Abstract
We study a regular elliptic boundary-value problem in a bounded domain with smooth boundary. We prove that the operator of this problem is a Fredholm one in a two-sided improved scale of functional Hilbert spaces and that it generates there a complete collection of isomorphisms. Elements of this scale are Hörmander-Volevich-Paneyakh isotropic spaces and some their modi.cations. An a priori estimate for a solution is obtained and its regularity is investigated.
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References
J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications [Russian translation], Mir, Moscow (1971).
Yu. M. Berezanskii, S. G. Krein, and Ya. A. Roitberg, “Theorem on homeomorphisms and a local increase in smoothness up to the boundary for solutions of elliptic equations,” Dokl. Akad. Nauk SSSR, 148, No. 4, 745–748 (1963).
Ya. A. Roitberg, “Elliptic problems with inhomogeneous boundary conditions and a local increase in smoothness up to the boundary for generalized solutions,” Dokl. Akad. Nauk SSSR, 157, No. 4, 798–801 (1964).
Yu. M. Berezanskii, Expansion in Eigenfunctions of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965).
Ya. A. Roitberg, Elliptic Boundary Value Problems in the Spaces of Distributions, Kluwer, Dordrecht (1996).
A. Kozhevnikov, “Complete scale of isomorphisms for elliptic pseudodifferential boundary-value problems,” J. London Math. Soc. (2nd series), 64, No. 2, 409–422 (2001).
L. Hörmander, Linear Partial Differential Operators [Russian translation], Mir, Moscow (1965).
L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 2: Differential Operators with Constant Coefficients [Russian translation], Mir, Moscow (1986).
L. R. Volevich and B. P. Paneyakh, “On some spaces of generalized functions and imbedding theorems,” Usp. Mat. Nauk, 20, No. 1, 3–74 (1965).
B. Paneyakh, The Oblique Derivative Problem. The Poincaré Problem, Wiley, Berlin (2000).
V. A. Mikhailets and A. A. Murach, “Improved scales of spaces and elliptic boundary-value problems. I,” Ukr. Math. J., 58, No. 2, 244–262 (2006).
V. A. Mikhailets and A. A. Murach, “Improved scales of spaces and elliptic boundary-value problems. II,” Ukr. Math. J., 58, No 3, 398–417 (2006).
Sh. A. Alimov, V. A. Il’in, and E. M. Nikishin, “Convergence problems of multiple trigonometric series and spectral decompositions. I,” Russ. Math. Surv., 31, No. 6, 29–86 (1976).
V. A. Mikhailets, “Asymptotics of the spectrum of elliptic operators and boundary conditions,” Sov. Math. Dokl., 266, No. 5, 464–468 (1982).
V. A. Mikhailets, “A precise estimate of the remainder in the spectral asymptotics of general elliptic boundary problems,” Funct. Anal. Appl., 23, No. 2, 137–139 (1989).
G. A. Kalyabin and P. I. Lizorkin, “Spaces of functions of generalized smoothness,” Math. Nachr., 133, No. 1, 7–32 (1987).
D. D. Haroske and S. D. Moura, “Continuity envelopes of spaces of generalized smoothness, entropy and approximation numbers,” J. Approxim. Theory, 128, 151–174 (2004).
W. Farkas and H.-G. Leopold, “Characterizations of function spaces of generalized smoothness,” Ann. Mat. Pura Appl., 185, No. 1, 1–62 (2006).
G. Shlenzak, “Elliptic problems in an improved scale of spaces,” Vestn. Mosk. Univ., Ser. 1, Mat., Mekh., 29, No. 4, 48–58 (1974).
V. A. Mikhailets and A. A. Murach, “Improved scales of spaces and elliptic boundary-value problems. III,” Ukr. Math. J., 59, No. 5, 744–765 (2007).
V. A. Mikhailets and A. A. Murach, “Regular elliptic boundary-value problem for a homogeneous equation in a two-sided improved scale of spaces,” Ukr. Math. J., 58, No. 11, 1748–1767 (2006).
V. A. Mikhailets and A. A. Murach, “Elliptic operator with homogeneous regular boundary conditions in two-sided refined scale of spaces,” Ukr. Math. Bull., 3, No. 4, 529–560 (2006).
A. A. Murach, “Elliptic pseudodifferential operators in the improved scale of spaces on a closed manifold,” Ukr. Math. J., 59, No. 6, 874–893 (2007).
V. A. Mikhailets and A. A. Murach, Elliptic Systems of Pseudodifferential Equations in a Refined Scale on a Closed Manifold, Preprint arXiv:0711.2164v1 [math.AP] (2007).
V. A. Mikhailets and A. A. Murach, Interpolation with a Function Parameter and Refined Scale of Spaces, Preprint arXiv:0712.1135v1 [math.AP] (2007).
A. A. Murach, “Elliptic boundary-value problems in complete scales of spaces of the Lizorkin-Triebel type,” Dokl. Nats. Akad. Nauk Ukr., No. 12, 36–39 (1994).
A. A. Murach, “Elliptic boundary-value problems in complete scales of Nikol’skii-type spaces,” Ukr. Math. J., 46, No. 12, 1827–1835 (1994).
S. G. Krein (editor), Functional Analysis [in Russian], Nauka, Moscow (1972).
M. S. Agranovich, “Elliptic boundary problems,” in: Encycl. Math. Sci., 79. Part. Different. Equat., Springer, Berlin (1997), pp. 1–144.
E. Seneta, Regularly Varying Functions, Springer, Berlin (1976).
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Springer, Berlin (1976).
V. A. Mikhailets and A. A. Murach, “Interpolation of spaces with functional parameter and spaces of differentiable functions,” Dopov. Nats. Akad. Nauk Ukr., No. 6, 13–18 (2006).
G. Geymonat, “Sui problemi ai limiti per i sistemi lineari ellittici,” Ann. Mat. Pura Appl., Ser. 4, 69, 207–284 (1965).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 4, pp. 497–520, April, 2008.
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Mikhailets, V.A., Murach, A.A. Elliptic boundary-value problem in a two-sided improved scale of spaces. Ukr Math J 60, 574–597 (2008). https://doi.org/10.1007/s11253-008-0074-z
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DOI: https://doi.org/10.1007/s11253-008-0074-z