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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[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.  Google Scholar

[8]

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

[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. Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[38]

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

[39]

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

[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.   Google Scholar

[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.  Google Scholar

[42]

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

[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.   Google Scholar

[44]

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

[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].  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

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

[5]

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

[6]

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

[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.  Google Scholar

[8]

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

[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. Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[13]

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

[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.  Google Scholar

[15]

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

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[38]

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

[39]

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

[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.   Google Scholar

[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.  Google Scholar

[42]

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

[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.   Google Scholar

[44]

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

[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].  Google Scholar

[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.  Google Scholar

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