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Funktsional. Anal. i Prilozhen., 2013, Volume 47, Issue 2, Pages 2–17 (Mi faa3109)  

This article is cited in 14 scientific papers (total in 14 papers)

Fractional Powers of Operators Corresponding to Coercive Problems in Lipschitz Domains

M. S. Agranovicha, A. M. Selitskiib

a Moscow State Institute of Electronics and Mathematics — Higher School of Economics
b Dorodnitsyn Computing Centre of the Russian Academy of Sciences, Moscow

Abstract: Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$, $n\ge2$, and let $L$ be a second-order matrix strongly elliptic operator in $\Omega$ written in divergence form. There is a vast literature dealing with the study of domains of fractional powers of operators corresponding to various problems (beginning with the Dirichlet and Neumann problems) with homogeneous boundary conditions for the equation $Lu=f$, including the solution of the Kato square root problem, which arose in 1961. Mixed problems and a class of problems for higher-order systems have been covered as well.
We suggest a new abstract approach to the topic, which permits one to obtain the results that we deem to be most important in a much simpler and unified way and cover new operators, namely, classical boundary operators on the Lipschitz boundary $\Gamma=\partial\Omega$ or part of it. To this end, we simultaneously consider two well-known operators associated with the boundary value problem.

Keywords: Lipschitz domain, strongly elliptic system, coercive problem, Kato's square root problem

DOI: https://doi.org/10.4213/faa3109

Full text: PDF file (230 kB)
References: PDF file   HTML file

English version:
Functional Analysis and Its Applications, 2013, 47:2, 83–95

Bibliographic databases:

UDC: 517.98+517.95
Received: 17.01.2013

Citation: M. S. Agranovich, A. M. Selitskii, “Fractional Powers of Operators Corresponding to Coercive Problems in Lipschitz Domains”, Funktsional. Anal. i Prilozhen., 47:2 (2013), 2–17; Funct. Anal. Appl., 47:2 (2013), 83–95

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. M. Selitskii, “Space of Initial Data for the Second Boundary-Value Problem for a Parabolic Differential-Difference Equation in Lipschitz Domains”, Math. Notes, 94:3 (2013), 444–447  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. Selitskii A.M., “l (P) -Solvability of Parabolic Problems With An Operator Satisfying the Kato Conjecture”, Differ. Equ., 51:6 (2015), 776–782  crossref  mathscinet  zmath  isi  scopus
    3. A. L. Skubachevskii, “Boundary-value problems for elliptic functional-differential equations and their applications”, Russian Math. Surveys, 71:5 (2016), 801–906  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. A. A. Shkalikov, “Perturbations of self-adjoint and normal operators with discrete spectrum”, Russian Math. Surveys, 71:5 (2016), 907–964  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. Agranovich M.S., “Spectral problems in Sobolev-type Banach spaces for strongly elliptic systems in Lipschitz domains”, Math. Nachr., 289:16 (2016), 1968–1985  crossref  mathscinet  zmath  isi  scopus
    6. Gurevich P., Vaeth M., “Stability for Semilinear Parabolic Problems in $L_2$ and $W^{1,2}$”, Z. Anal. ihre. Anwend., 35:3 (2016), 333–357  crossref  mathscinet  zmath  isi  elib  scopus
    7. A. Bonito, J. E. Pasciak, “Numerical approximation of fractional powers of regularly accretive operators”, IMA J. Numer. Anal., 37:3 (2017), 1245–1273  crossref  mathscinet  isi  scopus
    8. A. L. Skubachevskii, “The Kato conjecture for elliptic differential-difference operators with degeneration in a cylinder”, Dokl. Math., 97:1 (2018), 32–34  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    9. A. L. Skubachevskiǐ, “On a class of functional-differential operators satisfying the Kato conjecture”, St. Petersburg Math. J., 30:2 (2019), 329–346  mathnet  crossref  mathscinet  isi  elib
    10. A. L. Skubachevskii, “On a property of regularly accretive differential-difference operators with degeneracy”, Russian Math. Surveys, 73:2 (2018), 372–374  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    11. A. L. Skubachevskii, “Elliptic differential-difference operators with degeneration and the Kato square root problem”, Math. Nachr., 291:17-18 (2018), 2660–2692  crossref  mathscinet  zmath  isi
    12. Bonito A., Lei W., Pasciak J.E., “On Sinc Quadrature Approximations of Fractional Powers of Regularly Accretive Operators”, J. Numer. Math., 27:2 (2019), 57–68  crossref  isi
    13. Shaldanbayev A.Sh., Imanbayeva A.B., Beisebayeva A.Zh., Shaldanbayeva A.A., “On the Square Root of the Operator of Sturm-Liouville Fourth-Order”, News Natl. Acad. Sci. Rep. Kazakhstan-Ser. Phys.-Math., 3:325 (2019), 85–96  crossref  isi
    14. Shaldanbayev A.Sh., Shaldanbayeva A.A., Shaldanbay B.A., “On Square Root of Sturm-Liuville Operator”, News Natl. Acad. Sci. Rep. Kazakhstan-Ser. Phys.-Math., 3:325 (2019), 97–113  crossref  isi
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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