590 research outputs found
Mappings of least Dirichlet energy and their Hopf differentials
The paper is concerned with mappings between planar domains having least
Dirichlet energy. The existence and uniqueness (up to a conformal change of
variables in the domain) of the energy-minimal mappings is established within
the class of strong limits of homeomorphisms in the
Sobolev space , a result of considerable interest in the
mathematical models of Nonlinear Elasticity. The inner variation leads to the
Hopf differential and its trajectories.
For a pair of doubly connected domains, in which has finite conformal
modulus, we establish the following principle:
A mapping is energy-minimal if and only if
its Hopf-differential is analytic in and real along the boundary of .
In general, the energy-minimal mappings may not be injective, in which case
one observes the occurrence of cracks in . Nevertheless, cracks are
triggered only by the points in the boundary of where fails to be
convex. The general law of formation of cracks reads as follows:
Cracks propagate along vertical trajectories of the Hopf differential from
the boundary of toward the interior of where they eventually terminate
before making a crosscut.Comment: 51 pages, 4 figure
A neohookean model of plates
This article is about hyperelastic deformations of plates (planar domains) which minimize a neohookean-type energy. Particularly, we investigate a stored energy functional introduced by J. M. Ball [Proc. Roy. Soc. Edinb. Sect. A, 88 (1981), pp. 315-328]. The mappings under consideration are Sobolev homeomorphisms and their weak limits. They are monotone in the sense of C. B. Morrey. One major advantage of adopting monotone Sobolev mappings lies in the existence of the energy-minimal deformations. However, injectivity is inevitably lost, so an obvious question to ask is, what are the largest subsets of the reference configuration on which minimal deformations remain injective? The fact that such subsets have full measure should be compared with the notion of global invertibility, which deals with subsets of the deformed configuration instead. In this connection we present a Cantor-type construction to show that both the branch set and its image may have positive area. Another novelty of our approach lies in allowing the elastic deformations to be free along the boundary, known as frictionless problems
Hecke operators on period functions for
Matrix representations of Hecke operators on classical holomorphical cusp
forms and the corresponding period polynomials are well known. In this article
we derive representations of Hecke operators for vector valued period functions
for the congruence subgroups . For this we use an integral
transform from the space of vector valued cusp forms to the space of vector
valued period functions.Comment: v2: corrected version, submitted to Journal of Number, 29 pages, 1
figur
Language and the development of intercultural competence in an 'internationalised' university: staff and student perspectives
Within the currently diverse UK higher education environment, one important aspect of learning is the development of intercultural competence. The study that informs this paper investigated the ways intercultural competence was perceived as being enhanced or inhibited through current language and educational practices at a university that positions itself as internationally engaged and globally recognised. The project employed a multiple-case study design, examining eight academic programmes drawn from four different broad disciplinary groupings: social sciences, science, engineering, and management. Data were collected through individual, focus group and stimulated recall interviews, the latter using class observation recordings as a stimulus. The study revealed the ways in which language was exploited by both staff and students to convey particular meanings within an intercultural context. It was found that language choices, register and style were perceived as contributing to the pragmatic impact of either reinforcing barriers to or promoting intercultural competence development
Extremal Mappings of Finite Distortion
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135425/1/plms0655.pd
Eigenfunctions of the Laplacian and associated Ruelle operator
Let be a co-compact Fuchsian group of isometries on the Poincar\'e
disk \DD and the corresponding hyperbolic Laplace operator. Any
smooth eigenfunction of , equivariant by with real
eigenvalue , where , admits an integral
representation by a distribution \dd_{f,s} (the Helgason distribution) which
is equivariant by and supported at infinity \partial\DD=\SS^1. The
geodesic flow on the compact surface \DD/\Gamma is conjugate to a suspension
over a natural extension of a piecewise analytic map T:\SS^1\to\SS^1, the
so-called Bowen-Series transformation. Let be the complex Ruelle
transfer operator associated to the jacobian . M. Pollicott showed
that \dd_{f,s} is an eigenfunction of the dual operator for the
eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic
eigenfunction of for the eigenvalue 1, given by an
integral formula \psi_{f,s} (\xi)=\int \frac{J(\xi,\eta)}{|\xi-\eta|^{2s}}
\dd_{f,s} (d\eta), \noindent where is a -valued
piecewise constant function whose definition depends upon the geometry of the
Dirichlet fundamental domain representing the surface \DD/\Gamma
Fractional differentiability for solutions of nonlinear elliptic equations
We study nonlinear elliptic equations in divergence form
When
has linear growth in , and assuming that enjoys smoothness, local
well-posedness is found in for certain values of
and . In the particular case
, and ,
, we obtain for each
. Our main tool in the proof is a more general result, that
holds also if has growth in , , and
asserts local well-posedness in for each , provided that
satisfies a locally uniform condition
Doubly connected minimal surfaces and extremal harmonic mappings
The concept of a conformal deformation has two natural extensions:
quasiconformal and harmonic mappings. Both classes do not preserve the
conformal type of the domain, however they cannot change it in an arbitrary
way. Doubly connected domains are where one first observes nontrivial conformal
invariants. Herbert Groetzsch and Johannes C. C. Nitsche addressed this issue
for quasiconformal and harmonic mappings, respectively. Combining these
concepts we obtain sharp estimates for quasiconformal harmonic mappings between
doubly connected domains. We then apply our results to the Cauchy problem for
minimal surfaces, also known as the Bjorling problem. Specifically, we obtain a
sharp estimate of the modulus of a doubly connected minimal surface that
evolves from its inner boundary with a given initial slope.Comment: 35 pages, 2 figures. Minor edits, references adde
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