Speciality:
01.01.02 (Differential equations, dynamical systems, and optimal control)
Birth date:
5.01.1956
E-mail: , ,
Keywords: spectral theory of differential operators; elliptic eigenvalue problems with an indefinite weight function; weighted Sobolev spaces; boundary value problems for linear and nonlinear differential equations and systems; boundary value problems for operator-differential equations; operator theory.
Subject:
Necessary and sufficient and some sufficent conditions ensuring the Riesz basis property are obtained for the eigenfunctions and associated functions of the eigenvalue problems $Lu=\lambda Bu$, $x\in G\subset R^n$, $B_j u|_{\Gamma}=0$, $j=\overline{1,m}$, where $L$ is an elliptic, degenerate elliptic, or quasielliptic operator defined on a domain $G\subset R^n$ with boundary $\Gamma$, $B_j$'s are differential operators defined on $\Gamma$, and $Bu=g(x)u$, with $g(x)$ a measurable function assuming both positive and negative values in $G$. The basisness questions are studied in the weighted Lebesgue space endowed with the norm $\|u\|=\|u |g|^{1/2}\|_{L_{2}(G)}$. Similar results on the Riesz basis property are obtained for eigenelements and associated elements of linear selfadjoint pencils $Lu=\lambda Bu$. The questions of solvability of boundary value problems and qualitative properties of solutions are studied for the first order operator-differential equations $L(t)u=B(t)u_t$, where the operators $B(t):E\to E$ ($E$ is a complex Hilbert space) are symmetric at the interior points of the interval $(0,T)$ and selfadjoint at the points $0,T$, the operators $L(t)$ meet some conditions of the dissipativity type. The question on the interpolation is studied for the weighted Sobolev spaces endowed with the norm $\|u\|_{H_{p,\Psi}^m(\Omega)}^p= \int\limits_{\Omega}\sum\limits_{|\alpha|\le m}\omega_{\alpha}|D^{\alpha}u(x)|^p\,dx$. Here $\Psi=\{\omega_{\alpha}\}_{|\alpha|\le m}$ is a collection of positive continuous functions in $\Omega$. Under some conditions on $\omega_{\alpha}$, the spaces $(H_{p,\Psi}^m(\Omega),L_{p,\omega}(\Omega))_{1-s,p}$ are described ($\omega$ is also positive and continuous).
Main publications:
Egorov I. E., Pyatkov S. G., Popov S. V. Neklassicheskie operatorno-differentsialnye uravneniya. Novosibirsk: Nauka, 2000.
Pyatkov S. G. Riesz completeness of the eigenelements and associated elements of linear selfadjoint pencils // Russian Acad. Sci. Sb. Math., v. 81, no. 2, 1995, p. 343–361.
Pyatkov S. G. Interpolation of weighted Sobolev spaces // Sib. Advan. Math., v. 10, no. 4, 2000, p. 83–132.
Pyatkov S. G., Abasheeva N. L. Razreshimost kraevykh zadach dlya operatorno-differentsialnykh uravnenii smeshannogo tipa // Sib. matem. zhurnal, t. 41, # 6, 2000, s. 1419–1435.
Pyatkov S. G. Elliptic eigenvalue problems involving an indefinite weight functions // Sib. Advan. Math., v. 10, no. 4, 2000, p. 134–150.
