Coercive estimates, separability and coercive solvability of a nonlinear elliptic differential operator in a weight space

O. Kh. Karimov ; Z. Zh. Hakimova

Čebyševskij sbornik, Tome 24 (2023) no. 2, pp. 197-213.

Résumé

The work is devoted to establishing coercive estimates and proofs of separability theorems for a nonlinear elliptic differential operator of non-divergence form in a weighted space. On the basis of the obtained coercive estimates, the coercive solvability of a nonlinear elliptic differential second-order operator in the space $L_{2,\rho}(R^n)$ is investigated. The problem of "separability of differential expressions" was first studied by mathematicians V.N.Everitt and M. Girtz. They studied in detail the separability of the Sturm-Liouville operator. Further development of this theory belongs to K.H.Boymatov, M. Otelbaev and their students. Most of the published works on this theory relate to linear operators. There are only some papers that consider nonlinear differential operators, which are weak nonlinear perturbations of linear operators. The case when the operator under study is nonlinear, i.e. it cannot be represented as a weak perturbation of a linear operator, is considered only in some separate papers. The results obtained here also relate to this little-studied case. In this work, the coercive properties of a non-divergence nonlinear elliptic differential operator are studied $$ L[u]=-\sum_{i,j=1}^na_{ij}(x)\frac{\partial^2 u}{\partial x_i\partial x_j}+V(x,u)u(x), $$ in the weight space $L_{2,\rho}(R^n)$ and on the basis of coercive estimates, its separability in this space is proved. Based on the separability of the considered elliptic operator of nondivergent form, we study the coercive solvability of a nonlinear elliptic differential equation in a weighted Hilbert space $L_{2,\rho}(R^n)$.

Détail

Keywords: Elliptic operator, non-divergent type of operator, coercive estimates, nonlinearity, separability, solvability, Hilbert space,weight space.

Bibliographie

[1] Everitt W. N., Gierz M., “Some properties of the domains of certain differential operators”, Proc. London Math. Soc., 23 (1971), 301–324 | DOI | MR | Zbl

[2] Everitt W. N.,Gierz M., “On some properties of the powers of a family self-adjoint differential expressions”, Proc. London Math. Soc., 24 (1972), 149–170 | DOI | MR | Zbl

[3] Everitt W. N., Gierz M., “Some inequalities associated with certain differential operators”, Math. Z., 126 (1972), 308–326 | DOI | MR | Zbl

[4] Everitt W. N.,Gierz M., “Inequalities and separation for Schrodinger -type operators in $L_2(R^n)$”, Proc. Roy. Soc. Edinburg, Sect A, 79 (1977), 149–170 | MR

[5] Boimatov, K. Kh., “Theorems of separability”, Doklady Akad. Nauk SSSR, 213:5 (1973), 1009–1011 | Zbl

[6] Boimatov, K. Kh., “Separability theorems, weighted spaces and their applications”, Proc.of the Math. Inst. of the USSR Academy of Sciences im.Steklova, 170, 1984, 37–76 | Zbl

[7] Boimatov, K. Kh., “Coercive estimates and separability for second order elliptic differential equations”, Doklady Akad. Nauk SSSR, 301:5 (1988), 1033–1036

[8] Boimatov, K. Kh., Saripov, A., “Coercive properties of nonlinear Schrodinger and Dirac operators”, Dokl. Mathematics, 326:3 (1992), 393–398 | Zbl

[9] Boimatov, K. Kh., “Coercive estimates and separability theorems for differential operators of the second order”, Mathematical notes, 46:6 (1989), 110–112 | MR | Zbl

[10] Otelbaev, M., “Coercitive estimates and separability theorems for elliptic equations in $R^n$”, Proc.of the Math. Inst. of the USSR Academy of Sciences im.Steklova, 161, 1983, 195–217 | MR | Zbl

[11] Muratbekov, M. B., Muratbekov, M. M., Ospanov, K. N., “Coercive solvability of odd-order differential equations and its applications”, Dokl. Mathematics, 435:3 (2010), 310–313 | Zbl

[12] Zayed E. M. E., “Separation for the biharmonic differential operator in the Hilbert space associated with existence and uniqueness theorem”, J. Math. Anal. Appl., 337 (2008), 659–666 | DOI | MR | Zbl

[13] Zayed E. M. E., Salem Omram, “Separation for triple-harmonic differential operator in the Hilbert”, International J. Math. Combin., 4 (2010), 13–23 | Zbl

[14] Zayed E. M. E., Mohamed A. S., Atia H. A., “Inequalities and separation for the Laplace-Beltrami differential operator in Hilbert spaces”, J. Math. Anal. Appl., 336 (2007), 81–92 | DOI | MR | Zbl

[15] Zayed E. M. E., “Separation for an elliptic differential operators in a weighted its application to an existence and uniqueness theorem”, Dynamits of continuous, discrede and impulsive systems.Series A: Mathematical Analysis, 22 (2015), 409–421 | MR | Zbl

[16] Mohamed A. S., H. A, Atia, “Separation of the general second elliptic differential operator potential in the weighted weighted Hilbert spaces”, Applied Mathematics and Computation, 162 (2005), 155–163 | DOI | MR | Zbl

[17] Karimov, O. Kh., “On separation of second order nonlinear differential operators with matrix coefficients”, Izvestiya Akademii nauk Respubliki Tajikistan. Otdeleniye fiziko-matematicheskikh, khimicheskikh, geologicheskikh i tekhnicheskikh nauk, 2014, no. 4(157), 42–50 (in Russian)

[18] Karimov, O. Kh., “On separation of nonlinear second order nonlinear differential operators with matrix coefficients in a weighted space”, Doklady Akademii nauk Respubliki Tajikistan, 58:8 (2015), 665–673 (in Russian)

[19] Karimov. O. Kh., “Coercive properties and separability biharmonic operator with matrix potential”, Ufa mathematical journal, 9:1 (2017), 55–62 | Zbl

[20] Karimov, O. Kh., “On coercive solvability the schrodinger equation in a Hilbert space”, Doklady Akademii nauk Respubliki Tajikistan, 61:11–12 (2018), 829–836 (in Russian)

[21] Karimov O. Kh., “On the separation property of nonlinear second-order differential operators with matrix coefficients in weighted spaces”, Journal of mathematical sciences, 241:5 (2019), 589–595 | DOI | MR | Zbl