On relative boundedness of a class of degenerate differential operators in the Lebesgue space

  • Gadoev Makhmadrakhim G., gadoev@rambler.ru North-Eastern Federal University, Mirny Polytechnic Institute (branch), 5/1 Tikhonov Street, Mirny 678170, Yakutia, Russia
  • Iskhokov Faridun S., fariduniskhokov@mail.ru Academy of Sciences of the Republic of Tajikistan, A. Dzhuraev Mathematical Institute, 299/4 Aini Street, Dushanbe 734063, Tajikistan
Keywords: partial differential operator, non-power degeneracy, relative boundedness of operators, partition of the unit

Abstract

In the space $L_p(\Omega)$, where $1<p<+\infty$ and $\Omega$  is an arbitrary (bounded or unbounded) domain in $R^n$, we investigate relative boundedness for a class of higher order partial differential operators in non-divergent form. These operators have nonpower degeneracy on the whole boundary of $\Omega$  and degeneracy with respect to each of independent variables is characterized by different functions.

In the earlier published papers in this direction, as a rule, firstly the operator is defined in $\Omega$  and then functions characterizing degeneracies of the operator’s coefficients are defined in this domain.

In contrast to that, here we define $\Omega$  and these functions related to each other while fulfilling the “immersion condition” introduced by P. I. Lizorkin in [19]. In addition, differentiability of the functions by which we define degeneracy of the investigated operator is not required. Study of relative boundedness of differential operators is one of the modern directions in such operators theory with results theory of differentiable functions of many variables, the separation theory of differential operators, the spectral theory of differential operators, etc.

References

[1]Kato T., Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, Heidelberg, New York, 1995  mathscinet  zmath
[2]“Separability theorems, weighted spaces and their applications”, Proc. Steklov Inst. Math., 170 (1987), 39–81  mathnet  mathscinet  zmath
[3]Anderson T. G., Hinton D. B., “Relative boundedness and compactness theory for second-order differential operators”, J. Inequal. Appl., 1 (1997), 375–400  mathscinet  zmath
[4]Binding P., Hryntiv R., “Relative boundedness and relative compactness for linear operators in Banach space”, Proc. Amer. Math. Soc., 128:8 (2000), 2287–2290  crossref  mathscinet  zmath
[5]Otelbaev M., “On separation of elliptic operators”, Dokl. Akad. Nauk SSSR, 234:3 (1977), 540–543  mathscinet  zmath
[6]“Coercive estimates and separation theorems of elliptic operators in $R^n$”, Proc. Steklov Inst. Math., 161 (1984), 213–239  mathnet  mathscinet  zmath  zmath
[7]Muratbekov M., Otelbaev M., “On the existence of a resolvent and separability for a class of singular hyperbolic type differential operators on an unbounded domain”, Euras. Math. J., 7:1 (2016), 50–67  mathscinet  elib
[8]Boimatov K. Kh., “Separability theorems, weighted spaces and their applications to boundary value problems”, Dokl. Akad. Nauk SSSR, 247:3 (1979), 532–536  mathscinet
[9]Boimatov K. Kh., “Separation theorems”, Dokl. Akad. Nauk SSSR, 213:5 (1973), 1009–1011  mathnet  mathscinet
[10]Zorin V. A., “Method of investigation of essential and discrete spectra of Hamiltonian operator of hydogen-like atom in quantum mechanics of Kuryshkin”, Vestn. Ross. Univ. Druzhby Narodov, Ser. Prikl. Comput. Mat., 3:1 (2004), 121–131
[11]Brown R. C., Hinton D. B., “Relative form boundedness and compactness for a second-order differential operator”, J. Comput. Appl. Math., 171 (2004), 123–140  crossref  mathscinet  zmath  adsnasa  scopus
[12]Trigub R. M., “Comparison of linear differential operators”, Math. Notes, 82:3 (2007), 380–394  mathnet  crossref  crossref  mathscinet  mathscinet  zmath  elib  elib  scopus
[13]Behncke H., Nyamwala F. O., “Spectral analysis of higher order differential operators with unbounded coeffcients”, Math. Nachr., 285:1 (2012), 56–73  crossref  mathscinet  zmath  scopus
[14]Brown R. C., “Separation and disconjugacy”, J. Inequal. Pure Appl. Math., 4:3 (2003), 56, 1–16  mathscinet
[15]Ospanov K. N., Akhmetkaliyeva R. D., “Separation and the existence theorem for second order nonlinear differential equation”, Electr. J. Quantat. Theory Diff. Equ., 2012, no. 66, 1–12  mathscinet
[16]Zayed E. M. E., Omran S. A., “Separation for triple-harmonic differential operator in Hilbert space”, Int. J. Math. Combin., 4 (2010), 13–23  zmath
[17]Qi J., Sun H., “Relatively bounded and relatively compact perturbations for limit circle Hamiltonian systems”, Integr. Equ. Oper. Theory, 86:3 (2016), 359–375  crossref  mathscinet  zmath  elib  scopus
[18]Gadoev M. G. and Iskhokov F. S., “On invertibility of a class of degenerate differential operators in the Lebesgue space”, Mat. zamet. SVFU, 23:3 (2016), 3–26  zmath  elib
[19]Lizorkin P. I., “Estimate of mixed and intermediate derivatives in weighted $L_p$-norms”, Tr. Mat. Inst. Steklova, 156 (1983), 141–153  mathnet  mathscinet  zmath  zmath
[20]Gadoev M. G., Iskhokov F. S., “On some functional spaces, which norms are given by differential operators”, Proc. Int. Summer Math. Stechkin School-Conf. Function Theory (Aug. 15-25, 2016), Dushanbe, Tajikistan, 2016, 82–84
How to Cite
Gadoev, M. and Iskhokov, F. (&nbsp;) “On relative boundedness of a class of degenerate differential operators in the Lebesgue space”, Mathematical notes of NEFU, 25(1), pp. 3-14. doi: https://doi.org/10.25587/SVFU.2018.1.12764.
Section
Mathematics