Abstract
A criterion for the approximability of all solutions of the heat equation in a bounded cylindrical domain that belong to the Lebesgue class by more regular (e.g., Sobolev) solutions of the same equation in a bounded cylindrical domain with larger base is obtained. Namely, the complement of the smaller base to the larger one must have no (nonempty connected) compact components. As an important corollary, we prove a theorem on the existence of a doubly orthogonal basis for the corresponding pair of Hilbert spaces.
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References
B. F. Jones (Jr.), “An approximation theorem of Runge type for the heat equation,” Proc. Amer. Math. Soc. 52 (1), 289–292 (1975).
R. Diaz, “A Runge theorem for solutions of the heat equation,” Proc. Amer. Math. Soc. 80 (4), 643–646 (1980).
P. M. Gauthier and N. N. Tarkhanov, “Rational approximation and universality for a quasilinear parabolic equation,” J. Contemp. Math. Anal. 43, 353–364 (2008).
C. Runge, “Zur Theorie der eindeutigen analytischen Funktionen,” Acta Math. 6, 229–244 (1885).
S. N. Mergelyan, “Harmonic approximation and approximate solution of the Cauchy problem for the Laplace equation,” Uspekhi Mat. Nauk 11 (5(71)), 3–26 (1956).
B. Malgrange, “Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution,” Ann. Inst. Fourier (Grenoble) 6, 271–355 (1956).
N. Tarkhanov, The Analysis of Solutions of Elliptic Equations (Kluwer Acad. Publ., Dordrecht, 1997).
A. G. Vitushkin, “The analytic capacity of sets in problems of approximation theory,” Russian Math. Surveys 22 (6), 139–200 (1967).
V. P. Havin, “Approximation by analytic functions in the mean,” Dokl. Akad. Nauk SSSR 178 (5), 1025–1028 (1968).
L. I. Hedberg, “Nonlinear potential theory and Sobolev spaces,” in Nonlinear Analysis, Function Spaces and Applications, Teubner-Texte Math. (Teubner, Leipzig, 1986), Vol. 93, pp. 5–30.
I. F. Krasichkov, “Systems of functions with the dual orthogonality property,” Math. Notes 4 (5), 821–824 (1968).
A. N. Tikhonov and V. Ya. Arsenin, Methods for Solving Ill-Posed Problems (Nauka, Moscow, 1974) [in Russian].
N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations (Akademie Verlag, Berlin, 1995).
S. Bergman, The Kernel Function and Conformal Mapping (Amer. Math. Soc., Providence, RI, 1970).
L. A. Aizenberg and A. M. Kytmanov, “On the possibility of holomorphic extension into a domain of function defined on a connected piece of its boundary,” Sb. Math. 72 (2), 467–483 (1992).
A. A. Shlapunov and N. Tarkhanov, “Bases with double orthogonality in the Cauchy problem for systems with injective symbols,” Proc. London. Math. Soc. 71 (1), 1–54 (1995).
D. P. Fedchenko and A. A. Shlapunov, “On the Cauchy problem for the elliptic complexes in spaces of distributions,” Complex Var. Elliptic Equ. 59 (5), 651–679 (2014).
K. O. Makhmudov, O. I. Makhmudov and N. N. Tarkhanov, “A Nonstandard Cauchy Problem for the Heat Equation,” Math. Notes 102 (2), 250–260 (2017).
Ilya A. Kurilenko and Alexander A. Shlapunov, “On Carleman-type formulas for solutions to the heat equation,” J. Sib. Fed. Univ. Math. Phys. 12 (4), 421–433 (2019).
V. P. Mikhailov, Partial Differential Equations (Nauka, Moscow, 1976) [in Russian].
O. V. Besov, V. P. Il’in, and S. M. Nikol’skii, Integral Representations and Embedding Theorems (Nauka, Moscow, 1975) [in Russian].
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéare (Gauthier-Villars, Paris, 1969).
N. V. Krylov, Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, in Grad. Stud. in Math. (Amer. Math. Soc., Providence, RI, 2008), Vol. 96.
A. Friedman, Partial Differential Equations of Parabolic Type (Prentice-Hall, Englewood Cliffs, NJ, 1964).
L. I. Hedberg and T. H. Wolff, “Thin sets in nonlinear potential theory,” Ann. Inst. Fourier (Grenoble) 33 (4), 161–187 (1983).
A. G. Sveshnikov, A. N. Bogolyubov, and V. V. Kravtsov, Lectures on Mathematical Physics (Nauka, Moscow, 2004) [in Russian].
A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (Nauka, Moscow, 1966) [in Russian].
F. Bowman, Introduction to Bessel Functions (Dover Publ., New York, 1958).
A. A. Shlapunov, “Spectral decomposition of Green’s integrals and existence of \(W^{s,2}\)-solutions of matrix factorizations of the Laplace operator in a ball,” Rend. Sem. Mat. Univ. Padova 96, 237–256 (1996).
Funding
This work was supported by Sirius University of Science and Technology in the framework of the scientific project “Spectral and functional inequalities of mathematical physics and their applications.”
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Translated from Matematicheskie Zametki, 2022, Vol. 111, pp. 778–794 https://doi.org/10.4213/mzm13201.
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Shlapunov, A.A. On the Approximation of Solutions to the Heat Equation in the Lebesgue Class \(L^2\) by More Regular Solutions. Math Notes 111, 782–794 (2022). https://doi.org/10.1134/S0001434622050121
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DOI: https://doi.org/10.1134/S0001434622050121