We study irregular elliptic problems with boundary operators of higher orders and prove that these problems are Fredholm on appropriate pairs of the inner-product Hörmander spaces that form a two-sided refined Sobolev scale. We prove a theorem on the regularity of generalized solutions to the analyzed problems in these spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. D. Ventsel’, “On boundary conditions for many-dimensional diffusion processes,” Teor. Ver. Ee Primen., 4, 172–185 (1959).
V. N. Krasil’nikov, “On the solution of some boundary-contact problems of linear hydrodynamics,” Prikl. Mat. Mekh., 25, No. 4, 764–768 (1961).
V. A. Veshev and D. P. Kouzov, “On the influence of the medium on the vibration of plates connected at the right angle,” Akust. Zh., 23, No. 3, 368–377 (1977).
Ya. A. Roitberg, Elliptic Boundary-Value Problems in the Spaces of Distributions, Kluwer AP, Dordrecht (1996).
V. A. Kozlov, V. G. Maz’ya, and J. Rossmann, Elliptic Boundary-Value Problems in Domains with Point Singularities, American Mathematical Society, Providence, RI (1997).
L. Hörmander, Linear Partial Differential Operators, Springer, Berlin (1963).
L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. 2. Differential Operators with Constant Coefficients, Springer, Berlin (1983).
N. Jacob, Pseudodifferential Operators and Markov Processes, Vols. 1–3, Imperial College Press, London (2001, 2002, 2005).
V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation, and Elliptic Problems, De Gruyter, Berlin (2014).
F. Nicola and L. Rodino, Global Pseudodifferential Calculus on Euclidean Spaces, Birkhäuser, Basel (2010).
B. Paneah, The Oblique Derivative Problem. The Poincaré Problem, Wiley, Berlin (2000).
A. I. Stepanets, Methods of Approximation Theory, VSP, Utrecht (2005).
H. Triebel, The Structure of Functions, Birkhäuser, Basel (2001).
V. A. Mikhailets and A. A. Murach, “Elliptic operators in a refined scale of functional spaces,” Ukr. Mat. Zh., 57, No. 5, 689–696 (2005); English translation: Ukr. Math. J., 57, No. 5, 817–825 (2005).
V. A. Mikhailets and A. A. Murach, “Improved scales of spaces and elliptic boundary-value problems. II,” Ukr. Mat. Zh., 58, No. 3, 352–370 (2006); English translation: Ukr. Math. J. , 58, No. 3, 398–417 (2006).
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. Mat. Zh.,58, No. 11, 1536–1555 (2006); English translation: 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).
V. A. Mikhailets and A. A. Murach, “Improved scales of spaces and elliptic boundary-value problems. III,” Ukr. Mat. Zh., 59, No. 5, 679–701 (2007); English translation: Ukr. Math. J., 59, No. 5, 744–765 (2007).
V. A. Mikhailets and A. A. Murach, “Elliptic boundary-value problem in a two-sided improved scale of spaces,” Ukr. Math. Zh., 60, No. 4, 497–520 (2008); English translation: Ukr. Math. J., 60, No. 4, 574–597 (2008).
V. A. Mikhailets and A. A. Murach, “The refined Sobolev scale, interpolation, and elliptic problems,” Banach J. Math. Anal., 6, No. 2, 211–281 (2012).
A. V. Anop and A. A. Murach, “Parameter-elliptic problems and interpolation with a function parameter,” Meth. Funct. Anal. Top., 20, No. 2, 103–116 (2014).
A. V. Anop and A. A. Murach, “Regular elliptic boundary-value problems in the extended Sobolev scale,” Ukr. Mat. Zh., 66, No. 7, 867–883 (2014); English translation: Ukr. Math. J., 66, No. 7, 969–985 (2014).
A. V. Anop and T. M. Kasirenko, “Elliptic boundary-value problems in Hörmander spaces,” Meth. Funct. Anal. Topol., 22, No. 4, 295–310 (2016).
V. A. Mikhailets and A. A. Murach, “Extended Sobolev scale and elliptic operators,” Ukr. Mat. Zh., 65, No. 3, 392–404 (2013); English translation: Ukr. Math. J., 65, No. 3, 435–447 (2013).
V. A. Mikhailets and A. A. Murach, “Interpolation Hilbert spaces between Sobolev spaces,” Results Math., 67, No. 1, 135–152 (2015).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, No. 3, pp. 299–317, March, 2018.
Rights and permissions
About this article
Cite this article
Anop, A., Kasirenko, T.M. & Murach, O.O. Irregular Elliptic Boundary-Value Problems and Hörmander Spaces. Ukr Math J 70, 341–361 (2018). https://doi.org/10.1007/s11253-018-1504-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-018-1504-1