共查询到20条相似文献,搜索用时 31 毫秒
1.
In this paper, we first establish a constant rank theorem for the second fundamental form of the convex level sets of harmonic functions in space forms. Applying the deformation process, we prove that the level sets of the harmonic functions on convex rings in space forms are strictly convex. Moreover, we give a lower bound for the Gaussian curvature of the convex level sets of harmonic functions in terms of the Gaussian curvature of the boundary and the norm of the gradient on the boundary. 相似文献
2.
《偏微分方程通讯》2013,38(7-8):1267-1279
Abstract We study L 2 harmonic p-forms on conformally compact manifolds with a rather weak boundary regularity assumption. We proved that if the lower bound of the curvature operator is great than or equal to ?1 and the infimum of the L 2 spectrum of the Laplacian great than p(n ? p) for some p ≤ n/2, then there is no nontrivial L 2 harmonic p-form. 相似文献
3.
Gaussian Curvature Estimates for the Convex Level Sets of Solutions for Some Nonlinear Elliptic Partial Differential Equations
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Peihe Wang & Wei Zhang 《偏微分方程(英文版)》2012,25(3):239-275
We give lower bound estimates for the Gaussian curvature of convex level sets of minimal surfaces and the solutions to semilinear elliptic equations in terms of the norm of boundary gradient and the Gaussian curvature of the boundary. 相似文献
4.
Ana-Maria Matei 《Annals of Global Analysis and Geometry》2000,18(6):541-554
In this paper, we generalize to p-harmonic mapssome gap results known for harmonic maps. In particular, we prove that,under a certain level of energy depending on the curvature of the domainand target manifolds, the only p-harmonic maps are theconstant ones. The main tools are Bochner–Weitzenböck andReilly-type formulas involving the p-Laplace operator. 相似文献
5.
Juan A. Aledo Victorino Lozano José A. Pastor 《Mediterranean Journal of Mathematics》2010,7(3):263-270
We prove that the only compact surfaces of positive constant Gaussian curvature in
\mathbbH2×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} (resp. positive constant Gaussian curvature greater than 1 in
\mathbbS2×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}}) whose boundary Γ is contained in a slice of the ambient space and such that the surface intersects this slice at a constant
angle along Γ, are the pieces of a rotational complete surface. We also obtain some area estimates for surfaces of positive
constant Gaussian curvature in
\mathbbH2×\mathbbR{\mathbb{H}^{2}\times\mathbb{R}} and positive constant Gaussian curvature greater than 1 in
\mathbbS2×\mathbbR{\mathbb{S}^{2}\times\mathbb{R}} whose boundary is contained in a slice of the ambient space. These estimates are optimal in the sense that if the bounds
are attained, the surface is again a piece of a rotational complete surface. 相似文献
6.
S. B. Klimentov 《Journal of Mathematical Sciences》1994,69(1):862-868
It is proved that a regular Riemannian manifold diffeomorphic to a circle and having positive Gaussian curvature bounded from zero is immersible into a three-dimensional Euclidean space in the form of a regular surface if it has smallL
p (the norm of the gradient of Gaussian curvature), p > 2, or if it has a sufficiently small area (with any behavior of the geodesic boundary curvature).Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 51–59, 1991. 相似文献
7.
In this paper, we use the harmonic sequence to study the linearly full holomorphic two-spheres in complex Grassmann manifold
G(2, 4). We show that if the Gaussian curvature K (with respect to the induced metric) of a non-degenerate holomorphic two-sphere satisfies K ≤ 2 (or K ≥ 2), then K must be equal to 2. Simultaneously, we show that one class of the holomorphic two-spheres with constant curvature 2 is totally
geodesic. Concerning the degenerate holomorphic two-spheres, if its Gaussian curvature K ≤ 1 (or K ≥ 1), then K = 1. Moreover, we prove that all holomorphic two-spheres with constant curvature 1 in G(2, 4) must be U (4)-equivalent. 相似文献
8.
A representation of the sharp constant in a pointwise estimate for the absolute value of the directional derivative of a harmonic
function in a multidimensional ball is obtained under the assumption that the boundary values of the function belong to L
p
. This representation is specified in the cases of radial and tangential derivatives. It is proved for p = 1 and p = 2 that the maximum of the absolute value of the directional derivative of a harmonic function with a fixed L
p
-norm of its boundary values is attained at the radial direction. This confirms D. Khavinson’s conjecture for p = 1 and p = 2. Bibliography: 11 titles. 相似文献
9.
D. Azagra M. Jiménez-Sevilla F. Macià 《Calculus of Variations and Partial Differential Equations》2008,33(2):133-167
We consider the generalized evolution of compact level sets by functions of their normal vectors and second fundamental forms
on a Riemannian manifold M. The level sets of a function evolve in such a way whenever u solves an equation u
t
+ F(Du, D
2
u) = 0, for some real function F satisfying a geometric condition. We show existence and uniqueness of viscosity solutions to this equation under the assumptions
that M has nonnegative curvature, F is continuous off {Du = 0}, (degenerate) elliptic, and locally invariant by parallel translation. We then prove that this approach is geometrically
consistent, hence it allows to define a generalized evolution of level sets by very general, singular functions of their curvatures.
For instance, these assumptions on F are satisfied when F is given by the evolutions of level sets by their mean curvature (even in arbitrary codimension) or by their positive Gaussian
curvature. We also prove that the generalized evolution is consistent with the classical motion by the corresponding function
of the curvature, whenever the latter exists. When M is not of nonnegative curvature, the same results hold if one additionally requires that F is uniformly continuous with respect to D
2
u. Finally we give some counterexamples showing that several well known properties of the evolutions in are no longer true when M has negative sectional curvature.
D. Azagra was supported by grants MTM-2006-03531 and UCM-CAM-910626. M. Jimenez-Sevilla was supported by a fellowship of the
Ministerio de Educacion y Ciencia, Spain. F. Macià was supported by program “Juan de la Cierva” and projects MAT2005-05730-C02-02
of MEC (Spain) and PR27/05-13939 UCM-BSCH (Spain). 相似文献
10.
《Mathematische Nachrichten》2017,290(2-3):293-320
Basic aspects of the equiaffine geometry of level sets are developed systematically. As an application there are constructed families of 2n‐dimensional nondegenerate hypersurfaces ruled by n‐planes, having equiaffine mean curvature zero, and solving the affine normal flow. Each carries a symplectic structure with respect to which the ruling is Lagrangian. 相似文献
11.
B. Ammann 《manuscripta mathematica》2000,101(1):1-22
The Willmore conjecture states that any immersion of a 2-torus into euclidean space satisfies . We prove it under the condition that the L
p
-norm of the Gaussian curvature is sufficiently small.
Received: 10 June 1999 相似文献
12.
T. Arens 《Mathematical Methods in the Applied Sciences》2002,25(6):507-528
We consider the two‐dimensional problem of the scattering of a time‐harmonic wave, propagating in an homogeneous, isotropic elastic medium, by a rough surface on which the displacement is assumed to vanish. This surface is assumed to be given as the graph of a function ?∈C1,1(?). Following up on earlier work establishing uniqueness of solution to this problem, existence of solution is studied via the boundary integral equation method. This requires a novel approach to the study of solvability of integral equations on the real line. The paper establishes the existence of a unique solution to the boundary integral equation formulation in the space of bounded and continuous functions as well as in all Lp spaces, p∈[1, ∞] and hence existence of solution to the elastic wave scattering problem. Copyright © 2002 John Wiley & Sons, Ltd. 相似文献
13.
In this paper we prove new results for p harmonic functions, p≠2, 1<p<∞, in Lipschitz and starlike Lipschitz ring domains. In particular we prove the boundary Harnack inequality, Theorem 1, for the ratio of two positive p harmonic functions vanishing on a portion of the boundary of a Lipschitz domain, with constants only depending on p,n and the Lipschitz constant of the domain. For p capacitary functions, in starlike Lipschitz ring domains, we prove an even stronger result, Theorem 2, showing that the ratio is Hölder continuous up to the boundary. Moreover, for p capacitary functions in starlike Lipschitz ring domains we prove, Theorems 3 and 4, appropriate extensions to p≠2, 1<p<∞, of famous results of Dahlberg [12] and Jerison and Kenig [25] on the Poisson kernel associated to the Laplace operator (i.e. p=2). 相似文献
14.
Bochner's theorem that a compact Riemannian manifold with positive Ricci curvature has vanishing first cohomology group has various extensions to complete noncompact manifolds with Ricci possibly negative. One still has a vanishing theorem for L
2 harmonic one-forms if the infimum of the spectrum of the Laplacian on functions is greater than minus the infimum of the Ricci curvature. This result and its analogues for p-forms yield vanishing results for certain infinite volume hyperbolic manifolds. This spectral condition also imposes topological restrictions on the ends of the manifold. More refined results are obtained by taking a certain Brownian motion average of the Ricci curvature; if this average is positive, one has a vanishing theorem for the first cohomology group with compact supports on the universal cover of a compact manifold. There are corresponding results for L
2 harmonic spinors on spin manifolds. 相似文献
15.
Peter Lindqvist 《Israel Journal of Mathematics》1988,63(3):257-269
In the complex plane thep-harmonic equation div(|∇u|
p−2∇u) = 0, 1 <p < ∞, exhibits some features reminiscent of Function Theory. Our results about curvature in this structure complement known
facts about minimal surfaces and harmonic functions. Quasiregular mappings are used. 相似文献
16.
The Liouville property of a complete Riemannian manifold M (i.e., the question whether there exist non-trivial bounded harmonic functions on M) attracted a lot of attention. For Cartan–Hadamard manifolds the role of lower curvature bounds is still an open problem.
We discuss examples of Cartan–Hadamard manifolds of unbounded curvature where the limiting angle of Brownian motion degenerates
to a single point on the sphere at infinity, but where nevertheless the space of bounded harmonic functions is as rich as
in the non-degenerate case. To see the full boundary the point at infinity has to be blown up in a non-trivial way. Such examples
indicate that the situation concerning the famous conjecture of Greene and Wu about existence of non-trivial bounded harmonic
functions on Cartan–Hadamard manifolds is much more complicated than one might have expected.
相似文献
17.
Pascal Lefèvre Daniel Li Hervé Queffélec Luis Rodríguez-Piazza 《Journal d'Analyse Mathématique》2011,115(1):187-211
We investigate the behaviour of some thin sets of integers defined through random trigonometric polynomials when one replaces
Gaussian or Rademacher variables with p-stable ones, 1 < p < 2. We show that in one case, this behaviour is essentially the same as in the Gaussian case, whereas in another case, it
is entirely different. 相似文献
18.
Let X be a standard Markov process. We prove that a space inversion property of X implies the existence of a Kelvin transform of X‐harmonic, excessive and operator‐harmonic functions and that the inversion property is inherited by Doob h‐transforms. We determine new classes of processes having space inversion properties amongst transient processes satisfying the time inversion property. For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly. We treat in details the examples of free scaled power Bessel processes, non‐colliding Bessel particles, Wishart processes, Gaussian Ensemble and Dyson Brownian Motion. 相似文献
19.
Yong Hah Lee 《Potential Analysis》2005,23(1):83-97
In this paper, we describe the behavior of bounded energy finite solutions for certain nonlinear elliptic operators on a complete Riemannian manifold in terms of its p-harmonic boundary. We also prove that if two complete Riemannian manifolds are roughly isometric to each other, then their p-harmonic boundaries are homeomorphic to each other. In the case, there is a one to one correspondence between the sets of bounded energy finite solutions on such manifolds. In particular, in the case of the Laplacian, it becomes a linear isomorphism between the spaces of bounded harmonic functions with finite Dirichlet integral on the manifolds.
This work was supported by grant No. R06-2002-012-01001-0(2002) from the Basic Research Program of the Korea Science & Engineering Foundation. 相似文献
20.
We show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given
point determine the norm |?R|{{|\nabla{R}|}} of the covariant derivative of the Riemannian curvature tensor in that point. In particular, the spectra of small geodesic
spheres in a harmonic space determine whether the space is locally symmetric. For the proof we use the first few heat invariants
and consider certain coefficients in the radial power series expansions of the curvature invariants |R|2 and |Ric|2 of the geodesic spheres. Moreover, we obtain analogous results for geodesic balls with either Dirichlet or Neumann boundary
conditions. We also comment on the relevance of these results to constructions of Z.I. Szabó. 相似文献