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.
Biography
Graduated from Faculty of Mathematics and Physics of Far East State University (Vladivostok) in 1971 (department of mathematical analysis). Ph.D. thesis was defended in 1979. D.Sci. thesis was defended in 1999. A list of my works contains more than 30 titles.
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.
S. F. Dolbeeva, V. N. Pavlenko, S. V. Matveev, O. N. Dementiev, E. A. Sbrodova, A. A. Soloviev, V. I. Ukhobotov, V. E. Fedorov, A. A. Ershov, “Arlen Mikhaylovich Il'in. 90 years since the birth”, Chelyab. Fiz.-Mat. Zh., 7:2 (2022), 135–138
2018
2.
A. A. Soloviev, S. V. Repjevskij, “An estimate for the remainder in the expansion of the elliptic sine”, Trudy Inst. Mat. i Mekh. UrO RAN, 24:2 (2018), 256–265
2017
3.
A. A. Soloviev, “Complementary representation of polynomials over finite fields”, Chelyab. Fiz.-Mat. Zh., 2:2 (2017), 199–209
A. A. Soloviev, D. V. Chernikov, “Biorthogonal wavelet codes with prescribed code distance”, Diskr. Mat., 29:2 (2017), 96–108; Discrete Math. Appl., 28:3 (2018), 179–188
A. A. Soloviev, “A bound for the remainder term in the asymptotic expansion of the elliptic sine containing the first three terms”, Trudy Inst. Mat. i Mekh. UrO RAN, 23:2 (2017), 220–229
2014
7.
A. A. Soloviev, “Asymptotic behavior of solutions of the Hamer equation”, Algebra i Analiz, 26:3 (2014), 159–179; St. Petersburg Math. J., 26:3 (2015), 463–477
1998
8.
V. G. Maz'ya, A. A. Soloviev, “Integral equations of logarithmic potential theory on contours with a cusp in Hölder spaces”, Algebra i Analiz, 10:5 (1998), 85–142; St. Petersburg Math. J., 10:5 (1999), 791–832
A. A. Soloviev, “Integral equations for the Dirichlet and Neumann boundary value problems in a plane domain with a cusp on the boundary”, Mat. Zametki, 59:6 (1996), 881–892; Math. Notes, 59:6 (1996), 637–645
1994
10.
A. A. Soloviev, “О спектре произведения двух проекторов”, Vestnik Chelyabinsk. Gos. Univ., 1994, no. 2, 117–125
1992
11.
A. A. Soloviev, “Letter to the editors: "Estimates in $L^p$ of integral operators connected with spaces of analytic and harmonic functions" [Sibirsk. Mat. Zh. 26 (1985), no. 3, 168–191, 226]”, Sibirsk. Mat. Zh., 33:2 (1992), 218; Siberian Math. J., 33:2 (1992), 369
1991
12.
A. A. Soloviev, “On the index of the operator of the Dirichlet problem in a domain with a piecewise-smooth or Radon boundary”, Izv. Vyssh. Uchebn. Zaved. Mat., 1991, no. 11, 60–66; Soviet Math. (Iz. VUZ), 35:11 (1991), 61–66
A. A. Soloviev, “Асимптотика решений граничных интегральных уравнений теории упругости в плоской области с внешним пиком”, Vestnik Chelyabinsk. Gos. Univ., 1991, no. 1, 150
1989
14.
V. G. Maz'ya, A. A. Soloviev, “On an integral equation for the Dirichlet problem in a plane domain with cusps on the boundary”, Mat. Sb., 180:9 (1989), 1211–1233; Math. USSR-Sb., 68:1 (1991), 61–83
V. G. Maz'ya, A. A. Soloviev, “Solvability of an integral equation of the Dirichlet problem in a
plane domain with cusps on the boundary”, Dokl. Akad. Nauk SSSR, 298:6 (1988), 1312–1315; Dokl. Math., 37:1 (1988), 255–258
A. A. Soloviev, “Estimates in $L^p$ of integral operators connected with spaces of analytic and harmonic functions”, Sibirsk. Mat. Zh., 26:3 (1985), 168–191; Siberian Math. J., 26:3 (1985), 440–460
A. A. Soloviev, “Continuity of the harmonic projection in $L^p$-spaces”, Zap. Nauchn. Sem. LOMI, 126 (1983), 191–195
1978
18.
A. A. Soloviev, “Estimates in $L^p$ of the integral operators that are connected with spaces of analytic and harmonic functions”, Dokl. Akad. Nauk SSSR, 240:6 (1978), 1301–1304
2017
19.
S. M. Voronin, S. F. Dolbeeva, O. N. Dementiev, A. A. Ershov, M. G. Lepchinski, S. V. Matveev, N. B. Medvedeva, D. K. Potapov, E. A. Rozhdestvenskaya, E. A. Sbrodova, I. M. Sokolinskaya, A. A. Soloviev, V. I. Ukhobotov, V. E. Fedorov, “К 70-летию профессора Вячеслава Николаевича Павленко”, Chelyab. Fiz.-Mat. Zh., 2:4 (2017), 383–387
20.
S. F. Dolbeeva, V. N. Pavlenko, S. V. Matveev, O. N. Dementiev, A. V. Mel'nikov, E. A. Sbrodova, A. A. Soloviev, V. I. Ukhobotov, V. E. Fedorov, E. A. Fominykh, A. A. Ershov, “Arlen Mikhaylovich Il’in. Towards 85th birthday”, Chelyab. Fiz.-Mat. Zh., 2:1 (2017), 5–9
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