51 research outputs found
On the First Eigenvalues of Free Vibrating Membrane in Conformal Regular Domains
In 1961 G.Polya published a paper about the eigenvalues of vibrating
membrane. The "free vibrating membrane"' corresponds to the Neumann-Laplace
operator in bounded plane domains. In this paper we obtain estimates for the
first eigenvalue of this operator in a large class of domains that we call as
conformal regular domains, that includes convex domains, John domains etc... On
the base of our estimates we conjecture that the eigenvalues of the
Neumann-Laplace operator depend on the hyperbolic metrics of plane domains. We
propose a new method for the estimates that is based on weighted
Poincar\'e-Sobolev inequalities obtained by the authors recently.Comment: 21 page
Universal conformal weights on Sobolev spaces
The Riemann Mapping Theorem states existence of a conformal homeomorphism
of a simply connected plane domain with
non-empty boundary onto the unit disc . In the
first part of the paper we study embeddings of Sobolev spaces
into weighted Lebesgue spaces
with an {}"universal" weight that is Jacobian of
i.e. . Weighted Lebesgue spaces with such
weights depend only on a conformal structure of . By this reason we
call the weights conformal weights. In the second part of the paper we
prove compactness of embeddings of Sobolev spaces
into for any . With the help of Brennan's conjecture we extend these results to
Sobolev spaces . In this case is not
arbitrary and depends on and the summability exponent for Brennan's
conjecture. Applications to elliptic boundary value problems are demonstrated
in the last part of the paper.Comment: 18 pages Using comments of readers we corrected some misprints and
added additional explanations into proof
Composition Operators on Sobolev Spaces and Neumann Eigenvalues
In this paper we discuss applications of the geometric theory of composition
operators on Sobolev spaces to the spectral theory of non-linear elliptic
operators. The lower estimates of the first non-trivial Neumann eigenvalues of
the -Laplace operator in cusp domains , ,
are given.Comment: 16 page
Weighted Sobolev spaces and embedding theorems
In the present paper we study embedding operators for weighted Sobolev spaces
whose weights satisfy the well-known Muckenhoupt A_p-condition. Sufficient
conditions for boundedness and compactness of the embedding operators are
obtained for smooth domains and domains with boundary singularities. The
proposed method is based on the concept of 'generalized' quasiconformal
homeomorphisms (homeomorphisms with bounded mean distortion.) The choice of the
homeomorphism type depends on the choice of the corresponding weighted Sobolev
space. Such classes of homeomorphisms induce bounded composition operators for
weighted Sobolev spaces. With the help of these homeomorphism classes the
embedding problem for non-smooth domains is reduced to the corresponding
classical embedding problem for smooth domains. Examples of domains with
anisotropic H\"older singularities demonstrate sharpness of our machinery
comparatively with known results.Comment: 23 page
Sobolev spaces and mappings with bounded (P,Q)-distortion on Carnot groups
We study mappings with bounded (p,q)-distortion associated to Sobolev spaces
on Carnot groups. Mappings of such type have applications to the Sobolev type
embedding theory and classification of manifolds. For this class of mappings,
we obtain estimates of linear distortion, and a geometrical description. We
prove also Liouville type theorems and give some sufficient conditions for
removability of sets
Conformal Spectral Stability Estimates for the Neumann Laplacian
We study the eigenvalue problem for the Neumann-Laplace operator in conformal
regular planar domains . Conformal regular domains
support the Poincar\'e inequality and this allows us to estimate the variation
of the eigenvalues of the Neumann Laplacian upon domain perturbation via energy
type integrals. Boundaries of such domains can have any Hausdorff dimension
between one and two.Comment: 15 pages. arXiv admin note: substantial text overlap with
arXiv:1406.434
Sobolev Extension Operators and Neumann Eigenvalues
In this paper we apply estimates of the norms of Sobolev extension operators
to the spectral estimates of of the first nontrivial Neumann eigenvalue of the
Laplace operator in non-convex extension domains. As a consequence we obtain a
connection between resonant frequencies of free membranes and the
smallest-circle problem (initially proposed by J.~J.~Sylvester in 1857).Comment: 12 page
On the First Eigenvalue of the Degenerate -Laplace Operator in Non-Convex Domains
In this paper we obtain lower estimates of the first non-trivial eigenvalues
of the degenerate -Laplace operator, , in a large class of non-convex
domains. This study is based on applications of the geometric theory of
composition operators on Sobolev spaces that permits us to estimates constants
of Poincar\'e-Sobolev inequalities and as an application to derive lower
estimates of the first non-trivial eigenvalues for the Alhfors domains (i.e. to
quasidiscs). This class of domains includes some snowflakes type domains with
fractal boundaries.Comment: 20 pages, 4 figure
Spectral Properties of the Neumann-Laplace operator in Quasiconformal Regular Domains
In this paper we study spectral properties of the Neumann-Laplace operator in
planar quasiconformal regular domains . This study is
based on the quasiconformal theory of composition operators on Sobolev spaces.
Using the composition operators theory we obtain estimates of constants in
Poincar\'e-Sobolev inequalities and as a consequence lower estimates of the
first non-trivial eigenvalue of the Neumann-Laplace operator in planar
quasiconformal regular domains.Comment: 17 pages, 1 figure. arXiv admin note: text overlap with
arXiv:1701.0514
Sobolev homeomorphisms and Poincare inequality
We study global regularity properties of Sobolev homeomorphisms on
-dimensional Riemannian manifolds under the assumption of -integrability
of its first weak derivatives in degree . We prove that inverse
homeomorphisms have integrable first weak derivatives. For the case we
obtain necessary conditions for existence of Sobolev homeomorphisms between
manifolds. These necessary conditions based on Poincar\'e type inequality: As a corollary we obtain the following geometrical
necessary condition:
{\em If there exists a Sobolev homeomorphisms , , , a. e. in , of compact smooth
Riemannian manifold onto Riemannian manifold then the manifold
has finite geodesic diameter.}}Comment: In the first version, there was an inaccuracy in Theorem 4. In the
revised version added additional assumption
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