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Elliptic functional differential equations with degenerations. (English) Zbl 1471.35291
Summary: This review is devoted to differential-difference equations with degeneration in a bounded domain $$Q\subset\mathbb{R}^n$$ and applications (Kato conjecture, nonlocal boundary value problem). We consider differential-difference operators with degeneration of the second order and generalization to $$2m$$-order and the case where differential-difference operator contains several degenerate difference operators with degeneration. Generalized solutions of such equations may not belong even to the Sobolev space $$W^1_2(Q)$$.
##### MSC:
 35R10 Partial functional-differential equations 35J25 Boundary value problems for second-order elliptic equations 39A14 Partial difference equations 39A27 Boundary value problems for difference equations
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##### References:
 [1] Agranovich, M. S.; Selitskii, A. M., Fractional powers of operators corresponding to coercive problems in Lipschitz domains, Funct. Anal. Appl., 47, 83-95 (2013) · Zbl 1287.47030 [2] Auscher, P.; Hofman, S.; McIntosch, A.; Tchamitchian, P., The Kato square root problem for higher order elliptic operators and systems on $$\mathbb{R}^n$$, J. Evol. Equat., 1, 361-385 (2001) · Zbl 1019.35029 [3] Auscher, P.; Tchamitchian, P., Square roots of elliptic second order divergence operators on strongly Lipschitz domains: $$L^2$$, J. Anal. Math., 90, 1-12 (2003) · Zbl 1173.35420 [4] Axelsson, A.; Keith, S.; McIntosch, A., The Kato square root problem for mixed boundary value problems, J. London Math. Soc., 74, 113-130 (2006) · Zbl 1123.35013 [5] Bitsadze, A. V.; Samarskiy, A. A., On some simplest generalizations of linear elliptic boundary-value problems, Dokl. Akad. Nauk SSSR, 185, 739740 (1969) [6] Dunford, N.; Schwartz, J., Linear Operators. Part II: Spectral Theory. Self-Adjoint Operators in Hilbert Space (1963), Chichester: Wiley, Chichester [7] Fichera, G., Boundary Problems in Differential Equations (1960), Madison: Univ. of Wisconsin Press, Madison · Zbl 0122.33504 [8] Ivanova, E. P., Continuous dependence of solutions of boundary-value problems for differential-difference equations on translations of the independent variable, Sovrem. Mat. Fundam. Napravl., 59, 74-96 (2016) [9] Kato, T., “Fractional powers of dissipative operators, I,” J, Math. Soc. Jpn., 13, 246-274 (1961) · Zbl 0113.10005 [10] Kato, T., Fractional powers of dissipative operators, II, J. Math. Soc. Jpn., 14, 242-248 (1962) · Zbl 0108.11203 [11] Kato, T.; McLeod, J. B., Functional differential equation $$y^{\prime}(x)=ay(\lambda x)+by(x)$$, Bull. Am. Math. Soc., 77, 891-937 (1971) · Zbl 0236.34064 [12] Kato, T., Perturbation Theory for Linear Operators (1980), Berlin, Heidelberg, New York: Springer, Berlin, Heidelberg, New York [13] Keldysh, M. V., On some cases of degeneration of elliptic-type equations on the boundary of domain, Dokl. Akad. Nauk SSSR, 77, 181-183 (1951) [14] Krein, S. G., Linear Equations in Banach Space (1982), Basel: Birkhäuser, Basel · Zbl 0535.47008 [15] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations (Nauka, Moscow, 1964; Academic, New York, London, 1968). [16] Lions, J. L., Espaces d’interpolation et domaines de puissances fractionnaires d’opérateurs, J. Math. Soc. Jpn., 14, 233-241 (1962) · Zbl 0108.11202 [17] Lions, J.; Magenes, E., Nonhomogeneous Boundary Value Problems and Applications (1972), Berlin, Heidelberg: Springer, Berlin, Heidelberg · Zbl 0223.35039 [18] McIntosh, A., On the comparability of $$A^{1/2}$$, Proc. Am. Math. Soc., 32, 430-434 (1972) · Zbl 0248.47020 [19] Mikhaylov, V. P., Partial Differential Equations (1976), Moscow: Nauka, Moscow [20] Muravnik, A. B., Asymptotic properties of solutions of the Dirichlet problem in the half-plane for differential-difference elliptic equations, Mat. Zam., 100, 566-576 (2016) · Zbl 1361.35187 [21] Oleynik, O. A.; Radkevich, E. V., Second-Order Equations with Nonnegative Characteristic Form (1971), Moscow: VINITI, Moscow [22] Popov, V. A.; Skubachevskii, A. L., Sectorial differential difference operators with degeneration, Dokl. Math., 80, 716-719 (2009) · Zbl 1183.35268 [23] Popov, V. A.; Skubachevskii, A. L., A priori estimates for elliptic differential-difference operators with degeneration, J. Math. Sci., 190, 130-148 (2010) · Zbl 1292.35307 [24] Popov, V. A.; Skubachevskii, A. L., Smoothness of generalized solutions of elliptic differential-difference equations with degenerations, J. Math. Sci., 190, 135-146 (2013) · Zbl 1292.35306 [25] Popov, V. A.; Skubachevskii, A. L., Smoothness of generalized solutions of elliptic diference-diferential equations with degeneration near boundaries of subdomains, Russ. Math. Surv., 66, 1204-1206 (2011) · Zbl 1238.35173 [26] Popov, V. A.; Skubachevskii, A. L., On smoothness of solutions of some elliptic functional differential equations with degenerations, Russ. J. Math. Phys., 20, 492-507 (2013) · Zbl 1331.35358 [27] Popov, V. A., Traces of generalized solutions of elliptic differential-difference equations with degeneration, J. Math. Sci., 239, 840-854 (2019) · Zbl 1422.35065 [28] V. A. Popov, ‘‘Estimates of solutions of elliptic differential-difference equations with degeneration,’’ J. Math. Sci. (in press). [29] Rossovskii, L. E., Coercivity of functional differential equations, Mat. Zam., 59, 103-113 (1996) [30] Rossovskii, L. E., Elliptic functional differential equations with contractions and extensions of independent variables of the unknown function, Sovrem. Mat. Fundam. Napravl., 36, 125-142 (2014) [31] Skubachevskii, A. L., Nonlocal elliptic boundary-value problems with degeneration, Differ. Uravn., 19, 457-470 (1983) [32] Skubachevskii, A. L., The first boundary-value problem for strongly elliptic differential-difference equations, J. Differ. Equat., 63, 332-361 (1986) · Zbl 0598.35122 [33] Skubachevskii, A. L.; Tsvetkov, E. L., Secondary boundary-value problem for elliptic differential-difference equations, Differ. Equat., 25, 1245-1254 (1989) · Zbl 0703.35051 [34] Skubachevskii, A. L.; Tsvetkov, E. L., “General boundary-value problems for elliptic differential-difference equations,” Am. Math. Soc, Transl., Ser. 2, 193, 153-199 (1999) [35] A. L. Skubachevskii, Elliptic Functional Differential Equations and Applications, Vol. 91 of Operator Theory. Advances and Applications (Birkhäuser, Basel, Boston, Berlin, 1997). · Zbl 0946.35113 [36] Skubachevskii, A. L., Elliptic functional differential equations with degeneration, Tr. Mosk. Mat. Obs., 59, 240-285 (1997) [37] Skubachevskii, A. L., Boundary-value problems for elliptic functional-differential equations and their applications, Russ. Math. Surv., 71, 801-906 (2016) · Zbl 1364.35413 [38] A. L. Skubachevskii, ‘‘The Kato conjecture for elliptic differential-difference operators with degeneration in a cylinder,’’ Dokl. Math. 97 (32) (2018). https://doi.org/10.1134/S1064562418010106 · Zbl 1445.35138 [39] A. L. Skubachevskii, ‘‘Elliptic differential-difference operators with degeneration and the Kato square root problem,’’ Math. Nachr., 1-33 (2018). https://doi.org/10.1002/mana.201700475 · Zbl 1411.35103 [40] Skubachevskii, A. L., On a property of regularly accretive differential-difference operators with degeneracy, Russ. Math. Surv., 73, 189-190 (2018) · Zbl 06945057 [41] Skubachevskii, A. L., “On a class of functional-differential operators satisfying the Kato conjecture,” SPb, Math. J., 30, 329-346 (2019) · Zbl 07031967 [42] Skubachevskii, A. L.; Shamin, R. V., Second-order parabolic differential-difference equations, Dokl. Math., 64, 98-101 (2001) [43] Tasevich, A. L., Smoothness of generalized solutions of the Dirichlet problem for strong elliptic functional differential equations with orthotropic contractions, Sovrem. Mat. Fundam. Napravl., 58, 153-165 (2015) [44] Vishik, M. I., Boundary-value problems for elliptic equations degenerating on the boundary of domain, Mat. Sb., 35, 513-568 (1954)
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