4,326 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
Existence of energy-minimal diffeomorphisms between doubly connected domains
The paper establishes the existence of homeomorphisms between two planar
domains that minimize the Dirichlet energy. Specifically, among all
homeomorphisms f : R -> R* between bounded doubly connected domains such that
Mod (R) < Mod (R*) there exists, unique up to conformal authomorphisms of R, an
energy-minimal diffeomorphism. No boundary conditions are imposed on f.
Although any energy-minimal diffeomorphism is harmonic, our results underline
the major difference between the existence of harmonic diffeomorphisms and the
existence of the energy-minimal diffeomorphisms. The existence of globally
invertible energy-minimal mappings is of primary pursuit in the mathematical
models of nonlinear elasticity and is also of interest in computer graphics.Comment: 34 pages, no figure
Invertible harmonic mappings, beyond Kneser
We prove necessary and sufficient criteria of invertibility for planar
harmonic mappings which generalize a classical result of H. Kneser, also known
as the Rad\'{o}-Kneser-Choquet theorem.Comment: One section added. 15 page
Quantitative estimates on Jacobians for hybrid inverse problems
We consider -harmonic mappings, that is mappings whose components
solve a divergence structure elliptic equation , for . We investigate whether, with suitably prescribed
Dirichlet data, the Jacobian determinant can be bounded away from zero. Results
of this sort are required in the treatment of the so-called hybrid inverse
problems, and also in the field of homogenization studying bounds for the
effective properties of composite materials.Comment: 15 pages, submitte
Smooth equivalence of deformations of domains in complex euclidean spaces
We prove that two smooth families of 2-connected domains in \cc are
smoothly equivalent if they are equivalent under a possibly discontinuous
family of biholomorphisms. We construct, for , two smooth families of
smoothly bounded -connected domains in \cc, and for , two families
of strictly pseudoconvex domains in \cc^n, that are equivalent under
discontinuous families of biholomorphisms but not under any continuous family
of biholomorphisms. Finally, we give sufficient conditions for the smooth
equivalence of two smooth families of domains
Quasisymmetric distortion spectrum
We give improved bounds for the distortion of the Hausdorff dimension under
quasisymmetric maps in terms of the dilatation of their quasiconformal
extension. The sharpness of the estimates remains an open question and is shown
to be closely related to the fine structure of harmonic measure.Comment: 14 page
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