doi: 10.3934/cpaa.2021123
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Parabolic problems in generalized Sobolev spaces

1. 

National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", 37, Prosp. Peremohy, Kyiv, Ukraine, 03056

2. 

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

* Corresponding author

Received  January 2021 Revised  June 2021 Early access July 2021

Fund Project: This work was supported by the Grant H2020-MSCA-RISE-2019, project number 873071 (SOMPATY: Spectral Optimization: From Mathematics to Physics and Advanced Technology)

We consider a general inhomogeneous parabolic initial-boundary value problem for a $ 2b $-parabolic differential equation given in a finite multidimensional cylinder. We investigate the solvability of this problem in some generalized anisotropic Sobolev spaces. They are parametrized with a pair of positive numbers $ s $ and $ s/(2b) $ and with a function $ \varphi:[1,\infty)\to(0,\infty) $ that varies slowly at infinity. The function parameter $ \varphi $ characterizes subordinate regularity of distributions with respect to the power regularity given by the number parameters. We prove that the operator corresponding to this problem is an isomorphism on appropriate pairs of these spaces. As an application, we give a theorem on the local regularity of the generalized solution to the problem. We also obtain sharp sufficient conditions under which chosen generalized derivatives of this solution are continuous on a given set.

Citation: Valerii Los, Vladimir Mikhailets, Aleksandr Murach. Parabolic problems in generalized Sobolev spaces. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021123
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 [English translation in Russian Math. Surveys, 19 (1964), 53–157].  Google Scholar

[2]

Y. Ameur, Interpolation between Hilbert spaces, in Analysis of Operators on Function Spaces, Trends Math. (eds. A. Aleman etc.), Birkhäuser/Springer, Cham, (2019), 63–115. doi: 10.1007/978-3-030-14640-5_4.  Google Scholar

[3]

A. AnopR. Denk and A. Murach, Elliptic problems with rough boundary data in generalized Sobolev spaces, Commun. Pure Appl. Anal., 20 (2021), 697-735.  doi: 10.3934/cpaa.2020286.  Google Scholar

[4]

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

[5]

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

[6]

O. V. Besov, V. P. Il'in and S. M. Nikol'skij, Integral Representations of Functions and Embedding Theorems, (Russian) 2$^{nd}$ edition, Nauka, Moscow, 1996.  Google Scholar

[7] N. H. BinghamC. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1989.   Google Scholar
[8]

R. DenkM. Hieber and J. Prüss, Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224.  doi: 10.1007/s00209-007-0120-9.  Google Scholar

[9]

W. F. Donoghue, The interpolation of quadratic norms, Acta Math., 118 (1967), 251-270.  doi: 10.1007/BF02392483.  Google Scholar

[10]

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

[11]

S. D. Eidel'man and N. V. Zhitarashu, Parabolic Boundary Value Problems, Birkhäuser Verlag, Basel, 1998. doi: 10.1007/978-3-0348-8767-0.  Google Scholar

[12]

M. Fan, Qudratic interpolation and some operator inequalities, J. Math. Inequal., 5 (2011), 413-427.  doi: 10.7153/jmi-05-36.  Google Scholar

[13]

W. Farkas and H. G. Leopold, Characterisations of function spaces of generalized smoothness, Ann. Mat. Pura Appl., 185 (2006), 1-62.  doi: 10.1007/s10231-004-0110-z.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. II, Differential Operators with Constant Coefficients, Springer, Berlin, 1983.  Google Scholar

[18]

A. M. Il'in, A. S. Kalashnikov and O. A. Oleinik, Linear equations of the second order of parabolic type, (Russian) Uspekhi Mat. Nauk, 17 (1962), 3–146; English translation in Russian Math. Surveys, 17: 3 (1962), 1–143. Google Scholar

[19]

V. A. Il'in, The solvability of mixed problems for hyperbolic and parabolic equations, (Russian) Uspekhi Mat. Nauk, 15 (1960), 97–154; English translation in Russian Math. Surveys, 15: 1 (1960), 85–142. Google Scholar

[20]

S. G. Krein, Yu. L. Petunin and E. M. Semënov, Interpolation of Linear Operators, American Mathematical Society, Providence, RI, 1982.  Google Scholar

[21]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'tzeva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.  Google Scholar

[22]

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

[23]

V. M. Los, Anisotropic Hörmander spaces on the lateral surface of a cylinder, J. Math. Sci. (N. Y.), 217 (2016), 456-467.  doi: 10.1007/s10958-016-2985-9.  Google Scholar

[24]

V. M. Los, Theorems on isomorphisms for some parabolic initial-boundary-value problems in Hörmander spaces: limiting case, Ukrainian Math. J., 68 (2016), 894-909.  doi: 10.1007/s11253-016-1264-8.  Google Scholar

[25]

V. M. Los, Classical Solutions of Parabolic Initial-Boundary-Value Problems and Hörmander Spaces, Ukrainian Math. J., 68 (2017), 1412-1423.  doi: 10.1007/s11253-017-1303-0.  Google Scholar

[26]

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.  Google Scholar

[27]

V. Los and A. Murach, Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces, Open Math., 15 (2017), 57-76.  doi: 10.1515/math-2017-0008.  Google Scholar

[28]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser Verlag, Basel, 1995.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

F. Nicola and L. Rodino, Global Pseudo-Differential Calculus on Euclidean spaces, Birkhäser, Basel, 2010. Google Scholar

[33]

V. I. Ovchinnikov, The methods of orbits in interpolation theory, Math. Rep., 1 (1984), 349-515.   Google Scholar

[34]

B. Paneah, The Oblique Derivative Problem. The Poincaré problem, Wiley–VCH, Berlin, 2000.  Google Scholar

[35]

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

[36]

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; English translation in Amer. Math. Soc. Transl. (2), 57 (1966), 207–275.  Google Scholar

[37]

V. A. Solonnikov, Apriori estimates for solutions of second-order equations of parabolic type, (Russian) Tr. Mat. Inst. Steklova, 70 (1964), 133–212. Google Scholar

[38]

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

[39]

H. Triebel, The Structure of Functions, Birkhäser, Basel, 2001.  Google Scholar

[40]

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

[41]

N. V. Zhitarashu, Theorems on complete collection of isomorphisms in the $L_2$-theory of generalized solutions for one equation parabolic in Petrovski$\breve{l}$ sense, Mat. Sb., 128 (1985), 451-473.   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 [English translation in Russian Math. Surveys, 19 (1964), 53–157].  Google Scholar

[2]

Y. Ameur, Interpolation between Hilbert spaces, in Analysis of Operators on Function Spaces, Trends Math. (eds. A. Aleman etc.), Birkhäuser/Springer, Cham, (2019), 63–115. doi: 10.1007/978-3-030-14640-5_4.  Google Scholar

[3]

A. AnopR. Denk and A. Murach, Elliptic problems with rough boundary data in generalized Sobolev spaces, Commun. Pure Appl. Anal., 20 (2021), 697-735.  doi: 10.3934/cpaa.2020286.  Google Scholar

[4]

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

[5]

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

[6]

O. V. Besov, V. P. Il'in and S. M. Nikol'skij, Integral Representations of Functions and Embedding Theorems, (Russian) 2$^{nd}$ edition, Nauka, Moscow, 1996.  Google Scholar

[7] N. H. BinghamC. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge, 1989.   Google Scholar
[8]

R. DenkM. Hieber and J. Prüss, Optimal $L^p$-$L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224.  doi: 10.1007/s00209-007-0120-9.  Google Scholar

[9]

W. F. Donoghue, The interpolation of quadratic norms, Acta Math., 118 (1967), 251-270.  doi: 10.1007/BF02392483.  Google Scholar

[10]

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

[11]

S. D. Eidel'man and N. V. Zhitarashu, Parabolic Boundary Value Problems, Birkhäuser Verlag, Basel, 1998. doi: 10.1007/978-3-0348-8767-0.  Google Scholar

[12]

M. Fan, Qudratic interpolation and some operator inequalities, J. Math. Inequal., 5 (2011), 413-427.  doi: 10.7153/jmi-05-36.  Google Scholar

[13]

W. Farkas and H. G. Leopold, Characterisations of function spaces of generalized smoothness, Ann. Mat. Pura Appl., 185 (2006), 1-62.  doi: 10.1007/s10231-004-0110-z.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. II, Differential Operators with Constant Coefficients, Springer, Berlin, 1983.  Google Scholar

[18]

A. M. Il'in, A. S. Kalashnikov and O. A. Oleinik, Linear equations of the second order of parabolic type, (Russian) Uspekhi Mat. Nauk, 17 (1962), 3–146; English translation in Russian Math. Surveys, 17: 3 (1962), 1–143. Google Scholar

[19]

V. A. Il'in, The solvability of mixed problems for hyperbolic and parabolic equations, (Russian) Uspekhi Mat. Nauk, 15 (1960), 97–154; English translation in Russian Math. Surveys, 15: 1 (1960), 85–142. Google Scholar

[20]

S. G. Krein, Yu. L. Petunin and E. M. Semënov, Interpolation of Linear Operators, American Mathematical Society, Providence, RI, 1982.  Google Scholar

[21]

O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'tzeva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968.  Google Scholar

[22]

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

[23]

V. M. Los, Anisotropic Hörmander spaces on the lateral surface of a cylinder, J. Math. Sci. (N. Y.), 217 (2016), 456-467.  doi: 10.1007/s10958-016-2985-9.  Google Scholar

[24]

V. M. Los, Theorems on isomorphisms for some parabolic initial-boundary-value problems in Hörmander spaces: limiting case, Ukrainian Math. J., 68 (2016), 894-909.  doi: 10.1007/s11253-016-1264-8.  Google Scholar

[25]

V. M. Los, Classical Solutions of Parabolic Initial-Boundary-Value Problems and Hörmander Spaces, Ukrainian Math. J., 68 (2017), 1412-1423.  doi: 10.1007/s11253-017-1303-0.  Google Scholar

[26]

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.  Google Scholar

[27]

V. Los and A. Murach, Isomorphism theorems for some parabolic initial-boundary value problems in Hörmander spaces, Open Math., 15 (2017), 57-76.  doi: 10.1515/math-2017-0008.  Google Scholar

[28]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhauser Verlag, Basel, 1995.  Google Scholar

[29]

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

[30]

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

[31]

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

[32]

F. Nicola and L. Rodino, Global Pseudo-Differential Calculus on Euclidean spaces, Birkhäser, Basel, 2010. Google Scholar

[33]

V. I. Ovchinnikov, The methods of orbits in interpolation theory, Math. Rep., 1 (1984), 349-515.   Google Scholar

[34]

B. Paneah, The Oblique Derivative Problem. The Poincaré problem, Wiley–VCH, Berlin, 2000.  Google Scholar

[35]

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

[36]

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; English translation in Amer. Math. Soc. Transl. (2), 57 (1966), 207–275.  Google Scholar

[37]

V. A. Solonnikov, Apriori estimates for solutions of second-order equations of parabolic type, (Russian) Tr. Mat. Inst. Steklova, 70 (1964), 133–212. Google Scholar

[38]

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

[39]

H. Triebel, The Structure of Functions, Birkhäser, Basel, 2001.  Google Scholar

[40]

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

[41]

N. V. Zhitarashu, Theorems on complete collection of isomorphisms in the $L_2$-theory of generalized solutions for one equation parabolic in Petrovski$\breve{l}$ sense, Mat. Sb., 128 (1985), 451-473.   Google Scholar

[1]

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

[2]

Wenning Wei. On the Cauchy-Dirichlet problem in a half space for backward SPDEs in weighted Hölder spaces. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5353-5378. doi: 10.3934/dcds.2015.35.5353

[3]

Seung Jun Chang, Jae Gil Choi. Generalized transforms and generalized convolution products associated with Gaussian paths on function space. Communications on Pure & Applied Analysis, 2020, 19 (1) : 371-389. doi: 10.3934/cpaa.2020019

[4]

Alberto Fiorenza, Anna Mercaldo, Jean Michel Rakotoson. Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 893-906. doi: 10.3934/dcds.2002.8.893

[5]

Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space, II: Weighted Sobolev metrics and almost local metrics. Journal of Geometric Mechanics, 2012, 4 (4) : 365-383. doi: 10.3934/jgm.2012.4.365

[6]

M. Nakamura, Tohru Ozawa. The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 215-231. doi: 10.3934/dcds.1999.5.215

[7]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501

[8]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

[9]

Chérif Amrouche, Yves Raudin. Singular boundary conditions and regularity for the biharmonic problem in the half-space. Communications on Pure & Applied Analysis, 2007, 6 (4) : 957-982. doi: 10.3934/cpaa.2007.6.957

[10]

Giulia Cavagnari. Regularity results for a time-optimal control problem in the space of probability measures. Mathematical Control & Related Fields, 2017, 7 (2) : 213-233. doi: 10.3934/mcrf.2017007

[11]

Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space of surfaces. Journal of Geometric Mechanics, 2011, 3 (4) : 389-438. doi: 10.3934/jgm.2011.3.389

[12]

Sabri Bahrouni, Hichem Ounaies. Embedding theorems in the fractional Orlicz-Sobolev space and applications to non-local problems. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2917-2944. doi: 10.3934/dcds.2020155

[13]

María José Beltrán, José Bonet, Carmen Fernández. Classical operators on the Hörmander algebras. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 637-652. doi: 10.3934/dcds.2015.35.637

[14]

Luiz Gustavo Farah. Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1521-1539. doi: 10.3934/cpaa.2009.8.1521

[15]

G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279

[16]

Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51.

[17]

T. V. Anoop, Nirjan Biswas, Ujjal Das. Admissible function spaces for weighted Sobolev inequalities. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3259-3297. doi: 10.3934/cpaa.2021105

[18]

Rafael De La Llave, R. Obaya. Regularity of the composition operator in spaces of Hölder functions. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 157-184. doi: 10.3934/dcds.1999.5.157

[19]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597

[20]

Martins Bruveris. Completeness properties of Sobolev metrics on the space of curves. Journal of Geometric Mechanics, 2015, 7 (2) : 125-150. doi: 10.3934/jgm.2015.7.125

2020 Impact Factor: 1.916

Article outline

[Back to Top]