February  2021, 20(2): 697-735. doi: 10.3934/cpaa.2020286

Elliptic problems with rough boundary data in generalized Sobolev spaces

1. 

Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01024, Ukraine

2. 

University of Konstanz, Department of Mathematics and Statistics, 78457 Konstanz, Germany

* Corresponding author

Received  May 2020 Revised  September 2020 Published  February 2021 Early access  December 2020

Fund Project: The publication contains the results of studies conducted by the joint grant F81 of the National Research Fund of Ukraine and the German Research Society (DFG); competitive project F81/41686. This work was supported by the Grant H2020-MSCA-RISE-2019, project number 873071 (SOMPATY: Spectral Optimization: From Mathematics to Physics and Advanced Technology). The first author was supported by President of Ukraine's grant for competitive project F82/45932

We investigate regular elliptic boundary-value problems in boun\-ded domains and show the Fredholm property for the related operators in an extended scale formed by inner product Sobolev spaces (of arbitrary real orders) and corresponding interpolation Hilbert spaces. In particular, we can deal with boundary data with arbitrary low regularity. In addition, we show interpolation properties for the extended scale, embedding results, and global and local a priori estimates for solutions to the problems under investigation. The results are applied to elliptic problems with homogeneous right-hand side and to elliptic problems with rough boundary data in Nikolskii spaces, which allows us to treat some cases of white noise on the boundary.

Citation: Anna Anop, Robert Denk, Aleksandr Murach. Elliptic problems with rough boundary data in generalized Sobolev spaces. Communications on Pure and Applied Analysis, 2021, 20 (2) : 697-735. doi: 10.3934/cpaa.2020286
References:
[1]

M. S. Agranovich, Elliptic boundary problems, Encyclopaedia Math. Sci., 79 (1997), 1-144.  doi: 10.1007/978-3-662-06721-5_1.

[2]

A. V. Anop and T. M. Kasirenko, Elliptic boundary-value problems in Hörmander spaces, Methods Funct. Anal. Topology, 22 (2016), 295-310. 

[3]

A. V. AnopT. M. Kasirenko and O. O. Murach, Irregular elliptic boundary-value problems and Hörmander spaces, Ukranian Math. J., 70 (2018), 341-361.  doi: 10.1007/s11253-018-1504-1.

[4]

A. V. Anop and A. A. Murach, Regular elliptic boundary-value problems in the extended Sobolev scale, Ukrainian Math. J., 66 (2014), 969-985.  doi: 10.1007/s11253-014-0988-6.

[5]

A. V. Anop and A. A. Murach, To the theory of elliptic boundary-value problems in Hörmander spaces, Transactions of Institute of Mathematics of NAS of Ukraine, 12 (2015), 39-64. 

[6]

A. V. Anop and A. A. Murach, Some semi-homogeneous elliptic boundary-value problems in complete extended Sobolev scale, Transactions of Institute of Mathematics of NAS of Ukraine, 13 (2016), 27-54. 

[7]

A. V. Anop and A. A. Murach, Homogeneous elliptic equations in an extended Sobolev scale, Dopov. Nac. Akad. Nauk Ukr., (2018), 3–11. doi: 10.15407/dopovidi2018.03.003.

[8]

V. G. Avakumović, O jednom O-inverznom stavu, Rad Jugoslav. Akad. Znan. Umjet., 254 (1936), 167-186. 

[9]

Yu. M. Berezansky, Expansions in Eigenfunctions of Selfadjoint Operators, American Mathematical Society, Providence, RI, 1968.

[10]

J. Behrndt, S. Hassi and H. de. Snoo, Boundary Value Problems, Weyl Functions, and Differential Operators, Springer, Cham, 2020. doi: 10.1007/978-3-030-36714-5.

[11] N. H. BinghamC. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1989. 
[12]

V. V. Buldygin, K. H. Indlekofer, O. I. Klesov and J. G. Steinebach, Pseudo-Regularly Varying Functions and Generalized Renewal Processes, Springer, Cham, 2018. doi: 10.1007/978-3-319-99537-3.

[13]

J. FageotA. Fallah and M. Unser, Multidimensional Lévy white noise in weighted Besov spaces, Stochastic Process. Appl., 127 (2017), 1599-1621.  doi: 10.1016/j.spa.2016.08.011.

[14]

M. Faierman, Fredholm theory for an elliptic differential operator defined on $\mathbb R^n$ and acting on generalized Sobolev spaces, Commun. Pure Appl. Anal., 19 (2020), 1463-1483.  doi: 10.3934/cpaa.2020074.

[15]

C. Foiaş and J. L. Lions, Sur certains théorèmes d'interpolation, Acta Scient. Math. Szeged, 22 (1961), 269-282. 

[16]

F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, Proc. Sympos. Pure Math., 79 (2008), 105-173.  doi: 10.1090/pspum/079/2500491.

[17]

G. Geymonat, Sul problema di Dirichlet per le equazoni lineari ellittiche, Ann. Sci. Norm. Sup. Pisa, 16 (1962), 225-284. 

[18]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (revised 3rd edition), Springer, Berlin, 1998.

[19]

M. L. Gol'dman, Imbedding theorems for anisotropic Nikol'skiǐ–Besov spaces with moduli of continuity of a general type, Proc. Steklov Inst. Math., 170 (1987), 95-116. 

[20]

L. Hörmander, On the theory of general partial differential equations, Acta Math., 94 (1955), 161-248.  doi: 10.1007/BF02392492.

[21]

L. Hörmander, Linear Partial Differential Operators, Springer, Berlin, 1963.

[22]

L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. Ⅱ. Differential Operators with Constant Coefficients, Springer, Berlin, 1983. doi: 10.1007/978-3-642-96750-4.

[23]

L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. Ⅲ. Pseudo-Differential Operators, Springer, Berlin, 1985.

[24]

T. KasirenkoV. Mikhailets and A. Murach, Sobolev-like Hilbert spaces induced by elliptic operators, Complex Anal. Oper. Theor., 13 (2019), 1431-1440.  doi: 10.1007/s11785-018-00886-8.

[25]

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. doi: 10.1090/surv/052.

[26]

J. L. Lions and E. Magenes, Problémes aux limites non homogénes, Ⅱ, Ann. Inst. Fourier (Grenoble), 11 (1961), 137-178. 

[27]

J. L. Lions and E. Magenes, Problémes aux limites non homogénes, V, Ann. Scuola Norm. Sup. Pisa (3), 16 (1962), 1-44. 

[28]

J. L. Lions and E. Magenes, Problémes aux limites non homogénes, VI, J. Analyse Math., 11 (1963), 165-188.  doi: 10.1007/BF02789983.

[29]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications, vol. I, Springer, Berlin, 1972.

[30]

V. LosV. A. Mikhailets and A. A. Murach, An isomorphism theorem for parabolic problems in Hörmander spaces and its applications, Commun. Pure Appl. Anal., 16 (2017), 69-97.  doi: 10.3934/cpaa.2017003.

[31]

B. Malgrange, Sur une classe d'opératuers différentiels hypoelliptiques, Bull. Soc. Math., 85 (1957), 283-306. 

[32]

E. Magenes, Spazi di interpolazione ed equazioni a derivate parziali, Atti Ⅶ Congr. Un. Mat., (1965), 134–197.

[33]

W. Matuszewska, On a generalization of regularly increasing functions, Studia Math., 24 (1964), 271-279.  doi: 10.4064/sm-24-3-271-279.

[34]

V. A. Mikhailets and A. A. Murach, Elliptic operators in a refined scale of function spaces, Ukrainian. Math. J., 57 (2005), 817-825.  doi: 10.1007/s11253-005-0231-6.

[35]

V. A. Mikhailets and A. A. Murach, Refined scales of spaces, and elliptic boundary-value problems. Ⅱ, Ukrainian Math. J., 58 (2006), 398-417.  doi: 10.1007/s11253-006-0074-9.

[36]

V. A. Mikhailets and A. A Murach, Regular elliptic boundary-value problem for a homogeneous equation in a two-sided improved scale of spaces, Ukrainian Math. J., 58 (2006), 1748-1767.  doi: 10.1007/s11253-006-0166-6.

[37]

V. A. Mikhailets and A. A. Murach, Refined scales of spaces, and elliptic boundary-value problems. Ⅲ, Ukrainian Math. J., 59 (2007), 744-765.  doi: 10.1007/s11253-007-0048-6.

[38]

V. A. Mikhailets and A. A. Murach, Extended Sobolev scale and elliptic operators, Ukrainian Math. J., 65 (2013), 435-447.  doi: 10.1007/s11253-013-0787-5.

[39]

V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation and Elliptic Problems, De Gruyter, Berlin, 2014. doi: 10.1515/9783110296891.

[40]

V. A. Mikhailets and A. A. Murach, Interpolation Hilbert spaces between Sobolev spaces, Results Math., 67 (2015), 135-152.  doi: 10.1007/s00025-014-0399-x.

[41]

A. A. Murach, Extension of some Lions–Magenes theorems, Methods Funct. Anal. Topology, 15 (2009), 152-167. 

[42]

S. M. Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems [2nd edition], Springer-Verlag, New York–Heidelberg, 1975.

[43]

V. I. Ovchinnikov, The methods of orbits in interpolation theory, in Mathematical Reports. Vol. 1, Part. 2 (ed. J. Peetre), Harwood Academic Publishers, London, 1984.

[44]

J. Peetre, Another approach to elliptic boundary problems, Commun. Pure Appl. Math., 14 (1961), 711-731.  doi: 10.1002/cpa.3160140404.

[45]

J. Peetre, On interpolation functions. Ⅱ, Acta Sci. Math., 29 (1968), 91-92. 

[46]

A. Pliś, A smooth linear elliptic differential equation without any solution in a sphere, Commun. Pure Appl. Math., 14 (1961), 599-617.  doi: 10.1002/cpa.3160140331.

[47]

Ya. A. Roitberg, Elliptic problems with nonhomogeneous boundary conditions and local increase of smoothness up to the boundary for generalized solutions, Dokl. Math., 5 (1964), 1034-1038. 

[48]

Ya. Roitberg, Elliptic Boundary Value Problems in the Spaces of Distributions, Kluwer Academic Publishers, Dordrecht, 1996. doi: 10.1007/978-94-011-5410-9.

[49]

Ya. Roitberg, Boundary Value Problems in the Spaces of Distributions, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-015-9275-8.

[50]

R. Schnaubelt and M. Veraar, Stochastic equations with boundary noise, Progr. Nonlinear Differ. Equ. Appl., 80 (2011), 609-629.  doi: 10.1007/978-3-0348-0075-4_30.

[51]

R. T. Seeley, Singular integrals and boundary-value problems, Amer. J. Math., 88 (1966), 781-809.  doi: 10.2307/2373078.

[52]

E. Seneta, Regularly Varying Functions, Springer, Berlin, 1976.

[53]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators [2nd edition], Johann Ambrosius Barth, Heidelberg, 1995.

[54]

M. Veraar, Regularity of Gaussian white noise on the $d$-dimensional torus, Banach Center Publ., 95 (2011), 385-398.  doi: 10.4064/bc95-0-24.

[55]

L. R. Volevich and B. P. Paneah, Certain spaces of generalized functions and embedding theorems, Uspehi Mat. Nauk, 20 (1965), 3-74. 

[56]

T. N. Zinchenko and A. A. Murach, Douglis–Nirenberg elliptic systems in Hörmander spaces, Ukrainian Math. J., 64 (2012), 1672-1687.  doi: 10.1007/s11253-013-0743-4.

show all references

References:
[1]

M. S. Agranovich, Elliptic boundary problems, Encyclopaedia Math. Sci., 79 (1997), 1-144.  doi: 10.1007/978-3-662-06721-5_1.

[2]

A. V. Anop and T. M. Kasirenko, Elliptic boundary-value problems in Hörmander spaces, Methods Funct. Anal. Topology, 22 (2016), 295-310. 

[3]

A. V. AnopT. M. Kasirenko and O. O. Murach, Irregular elliptic boundary-value problems and Hörmander spaces, Ukranian Math. J., 70 (2018), 341-361.  doi: 10.1007/s11253-018-1504-1.

[4]

A. V. Anop and A. A. Murach, Regular elliptic boundary-value problems in the extended Sobolev scale, Ukrainian Math. J., 66 (2014), 969-985.  doi: 10.1007/s11253-014-0988-6.

[5]

A. V. Anop and A. A. Murach, To the theory of elliptic boundary-value problems in Hörmander spaces, Transactions of Institute of Mathematics of NAS of Ukraine, 12 (2015), 39-64. 

[6]

A. V. Anop and A. A. Murach, Some semi-homogeneous elliptic boundary-value problems in complete extended Sobolev scale, Transactions of Institute of Mathematics of NAS of Ukraine, 13 (2016), 27-54. 

[7]

A. V. Anop and A. A. Murach, Homogeneous elliptic equations in an extended Sobolev scale, Dopov. Nac. Akad. Nauk Ukr., (2018), 3–11. doi: 10.15407/dopovidi2018.03.003.

[8]

V. G. Avakumović, O jednom O-inverznom stavu, Rad Jugoslav. Akad. Znan. Umjet., 254 (1936), 167-186. 

[9]

Yu. M. Berezansky, Expansions in Eigenfunctions of Selfadjoint Operators, American Mathematical Society, Providence, RI, 1968.

[10]

J. Behrndt, S. Hassi and H. de. Snoo, Boundary Value Problems, Weyl Functions, and Differential Operators, Springer, Cham, 2020. doi: 10.1007/978-3-030-36714-5.

[11] N. H. BinghamC. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1989. 
[12]

V. V. Buldygin, K. H. Indlekofer, O. I. Klesov and J. G. Steinebach, Pseudo-Regularly Varying Functions and Generalized Renewal Processes, Springer, Cham, 2018. doi: 10.1007/978-3-319-99537-3.

[13]

J. FageotA. Fallah and M. Unser, Multidimensional Lévy white noise in weighted Besov spaces, Stochastic Process. Appl., 127 (2017), 1599-1621.  doi: 10.1016/j.spa.2016.08.011.

[14]

M. Faierman, Fredholm theory for an elliptic differential operator defined on $\mathbb R^n$ and acting on generalized Sobolev spaces, Commun. Pure Appl. Anal., 19 (2020), 1463-1483.  doi: 10.3934/cpaa.2020074.

[15]

C. Foiaş and J. L. Lions, Sur certains théorèmes d'interpolation, Acta Scient. Math. Szeged, 22 (1961), 269-282. 

[16]

F. Gesztesy and M. Mitrea, Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains, Proc. Sympos. Pure Math., 79 (2008), 105-173.  doi: 10.1090/pspum/079/2500491.

[17]

G. Geymonat, Sul problema di Dirichlet per le equazoni lineari ellittiche, Ann. Sci. Norm. Sup. Pisa, 16 (1962), 225-284. 

[18]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order (revised 3rd edition), Springer, Berlin, 1998.

[19]

M. L. Gol'dman, Imbedding theorems for anisotropic Nikol'skiǐ–Besov spaces with moduli of continuity of a general type, Proc. Steklov Inst. Math., 170 (1987), 95-116. 

[20]

L. Hörmander, On the theory of general partial differential equations, Acta Math., 94 (1955), 161-248.  doi: 10.1007/BF02392492.

[21]

L. Hörmander, Linear Partial Differential Operators, Springer, Berlin, 1963.

[22]

L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. Ⅱ. Differential Operators with Constant Coefficients, Springer, Berlin, 1983. doi: 10.1007/978-3-642-96750-4.

[23]

L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. Ⅲ. Pseudo-Differential Operators, Springer, Berlin, 1985.

[24]

T. KasirenkoV. Mikhailets and A. Murach, Sobolev-like Hilbert spaces induced by elliptic operators, Complex Anal. Oper. Theor., 13 (2019), 1431-1440.  doi: 10.1007/s11785-018-00886-8.

[25]

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. doi: 10.1090/surv/052.

[26]

J. L. Lions and E. Magenes, Problémes aux limites non homogénes, Ⅱ, Ann. Inst. Fourier (Grenoble), 11 (1961), 137-178. 

[27]

J. L. Lions and E. Magenes, Problémes aux limites non homogénes, V, Ann. Scuola Norm. Sup. Pisa (3), 16 (1962), 1-44. 

[28]

J. L. Lions and E. Magenes, Problémes aux limites non homogénes, VI, J. Analyse Math., 11 (1963), 165-188.  doi: 10.1007/BF02789983.

[29]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications, vol. I, Springer, Berlin, 1972.

[30]

V. LosV. A. Mikhailets and A. A. Murach, An isomorphism theorem for parabolic problems in Hörmander spaces and its applications, Commun. Pure Appl. Anal., 16 (2017), 69-97.  doi: 10.3934/cpaa.2017003.

[31]

B. Malgrange, Sur une classe d'opératuers différentiels hypoelliptiques, Bull. Soc. Math., 85 (1957), 283-306. 

[32]

E. Magenes, Spazi di interpolazione ed equazioni a derivate parziali, Atti Ⅶ Congr. Un. Mat., (1965), 134–197.

[33]

W. Matuszewska, On a generalization of regularly increasing functions, Studia Math., 24 (1964), 271-279.  doi: 10.4064/sm-24-3-271-279.

[34]

V. A. Mikhailets and A. A. Murach, Elliptic operators in a refined scale of function spaces, Ukrainian. Math. J., 57 (2005), 817-825.  doi: 10.1007/s11253-005-0231-6.

[35]

V. A. Mikhailets and A. A. Murach, Refined scales of spaces, and elliptic boundary-value problems. Ⅱ, Ukrainian Math. J., 58 (2006), 398-417.  doi: 10.1007/s11253-006-0074-9.

[36]

V. A. Mikhailets and A. A Murach, Regular elliptic boundary-value problem for a homogeneous equation in a two-sided improved scale of spaces, Ukrainian Math. J., 58 (2006), 1748-1767.  doi: 10.1007/s11253-006-0166-6.

[37]

V. A. Mikhailets and A. A. Murach, Refined scales of spaces, and elliptic boundary-value problems. Ⅲ, Ukrainian Math. J., 59 (2007), 744-765.  doi: 10.1007/s11253-007-0048-6.

[38]

V. A. Mikhailets and A. A. Murach, Extended Sobolev scale and elliptic operators, Ukrainian Math. J., 65 (2013), 435-447.  doi: 10.1007/s11253-013-0787-5.

[39]

V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation and Elliptic Problems, De Gruyter, Berlin, 2014. doi: 10.1515/9783110296891.

[40]

V. A. Mikhailets and A. A. Murach, Interpolation Hilbert spaces between Sobolev spaces, Results Math., 67 (2015), 135-152.  doi: 10.1007/s00025-014-0399-x.

[41]

A. A. Murach, Extension of some Lions–Magenes theorems, Methods Funct. Anal. Topology, 15 (2009), 152-167. 

[42]

S. M. Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems [2nd edition], Springer-Verlag, New York–Heidelberg, 1975.

[43]

V. I. Ovchinnikov, The methods of orbits in interpolation theory, in Mathematical Reports. Vol. 1, Part. 2 (ed. J. Peetre), Harwood Academic Publishers, London, 1984.

[44]

J. Peetre, Another approach to elliptic boundary problems, Commun. Pure Appl. Math., 14 (1961), 711-731.  doi: 10.1002/cpa.3160140404.

[45]

J. Peetre, On interpolation functions. Ⅱ, Acta Sci. Math., 29 (1968), 91-92. 

[46]

A. Pliś, A smooth linear elliptic differential equation without any solution in a sphere, Commun. Pure Appl. Math., 14 (1961), 599-617.  doi: 10.1002/cpa.3160140331.

[47]

Ya. A. Roitberg, Elliptic problems with nonhomogeneous boundary conditions and local increase of smoothness up to the boundary for generalized solutions, Dokl. Math., 5 (1964), 1034-1038. 

[48]

Ya. Roitberg, Elliptic Boundary Value Problems in the Spaces of Distributions, Kluwer Academic Publishers, Dordrecht, 1996. doi: 10.1007/978-94-011-5410-9.

[49]

Ya. Roitberg, Boundary Value Problems in the Spaces of Distributions, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-015-9275-8.

[50]

R. Schnaubelt and M. Veraar, Stochastic equations with boundary noise, Progr. Nonlinear Differ. Equ. Appl., 80 (2011), 609-629.  doi: 10.1007/978-3-0348-0075-4_30.

[51]

R. T. Seeley, Singular integrals and boundary-value problems, Amer. J. Math., 88 (1966), 781-809.  doi: 10.2307/2373078.

[52]

E. Seneta, Regularly Varying Functions, Springer, Berlin, 1976.

[53]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators [2nd edition], Johann Ambrosius Barth, Heidelberg, 1995.

[54]

M. Veraar, Regularity of Gaussian white noise on the $d$-dimensional torus, Banach Center Publ., 95 (2011), 385-398.  doi: 10.4064/bc95-0-24.

[55]

L. R. Volevich and B. P. Paneah, Certain spaces of generalized functions and embedding theorems, Uspehi Mat. Nauk, 20 (1965), 3-74. 

[56]

T. N. Zinchenko and A. A. Murach, Douglis–Nirenberg elliptic systems in Hörmander spaces, Ukrainian Math. J., 64 (2012), 1672-1687.  doi: 10.1007/s11253-013-0743-4.

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