共查询到20条相似文献,搜索用时 46 毫秒
1.
For any α∈(0,2), a truncated symmetric α-stable process in ℝ
d
is a symmetric Lévy process in ℝ
d
with no diffusion part and with a Lévy density given by c|x|−d−α
1{|x|<1} for some constant c. In (Kim and Song in Math. Z. 256(1): 139–173, [2007]) we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary
Harnack principle is valid for the positive harmonic functions of this process in any bounded convex domain and showed that
the Martin boundary of any bounded convex domain with respect to this process is the same as the Euclidean boundary. However,
for truncated symmetric stable processes, the boundary Harnack principle is not valid in non-convex domains. In this paper,
we show that, for a large class of not necessarily convex bounded open sets in ℝ
d
called bounded roughly connected κ-fat open sets (including bounded non-convex κ-fat domains), the Martin boundary with respect to any truncated symmetric stable process is still the same as the Euclidean
boundary. We also show that, for truncated symmetric stable processes a relative Fatou type theorem is true in bounded roughly
connected κ-fat open sets.
The research of P. Kim is supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic
Research Promotion Fund) (KRF-2007-331-C00037).
The research of R. Song is supported in part by a joint US-Croatia grant INT 0302167. 相似文献
2.
Summary. We prove an optimal a priori error estimate for the p-version of the boundary element method with hypersingular operators on piecewise plane open surfaces. The solutions of problems on open surfaces typically exhibit a singular behavior at the edges and corners of the surface which prevent an approximation analysis in H1. We analyze the approximation by polynomials of typical singular functions in fractional order Sobolev spaces thus giving, as an application, the optimal rate of convergence of the p-version of the boundary element method. This paper extends the results of [C. Schwab, M. Suri, The optimal p-version approximation of singularities on polyhedra in the boundary element method, SIAM J. Numer. Anal., 33 (1996), pp. 729–759] who only considered closed surfaces where the solution is in H1.Mathematics Subject Classification (2000): 41A10, 65N15, 65N38Financed by the FONDAP Program in Applied Mathematics, Chile.Supported by the FONDAP Program in Applied Mathematics and Fondecyt project no. 1010220, both Chile. 相似文献
3.
A. J. Tromba 《Journal of Fixed Point Theory and Applications》2011,10(2):253-277
This is the second in a series of two papers discussing the elementary but beautiful and fundamental question (open for some
eighty years) of whether or not a minimal surface spanning a sufficiently smooth curve, which is a local minimizer, is immersed
up to and including the boundary. We show that C
k
minimizers of energy or area cannot have nonexceptional boundary branch points. 相似文献
4.
Let a noncompact Riemann surface R of positive finite genus g be given. If f : R → R′ is a conformal mapping of R into a compact Riemann surface R′ of genus g, we have a realization of the ideal boundary of R on the surface R′. We consider (for the fixed R) all the possible R′ and the associated conformal mappings, and study how large the realized boundary can be. To this aim we pass to the (common)
universal space ℂ
g
of the Jacobi variety of any R′ and show that the image sets of the ideal boundary of R in ℂ
g
are uniformly bounded.
相似文献
5.
Igor Rivin 《Geometriae Dedicata》2001,87(1-3):345-360
We study simple closed geodesics on a hyperbolic surface of genus g with b geodesic boundary components and c cusps. We show that the number of such geodesics of length at most L is of order L
6g+2b+2c–6. This answers a long-standing open question. 相似文献
6.
Guozheng Yan 《Mathematical Methods in the Applied Sciences》2011,34(15):1813-1821
We consider the scattering of an electromagnetic time‐harmonic plane wave by an infinite cylinder having a mixed open crack (or arc) in R2 as the cross section. The crack is made up of two parts, and one of the two parts is (possibly) coated by a material with surface impedance λ. We transform the scattering problem into a system of boundary integral equations by adopting a potential approach, and establish the existence and uniqueness of a weak solution to the system by the Fredholm theory. Copyright © 2011 John Wiley & Sons, Ltd. 相似文献
7.
Sharp two-sided estimates for Green functions of censored α-stable process Y in a bounded C
1,1
open set D are obtained, where α (1, 2). It is shown that the Martin boundary and minimal Martin boundary of Y can all be identified with the Euclidean boundary of D. Sharp two-sided estimates for the Martin kernel of Y are also derived.
Received: 27 January 2002 / Revised version: 10 June 2002 / Published online: 24 October 2002
This research is supported in part by NSF Grant DMS-0071486.
Mathematics Subject Classification (2002): Primary: 60J45, 31C35; Secondary: 60G52, 31C15
Keywords or phrases: Censored stable process – Green function – Capacity – Martin boundary – Martin kernel – Harmonic function 相似文献
8.
Tiba 《Applied Mathematics and Optimization》2008,47(1):45-58
Abstract. We prove that for bounded open sets Ω with continuous boundary, Sobolev spaces of type W
0
l,p
(Ω ) are characterized by the zero extension outside of Ω . Combining this with a compactness result for domains of class C, we obtain a general existence theorem for shape optimization problems governed by nonlinear nonhomogenous Dirichlet boundary
value problems of arbitrary order, in arbitrary dimension and with general cost functionals. 相似文献
9.
Tiba 《Applied Mathematics and Optimization》2003,47(1):45-58
Abstract. We prove that for bounded open sets Ω with continuous boundary, Sobolev spaces of type W
0
l,p
(Ω ) are characterized by the zero extension outside of Ω . Combining this with a compactness result for domains of class C, we obtain a general existence theorem for shape optimization problems governed by nonlinear nonhomogenous Dirichlet boundary
value problems of arbitrary order, in arbitrary dimension and with general cost functionals. 相似文献
10.
Upper bounds are obtained for the heat content of an open set D in a geodesically complete Riemannian manifold M with Dirichlet boundary condition on ?D, and non-negative initial condition. We show that these upper bounds are close to being sharp if (i) the Dirichlet-Laplace-Beltrami operator acting in L 2(D) satisfies a strong Hardy inequality with weight δ2, (ii) the initial temperature distribution, and the specific heat of D are given by δ?α and δ?β respectively, where δ is the distance to ?D, and 1 < α <2, 1 < β <2. 相似文献
11.
We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and
in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving
the equation Δu-e
2u=Ko(z) on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factore
2u giving the Poincaré metric when the Riemann surface has smoothly bounded compact closure, and also a curvature equation proof
of Koebe's disk theorem.
Research supported in part by NSF Grant DMS-9971975 and also at MSRI by NSF grant DMS-9701755.
Research supported in part by NSF Grant DMS-9877077 相似文献
12.
By the Karamata regular variation theory and the method of lower and upper solutions, we establish the asymptotic behavior of boundary blow-up solutions of the quasilinear elliptic equation div(|u|p−2u)=b(x)f(u) in a bounded ΩRN subject to the singular boundary condition u(x)=∞, where the weight b(x) is non-negative and non-trivial in Ω, which may be vanishing on the boundary or go to unbounded, the nonlinear term f is a Γ-varying function at infinity, whose variation at infinity is not regular. 相似文献
13.
We present several constructions of a ``censored stable process' in an open set DRn, i.e., a symmetric stable process which is not allowed to jump outside D.
We address the question of whether the process will approach the boundary of D in a finite time – we give sharp conditions for such approach in terms of the stability index and the ``thickness' of the boundary. As a corollary, new results are obtained concerning Besov spaces on non-smooth domains, including the critical exponent case.
We also study the decay rate of the corresponding harmonic functions which vanish on a part of the boundary. We derive a boundary Harnack principle in C1,1 open sets.
Research partially supported by NSF Grant DMS-0071486.Mathematics Subject Classification (2000): Primary 60G52, Secondary 60G17, 60J45 相似文献
14.
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
15.
We consider boundary roughness for the ``droplet' created when supercritical two-dimensional Bernoulli percolation is conditioned to have an open dual circuit surrounding the origin and enclosing an area at least l2, for large l. The maximum local roughness is the maximum inward deviation of the droplet boundary from the boundary of its own convex hull; we show that for large l this maximum is at least of order l1/3(logl)–2/3. This complements the upper bound of order l1/3(logl)–2/3 proved in [Al3] for the average local roughness. The exponent 1/3 on l here is in keeping with predictions from the physics literature for interfaces in two dimensions.
The research of the first author was supported by NSF grant DMS-9802368. The research of the second author was supported by NSF grants DMS-9802368 and DMS-0103790.Mathematics Subject Classification (2000): Primary 60K35; Secondary 82B20, 82B43 相似文献
16.
17.
Yves Martinez-Maure 《Archiv der Mathematik》2002,79(6):489-498
LetC be a convex curve of constant width and of classC
4
+
. It is known thatC has at least 6 vertices and its interior contains either a point through which infinitely many normals pass or an open set
of points through each of which pass at least 6 normals. If all its vertices are nondegenerate, then: (i)C has exactly 6 vertices if, and only if, its evolute is the boundary of a topological disc through each interior point of
which pass at least 6 normals; (ii) ifC has more than 6 vertices, then there exists an open set of points through each of which pass at least 10 normals. The proof:
(i) expresses the number of normals passing through a point as a function of the index with respect to the evolute; (ii) relates
this index to the number of singularities of the evolute (i.e. of vertices). Furthermore, we give formulas for counting singularities
of generic hedgehogs in ℝ2 and ℝ3.
相似文献
18.
Fernando Schwartz 《Annales Henri Poincare》2011,12(1):67-76
We consider asymptotically flat Riemannian manifolds with non-negative scalar curvature that are conformal to
\mathbbRn\ W, n 3 3{\mathbb{R}^{n}{\setminus} \Omega, n\ge 3}, and so that their boundary is a minimal hypersurface. (Here,
W ì \mathbbRn{\Omega\subset \mathbb{R}^{n}} is open bounded with smooth mean-convex boundary.) We prove that the ADM mass of any such manifold is bounded below by
\frac12(V/bn)(n-2)/n{\frac{1}{2}\left(V/\beta_{n}\right)^{(n-2)/n}}, where V is the Euclidean volume of Ω and β
n
is the volume of the Euclidean unit n-ball. This gives a partial proof to a conjecture of Bray and Iga (Commun. Anal. Geom. 10:999–1016, 2002). Surprisingly, we do not require the boundary to be outermost. 相似文献
19.
A. A. Arkhipova 《Journal of Mathematical Sciences》2009,159(4):391-410
A variational problem with an obstacle for a certain class of quadratic functionals is considered. Admissible vector-valued
functions are assumed to satisfy the Dirichlet boundary condition, and the obstacle is a given smooth (N − 1)-dimensional surface S in ℝ
N
. The surface S is not necessarily bounded.
It is proved that any minimizer u of such an obstacle problem is a partially smooth function up to the boundary of a prescribed domain. It is shown that the
(n − 2)-Hausdorff measure of the set of singular points is zero. Moreover, u is a weak solution of a quasilinear system with two kinds of quadratic nonlinearities in the gradient. This is proved by
a local penalty method. Bibliography: 25 titles.
Dedicated to V. A. Solonnikov on the occasion of his jubilee
Published in Zapiski Nauchnykh. Seminarov POMI, Vol. 362, 2008, pp. 15–47. 相似文献
20.
Given an open bounded connected subset Ω of ℝn, we consider the overdetermined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary
data to the elliptic equation −div(A(|∇u|)∇u)=1 in Ω. We prove that, if this problem admits a solution in a suitable weak sense, then Ω is a ball. This is obtained under
fairly general assumptions on Ω and A. In particular, A may be degenerate and no growth condition is required. Our method of proof is quite simple. It relies on a maximum principle
for a suitable P-function, combined with some geometric arguments involving the mean curvature of ∂Ω. 相似文献