Open Access

# Elliptic problems in the sense of B. Lawruk on two-sided refined scales of spaces

### Abstract

We investigate elliptic boundary-value problems with additional unknown functions on the boundary of a Euclidean domain. These problems were introduced by Lawruk. We prove that the operator corresponding to such a problem is bounded and Fredholm on two-sided refined scales built on the base of inner product isotropic H\"ormander spaces. The regularity of the distributions forming these spaces are characterized by a real number and an arbitrary function that varies slowly at infinity in the sense of Karamata. For the generalized solutions to the problem, we prove theorems on a priori estimates and local regularity in these scales. As applications, we find new sufficient conditions under which the solutions have continuous classical derivatives of a prescribed order.

Key words: Elliptic boundary-value problem, slowly varying function, H¨ormander space, two-sided refined scale, Fredholm operator, a priori estimate for solutions, local regularity of solutions.

### Article Information

 Title Elliptic problems in the sense of B. Lawruk on two-sided refined scales of spaces Source Methods Funct. Anal. Topology, Vol. 21 (2015), no. 1, 6-21 MathSciNet 3407917 zbMATH 06533464 Milestones Received 25/11/2014 Copyright The Author(s) 2015 (CC BY-SA)

### Authors Information

I. S. Chepurukhina
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

A. A. Murach
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine 25/11/2014

### Citation Example

Iryna S. Chepurukhina and Aleksandr A. Murach, Elliptic problems in the sense of B. Lawruk on two-sided refined scales of spaces, Methods Funct. Anal. Topology 21 (2015), no. 1, 6-21.

### BibTex

@article {MFAT765,
AUTHOR = {Chepurukhina, Iryna S. and Murach, Aleksandr A.},
TITLE = {Elliptic problems in the sense of  B. Lawruk on two-sided refined scales of spaces},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {21},
YEAR = {2015},
NUMBER = {1},
PAGES = {6-21},
ISSN = {1029-3531},
MRNUMBER = {3407917},
ZBLNUMBER = {06533464},
URL = {http://mfat.imath.kiev.ua/article/?id=765},
}

### References

1. Anna V. Anop, Aleksandr A. Murach, Parameter-elliptic problems and interpolation with a function parameter, Methods Funct. Anal. Topology 20 (2014), no. 2, 103-116.  MathSciNet
2. A. V. Anop, A. A. Murach, Regular elliptic boundary-value problems in the extended Sobolev scale, Ukrainian Math. J. 66 (2014), no. 7, 969-985.  MathSciNet CrossRef
3. A. G. Aslanyan, D. G. Vasil′ev, V. B. Lidskii, Frequencies of free oscillations of a thin shell that is interacting with a fluid, Funktsional. Anal. i Prilozhen. 15 (1981), no. 3, 1-9.  MathSciNet
4. Ju. M. Berezans′kii, Expansions in eigenfunctions of selfadjoint operators, American Mathematical Society, Providence, R.I., 1968.  MathSciNet
5. N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular variation, Cambridge University Press, Cambridge, 1989.  MathSciNet
6. C. Foias, J.-L. Lions, Sur certains theor\`emes dinterpolation, Acta Sci. Math. Szeged 22 (1961), 269-282.  MathSciNet
7. I. S. Chepurukhina, On some classes of elliptic boundary-value problems in spaces of generalized smoothness, Differential equations and related topics, Zb. prac Inst. mat. NAN Ukr., Kyiv 11 (2014), no. 2, pp. 284-304. (Ukrainian)
8. P. G. Ciarlet, Plates and junctions in elastic multi-structures, Masson, Paris; Springer-Verlag, Berlin, 1990.  MathSciNet
9. Lars Hormander, Linear partial differential operators, Springer Verlag, Berlin-New York, 1976.  MathSciNet
10. Lars Hormander, The analysis of linear partial differential operators. II, Springer-Verlag, Berlin, 2005.  MathSciNet
11. J. Karamata, Sur certains "Tauberian theorems" de M. M. Hardy et Littlewood, Mathematica (Cluj) 3 (1930), 33-48.
12. V. A. Kozlov, V. G. Maz′ya, J. Rossmann, Elliptic boundary value problems in domains with point singularities, American Mathematical Society, Providence, RI, 1997.  MathSciNet
13. B. Lawruk, Parametric boundary-value problems for elliptic systems of linear differential equations. I. Construction of conjugate problems, Bull. Acad. Polon. Sci. S\er. Sci. Math. Astronom. Phys. 11 (1963), no. 5, 257-267. (Russian)
14. B. Lawruk, Parametric boundary-value problems for elliptic systems of linear differential equations. II. A boundary-value problem for a half-space, Bull. Acad. Polon. Sci. S\er. Sci. Math. Astronom. Phys. 11 (1963), no. 5, 269-278. (Russian)
15. B. Lawruk, Parametric boundary-value problems for elliptic systems of linear differential equations. III. Conjugate boundary problem for a half-space, Bull. Acad. Polon. Sci. S\er. Sci. Math. Astronom. Phys. 13 (1965), no. 2, 105-110. (Russian)
16. V. A. Mikhailets, A. A. Murach, Elliptic operators in a refined scale of function spaces, Ukrain. Mat. Zh. 57 (2005), no. 5, 689-696.  MathSciNet CrossRef
17. V. A. Mikhailets, A. A. Murach, Refined scales of spaces, and elliptic boundary value problems. II, Ukrain. Mat. Zh. 58 (2006), no. 3, 352-370.  MathSciNet CrossRef
18. V. A. Mikhailets, A. A. Murach, A regular elliptic boundary value problem for a homogeneous equation in a two-sided refined scale of spaces, Ukrain. Mat. Zh. 58 (2006), no. 11, 1536-1555.  MathSciNet CrossRef
19. V. A. Mikhailets, A. A. Murach, Refined scales of spaces, and elliptic boundary value problems. III, Ukrain. Mat. Zh. 59 (2007), no. 5, 679-701.  MathSciNet CrossRef
20. V. A. Mikhailets, A. A. Murach, An elliptic boundary value problem in a two-sided refined scale of spaces, Ukrain. Mat. Zh. 60 (2008), no. 4, 497-520.  MathSciNet CrossRef
21. Vladimir A. Mikhailets, Aleksandr A. Murach, Interpolation with a function parameter and refined scale of spaces, Methods Funct. Anal. Topology 14 (2008), no. 1, 81-100.  MathSciNet
22. Vladimir A. Mikhailets, Aleksandr A. Murach, Hormander spaces, interpolation, and elliptic problems, De Gruyter, Berlin, 2014.  MathSciNet CrossRef
23. Vladimir A. Mikhailets, Aleksandr A. Murach, The refined Sobolev scale, interpolation, and elliptic problems, Banach J. Math. Anal. 6 (2012), no. 2, 211-281.  MathSciNet CrossRef
24. Vladimir A. Mikhailets, Aleksandr A. Murach, Hormander spaces, interpolation, and elliptic problems, De Gruyter, Berlin, 2014.  MathSciNet CrossRef
25. Sergei Nazarov, Konstantin Pileckas, On noncompact free boundary problems for the plane stationary Navier-Stokes equations, J. Reine Angew. Math. 438 (1993), 103-141.  MathSciNet
26. J. Peetre, On interpolation functions. II, Acta Sci. MAth. (Szeged) 29 (1968), 91-92.  MathSciNet
27. Ja. A. Roitberg, Elliptic problems with non-homogeneous boundary conditions and local increase of smoothness of generalized solutions up to the boundary, Dokl. Akad. Nauk SSSR 157 (1964), 798-801.  MathSciNet
28. Ja. A. Roitberg, A theorem on the homeomorphisms induced in $L_p$ by elliptic operators and the local smoothing of generalized solutions, Ukrain. Mat. \v Z. 17 (1965), no. 5, 122-129.  MathSciNet
29. Yakov Roitberg, Elliptic boundary value problems in the spaces of distributions, Kluwer Academic Publishers Group, Dordrecht, 1996.  MathSciNet CrossRef
30. Yakov Roitberg, Boundary value problems in the spaces of distributions, Kluwer Academic Publishers, Dordrecht, 1999.  MathSciNet CrossRef
31. Eugene Seneta, Regularly varying functions, Springer-Verlag, Berlin-New York, 1976.  MathSciNet
32. G. Slenzak, Elliptic problems in a refined scale of spaces, Vestnik Moskov. Univ. Ser. I Mat. Meh. 29 (1974), no. 4, 48-58.  MathSciNet
33. L. R. Volevic, B. P. Panejah, Some spaces of generalized functions and embedding theorems, Uspehi Mat. Nauk 20 (1965), no. 1 (121), 3-74.  MathSciNet