Reularizagtion of the ill-posed Cauchy problem for elliptic systems and complexes;
acyclicity of differential complexes;
boundary value problems for elliptic systems
(the Cauchy problem, Dirichlet problem, mixed problems);
duality in spaces of solutions to elliptic systems;
completeness of the root functions of non-coercive mixed problems for elliptic equations in
the Sobolev–Slobodetskii spaces;
boundary problems for parabolic equations.
Main publications:
Shlapunov A., Tarkhanov N., “On completeness of root functions
of Sturm–Liouville problems
with discontinuous boundary operators”, Journal of Differential Equations, 255 (2013), 3302–3335
Shlapunov A., Tarkhanov N., “Mixed problems with a parameter”, Russian J. Math. Phys., 12:1 (2005), 97–124
Schulze B.-W., Shlapunov A. Tarkhanov N., “Green integrals on manifolds with cracks”, Annals of Global Analysis and Geometry, 24 (2003), 131–160
Shlapunov A., Tarkhanov N., “Duality by reproducing kernels”, International Journal of Math. and Math. Sciences, 2003:6 (2003), 1–78
Shlapunov A., Tarkhanov N., “Bases with double orthogonality in the Cauchy problem for systems with injective symbols”, The Proceedings of the London Mathematical Society, 71:1 (1995), 1–52
Arkadii B. Levskii, Alexander A. Shlapunov, “On the grothendieck duality for the space of holomorphic Sobolev functions”, J. Sib. Fed. Univ. Math. Phys., 17:4 (2024), 513–518
2.
Yu. A. Khoryakova, A. A. Shlapunov, “On Grothendieck-type duality for spaces of holomorphic functions of several variables”, Mat. Sb., 215:8 (2024), 120–140; Sb. Math., 215:8 (2024), 1114–1133
2023
3.
Ksenija V. Gagelgans, Alexander A. Shlapunov, “The Fredholm Navier–Stokes type equations for the de Rham complex over weighted Hölder spaces”, J. Sib. Fed. Univ. Math. Phys., 16:3 (2023), 283–299
4.
Ilya A. Kurilenko, Alexander A. Shlapunov, “On the ill-posed Cauchy problem for the polyharmonic heat equation”, J. Sib. Fed. Univ. Math. Phys., 16:2 (2023), 194–203
2022
5.
A. A. Shlapunov, “On the Approximation of Solutions to the Heat Equation in the Lebesgue Class $L^2$ by More Regular Solutions”, Mat. Zametki, 111:5 (2022), 778–794; Math. Notes, 111:5 (2022), 782–794
P. Yu. Vilkov, I. A. Kurilenko, A. A. Shlapunov, “Approximation and Carleman formulas for solutions to parabolic Lamé-type operators in cylindrical domains”, Sibirsk. Mat. Zh., 63:6 (2022), 1224–1236; Siberian Math. J., 63:6 (2022), 1049–1059
A. A. Shlapunov, N. N. Tarkhanov, “Inverse image of precompact sets and regular solutions to the Navier-Stokes equations”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 32:2 (2022), 278–297
A. A. Shlapunov, N. Tarkhanov, “An open mapping theorem for the Navier-Stokes type equations associated with the de Rham complex over ${\mathbb R}^n$”, Sib. Èlektron. Mat. Izv., 18:2 (2021), 1433–1466
K. V. Gagelgans, A. A. Shlapunov, “On the de Rham complex on a scale of anisotropic weighted Hölder spaces”, Sib. Èlektron. Mat. Izv., 17 (2020), 428–444
Ilya A. Kurilenko, Alexander A. Shlapunov, “On Carleman-type formulas for solutions to the heat equation”, J. Sib. Fed. Univ. Math. Phys., 12:4 (2019), 421–433
Azal Mera, Alexander A. Shlapunov, Nikolai Tarkhanov, “Navier–Stokes equations for elliptic complexes”, J. Sib. Fed. Univ. Math. Phys., 12:1 (2019), 3–27
K. V. Sidorova (Gagelgans), A. A. Shlapunov, “On the Closure of Smooth Compactly Supported Functions in Weighted Hölder Spaces”, Mat. Zametki, 105:4 (2019), 616–631; Math. Notes, 105:4 (2019), 604–617
Andrei A. Parfenov, Alexander A. Shlapunov, “On the Fredholm property for the steady Navier–Stokes equations in weighted Hölder spaces”, J. Sib. Fed. Univ. Math. Phys., 11:5 (2018), 659–662
A. N. Polkovnikov, A. A. Shlapunov, “Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions”, Sibirsk. Mat. Zh., 58:4 (2017), 870–884; Siberian Math. J., 58:4 (2017), 676–686
Yulia L. Cherepanova, Alexander A. Shlapunov, “On an analogue of the Riemann–Hilbert problem for a non-linear perturbation of the Cauchy–Riemann operator”, J. Sib. Fed. Univ. Math. Phys., 9:4 (2016), 427–431
2015
16.
N. Tarkhanov, A. A. Shlapunov, “Sturm–Liouville problems in weighted spaces in domains with nonsmooth edges. II”, Mat. Tr., 18:2 (2015), 133–204; Siberian Adv. Math., 26:4 (2016), 247–293
N. Tarkhanov, A. A. Shlapunov, “Sturm–Liouville problems in weighted spaces in domains with nonsmooth edges. I”, Mat. Tr., 18:1 (2015), 118–189; Siberian Adv. Math., 26:1 (2016), 30–76
Alexander N. Polkovnikov, Aleksander A. Shlapunov, “On the spectral properties of a non-coercive mixed problem associated with $\overline\partial$-operator”, J. Sib. Fed. Univ. Math. Phys., 6:2 (2013), 247–261
Alexander A. Shlapunov, “Boundary problems for Helmholtz equation and the Cauchy problem for Dirac operators”, J. Sib. Fed. Univ. Math. Phys., 4:2 (2011), 217–228
Ivan V. Shestakov, Alexander A. Shlapunov, “Negative Sobolev Spaces in the Cauchy Problem for the Cauchy–Riemann Operator”, J. Sib. Fed. Univ. Math. Phys., 2:1 (2009), 17–30
I. V. Shestakov, A. A. Shlapunov, “The Cauchy problem for operators with injective symbol in the Lebesgue space $L^2$ in a domain”, Sibirsk. Mat. Zh., 50:3 (2009), 687–702; Siberian Math. J., 50:3 (2009), 547–559
Ivan V. Shestakov, Alexander A. Shlapunov, “On the Cauchy Problem for Operators with Injective Symbols in Sobolev Spaces”, J. Sib. Fed. Univ. Math. Phys., 1:1 (2008), 52–62
D. P. Fedchenko, A. A. Shlapunov, “On the Cauchy problem for the multi-dimensional Cauchy-Riemann operator in the Lebesgue space $L^2$ in a domain”, Mat. Sb., 199:11 (2008), 141–160; Sb. Math., 199:11 (2008), 1715–1733
N. N. Tarkhanov, A. A. Shlapunov, “Green's formulas in complex analysis”, Itogi Nauki i Tekhniki. Sovrem. Mat. Pril. Temat. Obz., 108 (2006), 106–157; J. Math. Sci. (N. Y.), 120:6 (2004), 1868–1900
A. A. Shlapunov, “A Method for Constructing Solutions to Linear Systems of Partial Differential Equations”, Mat. Tr., 9:2 (2006), 191–204; Siberian Adv. Math., 17:2 (2007), 144–152
2005
27.
A. A. Shlapunov, “On some conditions for the solvability of overdetermined systems of differential equations in Sobolev spaces”, Izv. Vyssh. Uchebn. Zaved. Mat., 2005, no. 4, 70–80; Russian Math. (Iz. VUZ), 49:4 (2005), 67–76
A. A. Shlapunov, “On duality in spaces of polyharmonic functions”, Izv. Vyssh. Uchebn. Zaved. Mat., 2002, no. 8, 79–81; Russian Math. (Iz. VUZ), 46:8 (2002), 75–77
29.
A. A. Shlapunov, “On duality in spaces of solutions to elliptic systems”, Sibirsk. Mat. Zh., 43:4 (2002), 953–963; Siberian Math. J., 43:4 (2002), 769–777
A. A. Shlapunov, “On a olvability condition for systems with an injective symbol in terms of iterations of double layer potentials”, Sibirsk. Mat. Zh., 42:4 (2001), 952–963; Siberian Math. J., 42:4 (2001), 801–810
L. A. Aizenberg, O. V. Karepov, A. A. Shlapunov, “On the property of uniqueness and existence of a limit in Carleman's formulas”, Trudy Mat. Inst. Steklov., 203 (1994), 3–12; Proc. Steklov Inst. Math., 203 (1995), 1–9
1992
32.
N. N. Tarkhanov, A. A. Shlapunov, “Bases with double orthogonality in the Cauchy problem for systems
with an injective symbol”, Dokl. Akad. Nauk, 326:1 (1992), 45–49; Dokl. Math., 46:2 (1993), 225–230
A. A. Shlapunov, N. N. Tarkhanov, “On the Cauchy problem for holomorphic functions of the Lebesgue class $L^2$ in a domain”, Sibirsk. Mat. Zh., 33:5 (1992), 186–195; Siberian Math. J., 33:5 (1992), 914–922
Full list of publications
