January 2017, 16(1): 69-98. doi: 10.3934/cpaa.2017003

An isomorphism theorem for parabolic problems in Hörmander spaces and its applications

1. 

National Technical University of Ukraine "Kyiv Polytechnic Institute", Prospect Peremohy 37,03056, Kyiv-56, Ukraine

2. 

Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivska str. 3,01004 Kyiv, Ukraine

3. 

Chernihiv National Pedagogical University, Het'mana Polubotka str. 53,14013 Chernihiv, Ukraine

E-mail address: murach@imath.kiev.ua

Received  November 2015 Revised  July 2016 Published  November 2016

We investigate a general parabolic initial-boundary value problem with zero Cauchy data in some anisotropic Hörmander inner product spaces. We prove that the operators corresponding to this problem are isomorphisms between appropriate Hörmander spaces. As an application of this result, we establish a theorem on the local increase in regularity of solutions to the problem. We also obtain new sufficient conditions under which the generalized derivatives, of a given order, of the solutions should be continuous.

Citation: Valerii Los, Vladimir A. Mikhailets, Aleksandr A. Murach. An isomorphism theorem for parabolic problems in Hörmander spaces and its applications. Communications on Pure & Applied Analysis, 2017, 16 (1) : 69-98. doi: 10.3934/cpaa.2017003
References:
[1]

M. S. Agranovich and M. I. Vishik, Elliptic problems with parameter and parabolic problems of general form, (Russian), Uspehi Mat. Nauk, 19 (1964), 53-161.

[2]

A. V. Anop and A. A. Murach, Parameter-elliptic problems and interpolation with a function parameter, Methods Funct. Anal. Topology, 20 (2014), 103-116.

[3]

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.

[4]

Yu. M. Berezansky, Expansions in Eigenfunctions of Selfadjoint Operators, Transl. Math. Monogr. , vol. 17, American Mathematical Society, Providence, R. I. , 1968.

[5]

J. Bergh and J. Löfström, Interpolation Spaces, Grundlehren Math. Wiss. , band 223, SpringerVerlag, Berlin-New York, 1976.

[6]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia Math. Appl. , vol. 27, Cambridge University Press, Cambridge, 1989.

[7]

F. Cobos and D. L. Fernandez, Hardy-Sobolev spaces and Besov spaces with a function parameter, in Function Spaces and Applications (eds. M. Cwikel, J. Peetre, Y. Sagher and H. Wallin), Lecture Notes in Math. , vol. 1302, Springer, Berlin, (1988), 158-170. doi: 10.1007/BFb0078872.

[8]

S. D. Eidel'man, Parabolic Systems, North-Holland Publishing Co. , Amsterdam-London; Wolters-Noordhoff Publishing, Groningen, 1969.

[9]

S. D. Eidel'man, Parabolic equations, in Encyclopaedia Math. Sci. , vol. 63 (Partial Differential Equations, Ⅵ. Elliptic and Parabolic Operators) (eds. Yu. V. Egorov and M. A. Shubin), Springer, Berlin, (1994), 205-316.

[10]

S. D. Eidel'man and N. V. Zhitarashu, Parabolic Boundary Value Problems, Oper. Theory Adv. Appl. , vol. 101, Birkhäuser Verlag, Basel, 1998. doi: 10.1007/978-3-0348-8767-0.

[11]

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

[12]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall Inc. , Englewood Cliffs, N. J. , 1964.

[13]

L. Hörmander, Linear Partial Differential Operators, Grundlehren Math. Wiss. , band 116, Academic Press, Inc. , Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963.

[14]

L. Hörmander, The Analysis of Linear Partial Differential Operators, vol. Ⅱ, Differential Operators with Constant Coefficients, Grundlehren Math. Wiss. , band 257, Springer-Verlag, Berlin, 1983.

[15]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. Ⅲ, PseudoDifferential Operators, Grundlehren Math. Wiss. , band 274, Springer-Verlag, Berlin, 1985.

[16]

J. Karamata, Sur certains Tauberian theorems de M. M. Hardy et Littlewood, Mathematica (Cluj), 3 (1930), 33-48.

[17]

S. G. Krein, Yu. L. Petunin and E. M. Semënov, Interpolation of Linear Operators, Transl. Math. Monogr. , vol. 54, American Mathematical Society, Providence, R. I. , 1982.

[18]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'tzeva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. , vol. 23, American Mathematical Society, Providence, R. I. , 1968.

[19]

J. -L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications, vol. Ⅰ, Grundlehren Math. Wiss. , band 181, Springer-Verlag, New York-Heidelberg, 1972.

[20]

J. -L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications, vol. Ⅱ, Grundlehren Math. Wiss. , band 182, Springer-Verlag, New York-Heidelberg, 1972.

[21]

V. Los and A. A. Murach, Parabolic problems and interpolation with a function parameter, Methods Funct. Anal. Topology, 19 (2013), 146-160.

[22]

V. Los and A. A. Murach, Parabolic mixed problems in spaces of generalized smoothness, (Russian), Dopov. Nats. Acad. Nauk. Ukr. Mat. Prirodozn. Tehn. Nauki, 6 (2014), 23-31.

[23]

V. Los, Mixed problems for the two-dimensional heat-conduction equation in anisotropic Hörmander spaces, Ukrainian Math. J., 67 (2015), 735-747. doi: 10.1007/s11253-015-1111-3.

[24]

C. Merucci, Application of interpolation with a function parameter to Lorentz, Sobolev and Besov spaces, in Interpolation Spaces and Allied Topics in Analysis (eds. M. Cwikel and J. Peetre), Lecture Notes in Math. , vol. 1070, Springer, Berlin, (1984), 183-201. doi: 10.1007/BFb0099101.

[25]

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.

[26]

V. A. Mikhailets and A. A. Murach, Refined scales of spaces, and elliptic boundary-value problems. Ⅰ, Ukrainian Math. J., 58 (2006), 244-262. doi: 10.1007/s11253-006-0064-y.

[27]

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.

[28]

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.

[29]

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

[30]

V. A. Mikhailets and A. A. Murach, Interpolation with a function parameter and refined scale of spaces, Methods Funct. Anal. Topology, 14 (2008), 81-100.

[31]

V. A. Mikhailets and A. A. Murach, An elliptic boundary-value problem in a two-sided refined scale of spaces, Ukrainian. Math. J., 60 (2008), 574-597. doi: 10.1007/s11253-008-0074-z.

[32]

V. A. Mikhailets and A. A. Murach, The refined Sobolev scale, interpolation, and elliptic problems, Banach J. Math. Anal., 6 (2012), 211-281. doi: 10.15352/bjma/1342210171.

[33]

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.

[34]

V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation, and Elliptic Problems, De Gruyter Studies in Math. , vol. 60, De Gruyter, Berlin, 2014. doi: 10.1515/9783110296891.

[35]

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.

[36]

A. A. Murach, Elliptic pseudo-differential operators in a refined scale of spaces on a closed manifold, Ukrainian Math. J., 59 (2007), 874-893. doi: 10.1007/s11253-007-0056-6.

[37]

A. A. Murach and T. Zinchenko, Parameter-elliptic operators on the extended Sobolev scale, Methods Funct. Anal. Topology, 19 (2013), 29-39.

[38]

J. Peetre, On interpolation functions, Acta Sci. Math. (Szeged), 27 (1966), 167-171.

[39]

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

[40]

I. G. Petrovskii, On the Cauchy problem for systems of partial differential equations in the domain of non-anallytic functions, (Russian) Bull. Mosk. Univ., Mat. Mekh., 1 (1938), 1-72.

[41]

V. S. Rychkov, On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domain, J. London Math. Soc., 60 (1999), 237-257. doi: 10.1112/S0024610799007723.

[42]

E. Seneta, Regularly Varying Functions, Lecture Notes in Math. , vol. 508, Springer, Berlin, 1976.

[43]

L. N. Slobodeckii, Generalized Sobolev spaces and their application to boundary problems for partial differential equations, (Russian), Leningrad. Gos. Ped. Inst. Uchen. Zap., 197 (1958), 54-112.

[44]

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

[45]

L. R. Volevich and B. P. Paneah, Certain spaces of generalized functions and embedding theorems, (Russian), Uspekhi Mat. Nauk, 20 (1965), 3-74[English translation in Russian Math. Surveys, 20 (1965), 1-73].

[46]

T. N. Zinchenko and A. A. Murach, Douglis-Nirenberg elliptic systems in Hörmander spaces, Ukrainian Math. J., 64 (2013), 1672-1687. doi: 10.1007/s11253-013-0743-4.

show all references

References:
[1]

M. S. Agranovich and M. I. Vishik, Elliptic problems with parameter and parabolic problems of general form, (Russian), Uspehi Mat. Nauk, 19 (1964), 53-161.

[2]

A. V. Anop and A. A. Murach, Parameter-elliptic problems and interpolation with a function parameter, Methods Funct. Anal. Topology, 20 (2014), 103-116.

[3]

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.

[4]

Yu. M. Berezansky, Expansions in Eigenfunctions of Selfadjoint Operators, Transl. Math. Monogr. , vol. 17, American Mathematical Society, Providence, R. I. , 1968.

[5]

J. Bergh and J. Löfström, Interpolation Spaces, Grundlehren Math. Wiss. , band 223, SpringerVerlag, Berlin-New York, 1976.

[6]

N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Encyclopedia Math. Appl. , vol. 27, Cambridge University Press, Cambridge, 1989.

[7]

F. Cobos and D. L. Fernandez, Hardy-Sobolev spaces and Besov spaces with a function parameter, in Function Spaces and Applications (eds. M. Cwikel, J. Peetre, Y. Sagher and H. Wallin), Lecture Notes in Math. , vol. 1302, Springer, Berlin, (1988), 158-170. doi: 10.1007/BFb0078872.

[8]

S. D. Eidel'man, Parabolic Systems, North-Holland Publishing Co. , Amsterdam-London; Wolters-Noordhoff Publishing, Groningen, 1969.

[9]

S. D. Eidel'man, Parabolic equations, in Encyclopaedia Math. Sci. , vol. 63 (Partial Differential Equations, Ⅵ. Elliptic and Parabolic Operators) (eds. Yu. V. Egorov and M. A. Shubin), Springer, Berlin, (1994), 205-316.

[10]

S. D. Eidel'man and N. V. Zhitarashu, Parabolic Boundary Value Problems, Oper. Theory Adv. Appl. , vol. 101, Birkhäuser Verlag, Basel, 1998. doi: 10.1007/978-3-0348-8767-0.

[11]

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

[12]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall Inc. , Englewood Cliffs, N. J. , 1964.

[13]

L. Hörmander, Linear Partial Differential Operators, Grundlehren Math. Wiss. , band 116, Academic Press, Inc. , Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963.

[14]

L. Hörmander, The Analysis of Linear Partial Differential Operators, vol. Ⅱ, Differential Operators with Constant Coefficients, Grundlehren Math. Wiss. , band 257, Springer-Verlag, Berlin, 1983.

[15]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. Ⅲ, PseudoDifferential Operators, Grundlehren Math. Wiss. , band 274, Springer-Verlag, Berlin, 1985.

[16]

J. Karamata, Sur certains Tauberian theorems de M. M. Hardy et Littlewood, Mathematica (Cluj), 3 (1930), 33-48.

[17]

S. G. Krein, Yu. L. Petunin and E. M. Semënov, Interpolation of Linear Operators, Transl. Math. Monogr. , vol. 54, American Mathematical Society, Providence, R. I. , 1982.

[18]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'tzeva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. , vol. 23, American Mathematical Society, Providence, R. I. , 1968.

[19]

J. -L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications, vol. Ⅰ, Grundlehren Math. Wiss. , band 181, Springer-Verlag, New York-Heidelberg, 1972.

[20]

J. -L. Lions and E. Magenes, Non-Homogeneous Boundary-Value Problems and Applications, vol. Ⅱ, Grundlehren Math. Wiss. , band 182, Springer-Verlag, New York-Heidelberg, 1972.

[21]

V. Los and A. A. Murach, Parabolic problems and interpolation with a function parameter, Methods Funct. Anal. Topology, 19 (2013), 146-160.

[22]

V. Los and A. A. Murach, Parabolic mixed problems in spaces of generalized smoothness, (Russian), Dopov. Nats. Acad. Nauk. Ukr. Mat. Prirodozn. Tehn. Nauki, 6 (2014), 23-31.

[23]

V. Los, Mixed problems for the two-dimensional heat-conduction equation in anisotropic Hörmander spaces, Ukrainian Math. J., 67 (2015), 735-747. doi: 10.1007/s11253-015-1111-3.

[24]

C. Merucci, Application of interpolation with a function parameter to Lorentz, Sobolev and Besov spaces, in Interpolation Spaces and Allied Topics in Analysis (eds. M. Cwikel and J. Peetre), Lecture Notes in Math. , vol. 1070, Springer, Berlin, (1984), 183-201. doi: 10.1007/BFb0099101.

[25]

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.

[26]

V. A. Mikhailets and A. A. Murach, Refined scales of spaces, and elliptic boundary-value problems. Ⅰ, Ukrainian Math. J., 58 (2006), 244-262. doi: 10.1007/s11253-006-0064-y.

[27]

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.

[28]

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.

[29]

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

[30]

V. A. Mikhailets and A. A. Murach, Interpolation with a function parameter and refined scale of spaces, Methods Funct. Anal. Topology, 14 (2008), 81-100.

[31]

V. A. Mikhailets and A. A. Murach, An elliptic boundary-value problem in a two-sided refined scale of spaces, Ukrainian. Math. J., 60 (2008), 574-597. doi: 10.1007/s11253-008-0074-z.

[32]

V. A. Mikhailets and A. A. Murach, The refined Sobolev scale, interpolation, and elliptic problems, Banach J. Math. Anal., 6 (2012), 211-281. doi: 10.15352/bjma/1342210171.

[33]

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.

[34]

V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation, and Elliptic Problems, De Gruyter Studies in Math. , vol. 60, De Gruyter, Berlin, 2014. doi: 10.1515/9783110296891.

[35]

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.

[36]

A. A. Murach, Elliptic pseudo-differential operators in a refined scale of spaces on a closed manifold, Ukrainian Math. J., 59 (2007), 874-893. doi: 10.1007/s11253-007-0056-6.

[37]

A. A. Murach and T. Zinchenko, Parameter-elliptic operators on the extended Sobolev scale, Methods Funct. Anal. Topology, 19 (2013), 29-39.

[38]

J. Peetre, On interpolation functions, Acta Sci. Math. (Szeged), 27 (1966), 167-171.

[39]

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

[40]

I. G. Petrovskii, On the Cauchy problem for systems of partial differential equations in the domain of non-anallytic functions, (Russian) Bull. Mosk. Univ., Mat. Mekh., 1 (1938), 1-72.

[41]

V. S. Rychkov, On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domain, J. London Math. Soc., 60 (1999), 237-257. doi: 10.1112/S0024610799007723.

[42]

E. Seneta, Regularly Varying Functions, Lecture Notes in Math. , vol. 508, Springer, Berlin, 1976.

[43]

L. N. Slobodeckii, Generalized Sobolev spaces and their application to boundary problems for partial differential equations, (Russian), Leningrad. Gos. Ped. Inst. Uchen. Zap., 197 (1958), 54-112.

[44]

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

[45]

L. R. Volevich and B. P. Paneah, Certain spaces of generalized functions and embedding theorems, (Russian), Uspekhi Mat. Nauk, 20 (1965), 3-74[English translation in Russian Math. Surveys, 20 (1965), 1-73].

[46]

T. N. Zinchenko and A. A. Murach, Douglis-Nirenberg elliptic systems in Hörmander spaces, Ukrainian Math. J., 64 (2013), 1672-1687. doi: 10.1007/s11253-013-0743-4.

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