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1.
Let F be a local field, a nontrivial unitary additive character of F, and V a finite dimensional vector space over F. Let us say that a complex function on V is elementary if it has the form , where , Q is a rational function (the phase function), are polynomials, and multiplicative characters of F. For generic , this function canonically extends to a distribution on V (if char(F) = 0). Occasionally, the Fourier transform of an elementary function is also an elementary function (the basic example is
the Gaussian integral: k = 0, Q is a nondegenerate quadratic form). It is interesting to determine when exactly this happens. This question is the main subject
of our study. In the first part of this paper we show that for or , if the Fourier transform of an elementary function with phase function -Q such that is another elementary function with phase function , then is the Legendre transform of Q (the "semiclassical condition"). We study properties and examples of phase functions satisfying this condition, and give
a classification of phase functions such that both Q and are of the form f(x)/t, where f is a homogeneous cubic polynomial and t is an additional variable (this is one of the simplest possible situations). Unexpectedly, the proof uses Zak's classification
theorem for Severi varieties.? In the second part of the paper we give a necessary and sufficient condition for an elementary
function to have an elementary Fourier transform (in an appropriate "weak" sense) and explicit formulas for such Fourier transforms
in the case when Q and are monomials, over any local field F. We also describe a generalization of these results to the case of monomials of norms of finite extensions of F. Finally, we generalize some of the above results (including Fourier integration formulas) to the case when and Q comes from a prehomogeneous vector space. 相似文献
2.
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
3.
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
4.
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
5.
We present a characterization of ideal knots, i.e., of closed knotted curves of prescribed thickness with minimal length, where we use the notion of global curvature for the definition of thickness. We show with variational methods that for an ideal knot
, the normal vector
at a curve point
is given by the integral over all vectors
against a Radon measure, where
realizes the given thickness. As geometric consequences we obtain in particular, that points without contact lie on straight segments of
, and for points
with exactly one contact point
we have that
points exactly into the direction of
Moreover, isolated contact points lie on straight segments of
, and curved arcs of
consist of contact points only, all realizing the prescribed thickness with constant (maximal) global curvature.Received: 1 January 2003, Accepted: 12 March 2003, Published online: 1 July 2003Mathematics Subject Classification (2000):
53A04, 57M25, 74K05, 74M15, 92C40 相似文献
6.
This paper has two parts. In the first part, we study shift coordinates on a sphere S equipped with three distinguished points and a triangulation whose vertices are the distinguished points. These coordinates
parametrize a space
that we call an unfolded Teichmüller space. This space contains Teichmüller spaces of the sphere with
boundary components and
cusps (which we call generalized pairs of pants), for all possible values of
and
satisfying
. The parametrization of
by shift coordinates equips this space with a natural polyhedral structure, which we describe more precisely as a cone over
an octahedron in
. Each cone over a simplex of this octahedron is interpreted as a Teichmüller space of the sphere with
boundary components and
cusps, for fixed
and
, the sphere being furthermore equipped with an orientation on each boundary component. There is a natural linear action of
a finite group on
whose quotient is an augmented Teichmüller space in the usual sense. We describe several aspects of the geometry of the space
. Stretch lines and earthquakes can be defined on this space. In the second part of the paper, we use the shift coordinates
to obtain estimates on the behaviour of stretch lines in the Teichmüller space of a surface obtained by gluing hyperbolic
pairs of pants. We also use the shift coordinates to give formulae that express stretch lines in terms of Fenchel-Nielsen
coordinates. We deduce the disjointness of some stretch lines in Teichmüller space. We study in more detail the case of a
closed surface of genus 2.
Authors’ addresses: A. Papadopoulos, Institut de Recherche Mathématique Avancée, Université Louis Pasteur and CNRS, 7 rue
René Descartes, 67084 Strasbourg Cedex, France and Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany;
G. Théret, Institut de Recherche Mathématique Avancée, Université Louis Pasteur and CNRS, 7 rue René Descartes, 67084 Strasbourg
Cedex, France and Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, Building 1530, DK-8000 Aarhus C,
Denmark 相似文献
7.
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
8.
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
9.
P. Courilleau J. Mossino 《Calculus of Variations and Partial Differential Equations》2004,20(1):65-91
We study the limit behaviour of some nonlinear monotone equations, such as:
, in a domain
which is thin in some directions (e.g.
is a plate or a thin cylinder). After rescaling to a fixed domain
, the above equation is transformed into:
, with convenient operators
and
. Assuming that
and the inverse of
have particular forms and satisfy suitable compensated compactness assumptions, we prove a closure result, that is we prove that the limit problem has the same form. This applies in particular to the limit behaviour of nonlinear monotone equations in laminated plates.Received: 16 October 2002, Accepted: 12 June 2003, Published online: 22 September 2003Mathematics Subject Classification (2000):
35B27, 35B40, 74Q15 相似文献
10.
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
11.
Ovidiu Savin 《Calculus of Variations and Partial Differential Equations》2005,22(3):303-320
We consider the following obstacle problem for Monge-Ampere equation
and discuss the regularity of the free boundary
. We prove that
is
if f is bounded away from 0 and
, and it is C
1,1 if
.Received: 4 February 2003, Accepted: 3 March 2004, Published online: 16 July 2004 相似文献
12.
A Sobolev inequality with remainder term and critical equations on domains with topology for the polyharmonic operator 总被引:1,自引:0,他引:1
Thomas Bartsch Tobias Weth Michel Willem 《Calculus of Variations and Partial Differential Equations》2003,18(3):253-268
We prove a Sobolev inequality with remainder term for the imbedding
,
arbitrary, generalizing a corresponding result of Bianchi and Egnell for the case m = 1. We also show that the manifold of least energy solutions
of the equation
is a nondegenerate critical manifold for the corresponding variational integral. Finally we generalize the results of J. M. Coron on the existence of solutions of equations with critical exponent on domains with nontrivial topology to the biharmonic operator.Received: 21 March 2002, Accepted: 5 November 2002, Published online: 16 May 2003 相似文献
13.
We derive an explicit formula for the isoperimetric defect
of an arbitrary minimal surface
,in terms of a double integral over the surface of certain geometric quantities, together with a double boundary integral which always has the correct sign. As a by-product of these computations we show that the best known universal isoperimetric estimate, that
for any minimal surface
(due to L. Simon), may be improved to the universal estimate
.Received: 21 June 2001, Accepted: 16 June 2002, Published online: 5 September 2002 相似文献
14.
Yavdat Il’yasov Thomas Runst 《Calculus of Variations and Partial Differential Equations》2005,22(1):101-127
This paper concerns with a family of inhomogeneous Neumann boundary value problems having indefinite nonlinearities which depend on a real parameter
. We discuss the existence and the multiplicity of positive solutions with respect to
. Developing the fibering method further, we can introduce a constructive concept of the calculation of certain nonlocal intervals
, the so-called sufficient intervals of the existence. Then we are able to prove some new results on the existence and the multiplicity of positive solutions
for
.Received: 22 December 2003, Accepted: 29 January 2004, Published online: 16 July 2004Mathematics Subject Classification (2000):
35J70, 35J65, 47H17 相似文献
15.
Finn and Kosmodemyanskii, Jr. gave an example of a domain
containing a disk
, and of a family of domains
converging to
as
, such that the heights u
t
of capillary surfaces in vertical tubes with the sections
in a gravity field g satisfy
for every
, but for which u
1< u
0 over
for all g > 0. In subsequent work, Finn and Lee characterized the most general convex
that leads to such a discontinuous transition when
is a disk. It has been suggested that the cause for this curious behavior is related to the fact that in all cases considered the boundaries of the
have a discontinuity in their curvatures, that is bounded below in magnitude. In the present note we present an alternative form of the example, in which the domains
are disks concentric to
. Thus, the limited smoothness in the original example of the convergence to
of the approxim
ating domains cannot be viewed as the root cause of the anomaly. The procedure presented here leads to explicit bounds, which were not available in the earlier forms of the example.Received: 3 September 2002, Accepted: 17 February 2003, Published online: 1 July 2003Mathematics Subject Classification:
76B45, 53A10, 49Q10 相似文献
16.
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
17.
Min Ho Lee 《Monatshefte für Mathematik》2004,141(3):187-196
Let
be an equivariant holomorphic map of symmetric domains associated to a homomorphism
of semisimple algebraic groups defined over
. If
and
are torsion-free arithmetic subgroups with
, the map induces a morphism :
of arithmetic varieties and the rationality of is defined by using symmetries on
and
as well as the commensurability groups of and . An element
determines a conjugate equivariant holomorphic map
of which induces the conjugate morphism
of . We prove that is rational if is rational. 相似文献
18.
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
19.
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
20.
Given a finite subset
of an additive group
such as
or
, we are interested in efficient covering of
by translates of
, and efficient packing of translates of
in
. A set
provides a covering if the translates
with
cover
(i.e., their union is
), and the covering will be efficient if
has small density in
. On the other hand, a set
will provide a packing if the translated sets
with
are mutually disjoint, and the packing is efficient if
has large density.
In the present part (I) we will derive some facts on these concepts when
, and give estimates for the minimal covering densities and maximal packing densities of finite sets
. In part (II) we will again deal with
, and study the behaviour of such densities under linear transformations. In part (III) we will turn to
.
Authors’ address: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395,
USA
The first author was partially supported by NSF DMS 0074531. 相似文献