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1.
Min-Chun Hong Changyou Wang 《Calculus of Variations and Partial Differential Equations》2005,23(4):425-450
For
and
, we show that any minimizing biharmonic map from
to Sk is smooth off a closed set whose Hausdorff dimension is at most n-5. When n = 5 and k = 4, for a parameter
we introduce a
-relaxed energy
of the Hessian energy for maps in
so that each minimizer
of
is also a biharmonic map. We also establish the existence and partial regularity of a minimizer of
for
.Received: 5 April 2004, Accepted: 19 October 2004, Published online: 10 December 2004 相似文献
2.
Manfred Stoll 《Monatshefte für Mathematik》2005,144(2):131-139
Let B denote the unit ball in n, n 1, and let and
denote the volume measure and gradient with respect to the Bergman metric on B. In the paper we consider the weighted Dirichlet spaces
,
, and weighted Bergman spaces
,
,
, of holomorphic functions f on B for which
and
respectively are finite, where
and
The main result of the paper is the following theorem.Theorem 1. Let f be holomorphic on B and
.(a) If
for some
, then
for all p,
, with
.(b) If
for some p,
, then
for all
with
. Combining Theorem 1 with previous results of the author we also obtain the following.Theorem 2. Suppose
is holomorphic in B. If
for some p,
, and
, then
. Conversely, if
for some p,
, then the series in * converges. 相似文献
3.
Weak limits of graphs of smooth maps
with equibounded Dirichlet integral give rise to elements of the space
. We assume that the 2-homology group of
has no torsion and that the Hurewicz homomorphism
is injective. Then, in dimension n = 3, we prove that every element T in
, which has no singular vertical part, can be approximated weakly in the sense of currents by a sequence of smooth graphs {u
k
} with Dirichlet energies converging to the energy of T. We also show that the injectivity hypothesis on the Hurewicz map cannot be removed. We finally show that a similar topological obstruction on the target manifold holds for the approximation problem of the area functional.Received: 9 May 2003, Accepted: 5 June 2003, Published online: 25 February 2004 相似文献
4.
Relaxation problems for a functional of the type
are analyzed, where
is a bounded smooth open subset of
and g is a Carathéodory function. The admissible functions u are forced to satisfy a pointwise gradient constraint of the type
for a.e.
being, for every
, a bounded convex subset of
, in general varying with x not necessarily in a smooth way. The relaxed functionals
and
of G obtained letting u vary respectively in
, the set of the piecewise C
1-functions in
, and in
in the definition of G are considered. For both of them integral representation results are proved, with an explicit representation formula for the density of
. Examples are proposed showing that in general the two densities are different, and that the one of
is not obtained from g simply by convexification arguments. Eventually, the results are discussed in the framework of Lavrentieff phenomenon, showing by means of an example that deep differences occur between
and
. Results in more general settings are also obtained.Received: 18 December 2002, Accepted: 18 November 2003, Published online: 16 July 2004Mathematics Subject Classification (2000):
49J45, 49J10, 49J53This work is part of the European Research Training Network Homogenization and Multiple Scales (HMS 2000), under contract HPRN-2000-00109. It is also part of the 2003-G.N.A.M.P.A. Project Metodi Variazionali per Strutture Sottili, Frontiere Oscillanti ed Energie Vincolate. 相似文献
5.
Let
be an arbitrary real normed space of finite dimension d ≥ 2. We define the metric capacity of
as the maximal
such that every m-point metric space is isometric to some subset of
(with metric induced by
). We obtain that the metric capacity of
lies in the range from 3 to
, where the lower bound is sharp for all d, and the upper bound is shown to be sharp for d ∈ {2, 3}. Thus, the unknown sharp upper bound is asymptotically linear, since it lies in the range from d + 2 to
.
Research supported by the German Research Foundation, Project AV 85/1-1. 相似文献
6.
We define a bijection from Littlewood-Richardson tableaux to rigged configurations and show that it preserves the appropriate
statistics. This proves in particular a quasi-particle expression for the generalized Kostka polynomials labeled by a partition and a sequence of rectangles R. The generalized Kostka polynomials are q-analogues of multiplicities of the irreducible -module of highest weight in the tensor product . 相似文献
7.
Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums
, as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form
, where
is a continuous function with
,
runs over
, the set of Farey fractions of order Q in the unit interval [0,1] and
are consecutive elements of
. We show that the limit lim
Q→∞
A
h
(Q) exists and is independent of h. 相似文献
8.
We establish existence of nodal solutions to the pure critical exponent problem
in
u = 0 on
where
a bounded smooth domain which is invariant under an orthogonal involution of
We extend previous results for positive solutions due to Coron, Dancer, Ding, and Passaseo to existence and multiplicity results for solutions which change sign exactly once.Received: 4 April 2003, Accepted: 26 August 2003, Published online: 24 November 2003Mathematics Subject Classification (2000):
35J65, 35J20Research partially supported by PAPIIT, UNAM, under grant IN110902-3. 相似文献
9.
Jonathan Bevan 《Calculus of Variations and Partial Differential Equations》2005,23(3):347-372
We give examples of non-C1 one-homogeneous mappings
and non-negative strictly polyconvex functions f such that
where B denotes the unit ball in
. Such u are therefore singular minimizers of the corresponding strictly polyconvex functionals
in appropriate Sobolev spaces.
Received: 12 January 2004, Accepted: 13 September 2004, Published online: 10 December 2004 相似文献
10.
Angela Pistoia Olivier Rey 《Calculus of Variations and Partial Differential Equations》2003,18(3):243-251
Let
,
, be a bounded domain as defined by Flucher, Garroni and Müller [6], which has a singular point
such that the Robins function achieves its infimum at
. Considering the elliptic problem
in
; u = 0 on
, with p = (N + 2)/(N-2),
, and
a minimizing solution of
,
concentrates at
as
goes to zero.Received: 15 September 2002, Accepted: 5 November 2002, Published online: 16 May 2003Mathematics Subject Classification:
35J65Angela Pistoia: The author is supported by M.U.R.S.T., project Metodi variazionali e topologici nello studio di fenomeni non lineari 相似文献
11.
We prove a new approximation theorem, which enables us to show that the relaxed energy
of Sobolev mappings u from higher dimensional balls into S2 is given by
, provided their singular set is of Lebesgue measure zero. Here
is the mass of the minimal integer multiplicity connection associated to the singularity current Su of u. Using this approximation theorem, we prove a partial regularity theorem for minimizers of the relaxed energy functional.Received: 5 May 2004, Accepted: 19 October 2004, Published online: 10 December 2004 相似文献
12.
13.
For integers
, we consider
-valued Radon measures
on an open set
which satisfy
for all
. We show that under certain conditions,
]*> has an (n - p)-dimensional density everywhere, and the set of points of positive density is countably (n - p)-rectifiable. This simplifies the proofs of several rectifiability theorems involving varifolds with vanishing first variations, p-harmonic maps, or Yang-Mills connections.Received: 4 April 2002, Accepted: 16 June 2002, Published online: 5 September 2002Mathematics Subject Classification (1991):
49Q15, 49Q05, 58E20, 58E15 相似文献
14.
R. G. Novikov 《Selecta Mathematica, New Series》1997,3(2):245-302
We consider the Dirac-ZS-AKNS system (1) where (the space of functions with n derivatives in L
1), (2) We consider for (1) the transition matrix and, in addition, for the case of the Dirac system (i.e. for the selfadjoint case the scattering matrix We can divide main results of the present work into three parts. I. We show that the inverse scattering transform and the inverse Fourier transform give the same solution, up to smooth functions,
of the inverse scattering problem for (1). More preciseley, we show that, under condition (2) with , the following formulas are valid: (3) and, in addition, for the case of the Dirac system (4) where denotes the factor space. II. Using (3), (4), we give the characterization of the transition matrix and the scattering matrix for the case of the Dirac
system under condition (2) with
III. As applications of the results mentioned above, we show that 1) for any real-valued initial data , the Cauchy problem for the sh-Gordon equation has a unique solution such that and for any t > 0, 2) in addition, for , for such a solution the following formula is valid: where
denotes the space of functions locally integrable with n derivatives. We give also a review of preceding results. 相似文献
15.
A. Martinez 《Annales Henri Poincare》2002,3(4):739-756
We study the resonances of the semiclassical Schr?dinger operator
near a non-trapping energy level
in the case when the potential V is not necessarily analytic on all of
but only outside some compact set. Then we prove that for some
and for any C > 0, P admits no resonance in the domain
if V is
, and
if V is Gevrey with index s. Here
does not depend on h and the results are uniform with respect to h > 0 small enough.
Submitted 05/02/02, accepted 06/05/02
An erratum to this article is available at . 相似文献
16.
We prove partial regularity of vector-valued minimizers u of the polyconvex variational integral
, where
stands for the minors of the gradient Du. For the integrand, we assume f to be a continuous function of class C
2, strictly convex and of polynomial growth in the minors, and g to be a bounded Carathéodory function. We do not employ a Caccioppoli inequality.Received: 19 March 2002, Accepted: 24 October 2002, Published online: 16 May 2003Mathematics Subject Classification (2000):
49N60, 35J50 相似文献
17.
For a smooth harmonic map flow
with blow-up as
, it has been asked [5,6,7] whether the weak limit
is continuous. Recently, in [12], we showed that in general it need not be. Meanwhile, the energy function
, being weakly positive, smooth and weakly decreasing, has a continuous extension to [0,T]. Here we show that if this extension is also Hölder continuous, then the weak limit u(T) must also be Hölder continuous.Received: 1 September 2003, Accepted: 7 October 2003, Published online: 25 February 2004Version of 19/9/03. Partly supported by an EPSRC Advanced Research Fellowship. 相似文献
18.
Given an almost complex structure J in a cylinder of
(p > 1) together with a compatible symplectic form
and given an arbitrary J-holomorphic curve
without boundary in that cylinder, we construct an holomorphic perturbation of
, for the canonical complex structure J
0 of
, such that the distance between these two curves in W
1,2 and
norms, in a sub-cylinder, are controled by quantities depending on J,
and by the area of
only. These estimates depend neither on the topology nor on the conformal class of
. They are key tools in the recent proof of the regularity of 1-1 integral currents in [RT].Received: 2 October 2003, Accepted: 18 November 2003, Published online: 25 February 2004 相似文献
19.
We consider the following singularly perturbed semilinear elliptic problem:
where
is a bounded domain in R
N
with smooth boundary
,
is a small constant and f is some superlinear but subcritical nonlinearity. Associated with (I) is the energy functional
defined by
where
. Ni and Takagi ([29, 30]) proved that for a single boundary spike solution
, the following asymptotic expansion holds:
where c
1 > 0 is a generic constant,
is the unique local maximum point of
and
is the boundary mean curvature function at
. In this paper, we obtain a higher-order expansion of
where c
2, c
3 are generic constants and
is the scalar curvature at
. In particular c
3 > 0. Some applications of this expansion are given.Received: 14 January 2003, Accepted: 28 July 2003, Published online: 15 October 2003Mathematics Subject Classification (2000):
Primary 35B40, 35B45; Secondary 35J25 相似文献
20.
We study the regularizing effect of perimeter penalties for a problem of optimal compliance in two dimensions. In particular, we consider minimizers of
where
The sets
,
, and the force f are given. We show that if we consider only scalar valued u and constant
, or if we consider the elastic energy
, then
is
away from where
is pinned. In the scalar case, we also show that, for any
of class
,
is
. The proofs rely on a notion of weak outward curvature of
, which we can bound without considering properties of the minimizing fields, together with a bootstrap argument.Received: 5 March 2002, Accepted: 3 September 2002, Published online: 17 December 2002 相似文献