (n‐1)‐dimensional measure zero. Furthermore, we show this is optimal by constructing solutions to det D2u = 1 with singular set of Hausdorff dimension as close as we like to n‐1. As a consequence we obtain W2,1 regularity for the Monge‐Ampère equation with bounded right‐hand side and unique continuation for the Monge‐Ampère equation with sufficiently regular right‐hand side. © 2015 Wiley Periodicals, Inc. 相似文献
3.
Amos N. Koeller 《Calculus of Variations and Partial Differential Equations》2012,43(1-2):265-309
We consider n-dimensional hypersurfaces flowing by the mean curvature flow with Neumann free boundary conditions supported on a smooth support surface. Under assumptions mirroring those for the case of the mean curvature flow without boundary we show that the n-dimensional Hausdorff measure of the singular set is zero. 相似文献
4.
We extend both the weak separation condition and the finite type condition to include finite iterated function systems (IFSs)
of injective C
1 conformal contractions on compact subsets of
\mathbbRd{{\mathbb{R}}^d} . For conformal IFSs satisfying the bounded distortion property, we prove that the finite type condition implies the weak
separation condition. By assuming the weak separation condition, we prove that the Hausdorff and box dimensions of the attractor
are equal and, if the dimension of the attractor is α, then its α-dimensional Hausdorff measure is positive and finite. We
obtain a necessary and sufficient condition for the associated self-conformal measure μ to be singular. By using these we
give a first example of a singular invariant measure μ that is associated with a non-linear IFS with overlaps. 相似文献
5.
G. S. Weiss 《Calculus of Variations and Partial Differential Equations》2003,17(3):311-340
The equation where converges to the Dirac measure concentrated at with mass has been used as a model for the propagation of flames with high activation energy. For initial data that are bounded in
and have a uniformly bounded support, we study non-negative solutions of the Cauchy problem in as We show that each limit of is a solution of the free boundary problem in on (in the sense of domain variations and in a more precise sense). For a.e. time t the graph of u(t) has a unique tangent cone at -a.e. The free boundary is up to a set of vanishing measure the sum of a countably n-1-rectifiable set and of the set on which vanishes in the mean. The non-degenerate singular set is for a.e. time a countably n-1-rectifiable set. As key tools we introduce a monotonicity formula and, on the singular set, an estimate for the parabolic
mean frequency.
Received: 8 August 2001 / Accepted: 8 May 2002 / Published online: 5 September 2002
RID="a"
ID="a" Partially supported by a Grant-in-Aid for Scientific Research, Ministry of Education, Japan. 相似文献
6.
Atsushi Tachikawa 《Calculus of Variations and Partial Differential Equations》2003,16(2):217-224
In this paper, we consider the energy of maps from an Euclidean space into a Finsler space and study the partial regularity of energy minimizing maps. We show that the -dimensional Hausdorff measure of the singular set of every energy minimizing map is 0 for some , when m=3,4.
Received: 6 June 2001 / Accepted: 10 July 2001 / Published online: 12 October 2001 相似文献
7.
We provide general criteria for energy measures of regular Dirichlet forms on self-similar sets to be singular to Bernoulli type measures. In particular, every energy measure is proved to be singular to the Hausdorff measure for canonical Dirichlet forms on 2-dimensional Sierpinski carpets.Partially supported by Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Encouragement of Young Scientists, 15740089.Mathematics Subject Classification (2000): 28A80 (60G30, 31C25, 60J60) 相似文献
8.
Ming Li 《Calculus of Variations and Partial Differential Equations》1995,3(4):513-529
In this paper, we study energy minimizing harmonic maps into a complete Riemannian manifold. We prove that the singular set of such a map has Hausdorff dimension at mostn–2, wheren is the dimension of the domain. We will also give an example of an energy minimizing map from surface to surface that has a singular point. Thus then–2 dimension estimate is optimal, in contrast to then–3 dimension estimate of Schoen-Uhlenbeck [SU] for compact targets. 相似文献
9.
We prove that there exist self-similar sets of zero Hausdorff measure, but positive and finite packing measure, in their dimension;
for instance, for almost everyu ∈ [3, 6], the set of all sums ∑
0
8
a
n
4−n
a
n
4−n
with digits witha
n
∈ {0, 1,u} has this property. Perhaps surprisingly, this behavior is typical in various families of self-similar sets, e.g., for projections
of certain planar self-similar sets to lines. We establish the Hausdorff measure result using special properties of self-similar
sets, but the result on packing measure is obtained from a general complement to Marstrand’s projection theorem, that relates
the Hausdorff measure of an arbitrary Borel set to the packing measure of its projections.
Research of Y. Peres was partially supported by NSF grant #DMS-9803597.
Research of K. Simon was supported in part by the OTKA foundation grant F019099.
Research of B. Solomyak was supported in part by NSF grant #DMS 9800786, the Fulbright Foundation, and the Institute of Mathematics
at The Hebrew University of Jerusalem. 相似文献
10.
Igor Kukavica 《Journal d'Analyse Mathématique》1995,67(1):269-280
We establish sharp upper bounds on the (n−1)-dimensional Hausdorff measure of the zero (nodal) sets and on the maximal order of vanishing corresponding to eigenfunctions
of a regular elliptic problem on a bounded domain Ω ⊆ ℝ
n
with real-analytic boundary. The elliptic operator may be of an arbitrary even order, and its coefficients are assumed to
be real-analytic. This extends a result of Donnelly and Fefferman ([DF1], [DF3]) concerning upper bounds for nodal volumes
of eigenfunctions corresponding to the Laplacian on compact Riemannian manifolds with boundary. 相似文献
11.
Riassunto Si provano, utilizzando certe misure di ricoprimento, risultati di approssimazione in perimetro e in misura per gli aperti
limitati di perimetro finito, il cui perimetro coincide con la misura di Hausdorff (n−1)-dimensionale della frontiera.
Lavoro eseguito nell'ambito dello G.N.A.F.A. 相似文献
Summary We prove some results of approximation of a limited open Caccioppoli set, whose perimeter equals the (n−1)-dimensional Hausdorff measure of its boundary. A special geometric covering measure is employed.
Lavoro eseguito nell'ambito dello G.N.A.F.A. 相似文献
12.
We extend both the weak separation condition and the finite type condition to include finite iterated function systems (IFSs)
of injective C
1 conformal contractions on compact subsets of . For conformal IFSs satisfying the bounded distortion property, we prove that the finite type condition implies the weak
separation condition. By assuming the weak separation condition, we prove that the Hausdorff and box dimensions of the attractor
are equal and, if the dimension of the attractor is α, then its α-dimensional Hausdorff measure is positive and finite. We
obtain a necessary and sufficient condition for the associated self-conformal measure μ to be singular. By using these we
give a first example of a singular invariant measure μ that is associated with a non-linear IFS with overlaps.
The authors are supported in part by an HKRGC grant. 相似文献
13.
Jean Bourgain 《Israel Journal of Mathematics》1994,87(1-3):193-201
According to a result of K. Falconer (1985), the setD(A)={|x−y|;x, y ∈A} of distances for a Souslin setA of ℝ
n
has positive 1-dimensional measure provided the Hausdorff dimension ofA is larger than (n+1)/2.* We give an improvement of this statement in dimensionsn=2,n=3. The method is based on the fine theory of Fourier restriction phenomena to spheres. Variants of it permit further improvements
which we don’t plan to describe here. This research originated from some discussions with P. Mattila on the subject.
dimA >n/2 would be the optimal result forn ≥ 2. 相似文献
14.
Irmina Herburt 《Geometriae Dedicata》1994,49(2):221-230
We prove that a rigid set inR
n
remains rigid if we remove a countable subset of its interior. This gives us a method of obtaining (n – 1)-dimensional rigid sets inR
n
.Recently V. A. Aleksandrov announced that he had found a 1-dimensional rigid set inR
2. Our method is quite different and more general (for arbitrary dimensionn). 相似文献
15.
Kazuhiro Takimoto 《Journal of Mathematical Analysis and Applications》2005,309(1):227-237
We consider the removability of singular sets for the curvature equations of the form Hk[u]=ψ, which is determined by the kth elementary symmetric function, in an n-dimensional domain Ω. We prove that, for 1?k?n−1 and a compact set K whose (n−k)-dimensional Hausdorff measure is zero, any generalized solution to the curvature equation on Ω?K is always extendable to a generalized solution on the whole domain Ω. 相似文献
16.
Let u be a convex potential of the optimal transfer map from a convex open set X to a nonconvex open set Y in the plane. If u only has singularities whose sets of supports are one dimension, then under some mild assumptions on Y, we show that the singular set of u are disjoint union of countably many C 1 curves. Or we can say that the singular set is a C 1 manifold. 相似文献
17.
Ulrich Zähle 《Stochastic Processes and their Applications》1984,17(2):265-283
In the first main result the mean (m?n)-dimensional Hausdorff measure of the set of crossing points of a level y ? n by an m-dimensional continuous random vector field with values in R n, m?n, is computed. The second one deals with horizontal-window conditional (Palm) distributions for such random fields. For this purpose, a general concept of Palm measures is introduced, which contains both the ‘stationary’ and the ‘nonstationary’ one. 相似文献
18.
We prove that a region in a two-dimensional affine subspace of a normed space V has the least 2-dimensional Hausdorff measure among all compact surfaces with the same boundary. Furthermore, the 2-dimensional Hausdorff area density admits a convex extension to Λ2 V. The proof is based on a (probably) new inequality for the Euclidean area of a convex centrally-symmetric polygon. 相似文献
19.
Valentino Magnani 《manuscripta mathematica》2003,110(1):55-76
We obtain an intrinsic Blow-up Theorem for regular hypersurfaces on graded nilpotent groups. This procedure allows us to
represent explicitly the Riemannian surface measure in terms of the spherical Hausdorff measure with respect to an intrinsic
distance of the group, namely homogeneous distance. We apply this result to get a version of the Riemannian coarea forumula
on sub-Riemannian groups, that can be expressed in terms of arbitrary homogeneous distances. We introduce the natural class
of horizontal isometries in sub-Riemannian groups, giving examples of rotational invariant homogeneous distances and rotational
groups, where the coarea formula takes a simpler form. By means of the same Blow-up Theorem we obtain an optimal estimate
for the Hausdorff dimension of the characteristic set relative to C
1,1
hypersurfaces in 2-step groups and we prove that it has finite Q–2 Hausdorff measure, where Q is the homogeneous dimension of the group.
Received: 6 February 2002
Mathematics Subject Classification (2000): 28A75 (22E25) 相似文献
20.
R.Daniel MauldinMariusz Urbański 《Advances in Mathematics》2002,168(2):225-253
Let h be the Hausdorff dimension of the limit set of a conformal parabolic iterated function system in dimension d?2. In case the system of maps is finite, we provide necessary and sufficient conditions for the h-dimensional Hausdorff measure to be positive and finite and also, assuming the strong open set condition holds, characterize when the h-dimensional packing measure of the limit set is positive and finite. We also prove that the upper ball (box)-counting dimension and the Hausdorff dimension of this limit set coincide. As a byproduct we include a compact analysis of the behaviour of parabolic conformal diffeomorphisms in dimension 2 and separately in any dimension greater than or equal to 3. 相似文献