Speciality:
01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date:
03.02.1948
E-mail: Keywords: harmonic analysis; integral operators of harmonic and analytic functions; boundary integral equations of logarithmic potential theory and elasticity theory on nonregular contours; elliptic problems in domains with nonregular boundary.
Subject:
The full description of finitely connected plane domains $\Omega$ with a piecewise smooth boundary having that property the harmonic projector continuously maps the space $L^p(\Omega)$, $1<p<\infty$, onto the subspace of harmonic functions, was obtained. For these domains the theorem of the unique solvability in ${\stackrel{\circ}{W}^2_p}(\Omega) of the biharmonic equations $\Delta^2 u=f$, $f\in W^{-2}_p(\Omega), was proved. A number of paper (with V. G. Mazya) the boundary integral equations of the logarithmic potential theory are studied under the assumption that contour has a peak. For each equation a pair of function spaces such that the corresponding operator maps one of them onto another was found. The kernels of these operators was described and conditions for the triviality of this kernels was found. For domains with a peak the theorems on solvability of the classic Dirichlet and Neumann problems in appropriate pairs of functional spaces with $L^p$-metric was obtained.
Main publications:
Maz'ya V., Soloviev A. $L_p$-theory of a boundary integral equation on a cuspidal contour // Applicable Analysis, 1997, v. 65, p. 289–305.
Maz'ya V., Soloviev A. $L_p$-theory of boundary integral equation on a contour with outward peak // Integral Equations and Operator Theory, 1998, v. 32, p. 75–100.
Maz'ya V., Soloviev A. $L_p$-theory of boundary integral equation on a contour with inward peak // Zeitschrift fuer Analysis und ihre Anwendungen, 1998, v. 17, n. 3, p. 641–673.
Maz'ya V., Soloviev A. $L_p$-theory of direct boundary integral equations on a contour with peak // Mathematical aspect of boundary element methods, p. 203–214; In Research Notes in Mathematics, 414, Chapman \& Hall/CRC, London, 2000.
