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

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

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

Bibliographic databases:

UDC: 517.98+517.95

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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• http://mi.mathnet.ru/eng/faa3109
• https://doi.org/10.4213/faa3109
• http://mi.mathnet.ru/eng/faa/v47/i2/p2

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. Selitskii A.M., “l (P) -Solvability of Parabolic Problems With An Operator Satisfying the Kato Conjecture”, Differ. Equ., 51:6 (2015), 776–782
3. A. L. Skubachevskii, “Boundary-value problems for elliptic functional-differential equations and their applications”, Russian Math. Surveys, 71:5 (2016), 801–906
4. A. A. Shkalikov, “Perturbations of self-adjoint and normal operators with discrete spectrum”, Russian Math. Surveys, 71:5 (2016), 907–964
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
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
7. A. Bonito, J. E. Pasciak, “Numerical approximation of fractional powers of regularly accretive operators”, IMA J. Numer. Anal., 37:3 (2017), 1245–1273
8. A. L. Skubachevskii, “The Kato conjecture for elliptic differential-difference operators with degeneration in a cylinder”, Dokl. Math., 97:1 (2018), 32–34
9. A. L. Skubachevskiǐ, “On a class of functional-differential operators satisfying the Kato conjecture”, St. Petersburg Math. J., 30:2 (2019), 329–346
10. A. L. Skubachevskii, “On a property of regularly accretive differential-difference operators with degeneracy”, Russian Math. Surveys, 73:2 (2018), 372–374
11. A. L. Skubachevskii, “Elliptic differential-difference operators with degeneration and the Kato square root problem”, Math. Nachr., 291:17-18 (2018), 2660–2692
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
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
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
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