103 research outputs found
Sharp spectral stability estimates via the Lebesgue measure of domains for higher order elliptic operators
We prove sharp stability estimates for the variation of the eigenvalues of
non-negative self-adjoint elliptic operators of arbitrary even order upon
variation of the open sets on which they are defined. These estimates are
expressed in terms of the Lebesgue measure of the symmetric difference of the
open sets. Both Dirichlet and Neumann boundary conditions are considered
Stability estimates for resolvents, eigenvalues and eigenfunctions of elliptic operators on variable domains
We consider general second order uniformly elliptic operators subject to
homogeneous boundary conditions on open sets parametrized by
Lipschitz homeomorphisms defined on a fixed reference domain .
Given two open sets , we estimate the
variation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm
for finite values of , under
natural summability conditions on eigenfunctions and their gradients. We prove
that such conditions are satisfied for a wide class of operators and open sets,
including open sets with Lipschitz continuous boundaries. We apply these
estimates to control the variation of the eigenvalues and eigenfunctions via
the measure of the symmetric difference of the open sets. We also discuss an
application to the stability of solutions to the Poisson problem.Comment: 34 pages. Minor changes in the introduction and the refercenes.
Published in: Around the research of Vladimir Maz'ya II, pp23--60, Int. Math.
Ser. (N.Y.), vol. 12, Springer, New York 201
Sobolev Embedding Theorem for the Sobolev-Morrey spaces
In this paper we prove a Sobolev Embedding Theorem for Sobolev-Morrey spaces. The proof is based on the Sobolev Integral Representation Theorem and on a recent results on Riesz potentials in generalized Morrey spaces of Burenkov, Gogatishvili, Guliyev, Mustafaev and on estimates on the Riesz potentials. We mention that a Sobolev Embedding Theorem for Sobolev morrey spaces had been proved by Campanato, for a subspace of our Sobolev-Morrey space which corresponds to the closure of the set of smooth functions in our Sobolev-Morrey space. The methods of the present paper are considerably different from those of Campanato
О достаточном условии предкомпактности множеств в обобщенных пространствах Морри
В статье приведены достаточные условия предкомпактности множеств в обобщенных пространствах Морри Mpw(·)(Rn). Из доказанной теоремы в случае w(r) = r-λ ; 0≤λ≤ n/p ; вытекает известный результат для пространства Морри Mp λ (Rn), а в случае λ = 0 хорошо известная теорема Фреше - Колмогорова. Предварительно доказаны несколько лемм об оценке средних функций в обобщенном пространстве Морри. Эти леммы представляют самостоятельный интерес. Обсуждается необходимость полученных условий
Maximal operator in variable exponent generalized morrey spaces on quasi-metric measure space
We consider generalized Morrey spaces on quasi-metric measure spaces , in general unbounded, with variable exponent p(x) and a general function defining the Morrey-type norm. No linear structure of the underlying space X is assumed. The admission of unbounded X generates problems known in variable exponent analysis. We prove the boundedness results for maximal operator known earlier only for the case of bounded sets X. The conditions for the boundedness are given in terms of the so called supremal inequalities imposed on the function , which are weaker than Zygmund-type integral inequalities often used for characterization of admissible functions . Our conditions do not suppose any assumption on monotonicity of in r
Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces. I
Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces. II
The survey is aimed at providing detailed information about recent results in the problem of the boundedness in general Morrey-type spaces of various important operators of real analysis, namely of the maximal operator, fractional maximal operator, Riesz potential, singular integral operator, Hardy operator. The main focus is on the results which contain, for a certain range of the numerical parameters, necessary and sufficient conditions on the functional parameters characterizing general Morrey-type spaces, ensuring the boundedness of the aforementioned operators from one general Morrey-type space to another one. The major part of the survey is dedicated to the results obtained by the author jointly with his co-authores A. Gogatishvili, M. L. Goldman, D. K. Darbayeva, H. V. Guliyev, V. S. Guliyev, P. Jain, R. Mustafaev, E. D. Nursultanov, R. Oinarov, A. Serbetci, T. V. Tararykova. In Part I of the survey under discussion were the definition and basic properties of the local and global general Morrey-type spaces, embedding theorems, and the boundedness properties of the maximal operator. Part II of the survey contains discussion of boundedness properties of the fractional maximal operator, Riesz potential, singular integral operator, Hardy operator. All definitions and notation in Part II are the same as in Part I
LOCAL AND GLOBAL ESTIMATES OF DERIVATIVES OF THE NONHOMOGENEOUS HYPOELLIPTIC EQUATION SOLUTIONS
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