June  2016, 9(3): 847-868. doi: 10.3934/dcdss.2016032

Nonlocal elliptic problems in infinite cylinder and applications

1. 

Peoples' Friendship University of Russia, ul. Miklukho-Maklaya 6, Moscow, 117198, Russian Federation

Received  March 2015 Revised  August 2015 Published  April 2016

We consider a unique solvability of nonlocal elliptic problems in infinite cylinder in weighted spaces and in Hölder spaces. Using these results we prove the existence and uniqueness of classical solution for the Vlasov--Poisson equations with nonlocal conditions in infinite cylinder for sufficiently small initial data.
Citation: Alexander L. Skubachevskii. Nonlocal elliptic problems in infinite cylinder and applications. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 847-868. doi: 10.3934/dcdss.2016032
References:
[1]

M. S. Agranovich and M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type, Uspekhi Mat. Nauk, 19 (1964), 53-161; English transl.: Russian Math. Surveys, 19 (1964).

[2]

A. A. Arsen'ev, Existence in the large of a weak solution of Vlasov's system of equations, Zhurnal Vychisl. Matem. i Mat. Phys., 15 (1975), 136-147; English transl.: U.S.S.R. Comput. Math. Math. Phys., 15 (1975), 131-143.

[3]

J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics, J. Differential Equations, 25 (1977), 342-364. doi: 10.1016/0022-0396(77)90049-3.

[4]

O. V. Besov, Some properties of the space $H_p^{(r_1, \ldots, r_m)}$, Izv. Vysch. Uchebn. Zaven. Mat., 1 (1960), 16-23.

[5]

A. V. Bitsadze and A. A. Samarskii, Some elementary generalizations of linear elliptic boundary value problems, Dokl. Akad. Nauk SSSR, 185 (1969), 739-740; English transl.: Soviet Math. Dokl., 10 (1969).

[6]

P. M. Blekher, Operators that depend meromorphically on a parameter, Vestnik Moscov. Univ., Ser. I Math. Mekh., 24 (1969), 30-36; English transl.: Moscow Univ. Math. Bull., 24 (1969).

[7]

R. J. Di Perna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757. doi: 10.1002/cpa.3160420603.

[8]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Second Edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 224 Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[9]

Y. Guo, Regularity for the Vlasov equations in a half-space, Indiana Univ. Math. J., 43 (1994), 255-320. doi: 10.1512/iumj.1994.43.43013.

[10]

P. L. Gurevich, Elliptic problems with nonlocal boundary conditions and Feller semigroups, Sovrem. Mat. Fundam. Napravl., 38 (2010), 3-173; English transl.: J. Math. Sci. (N. Y.), 182 (2012), 255-440. doi: 10.1007/s10958-012-0746-y.

[11]

A. K. Gushchin and V. P. Mikhailov, On the solvability of nonlocal problems for a second-order elliptic equation, Math. Sb., 185 (1994), 121-160; English transl.: Russian Acad. Sci. Sb. Math., 81 (1995), 101-136. doi: 10.1070/SM1995v081n01ABEH003617.

[12]

H. J. Hwang and J. J. L. Velázquez, On global existence for the Vlasov-Poisson system in a half-space, J. Differential equations, 247 (2009), 1915-1948. doi: 10.1016/j.jde.2009.06.004.

[13]

K. Yu. Kishkis, The index of the Bitsadze-Samarskii problem for harmonic functions, Differentsial'nye Uravneniya, 24 (1988), 105-110; English transl.: Differential Equations, 24 (1988).

[14]

V. A. Kondratiev, Boundary value problems for elliptic equations in domains with conical and angular points, Trudy Moskov. Mat. Obshch., 16 (1967), 209-292; English transl.: Trans. Moscow Math. Soc., 16 (1967).

[15]

V. A. Kondratiev and O. A. Oleinik, On asymptotics of solutions of nonlinear second order elliptic equations in cylindrical domains, in Partial differential equations and functional analysis, Progr. Nonlinear Differential Equations Appl., 22, Birkhäuser Boston, Boston, MA, 1996, 160-173.

[16]

V. V. Kozlov, The generalized Vlasov kinetic equation, Uspekhi Mat. Nauk 63 (2008), 93-130; English transl.: Russian Math. Surveys, 63 (2008), 691-726. doi: 10.1070/RM2008v063n04ABEH004549.

[17]

A. M. Krall, The development of general differential and general differential-boundary systems, Rocky Mountain J. Math., 5 (1975), 493-542. doi: 10.1216/RMJ-1975-5-4-493.

[18]

K. Miyamoto, Fundamentals of Plasma Physics and Controlled Fusion, Iwanami Book Service Centre, Tokyo, 1997.

[19]

S. A. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries, de Guyter Exp. Math., 13, Walter de Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110848915.525.

[20]

I. Pankratova and A. Piatnitski, On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 935-970. doi: 10.3934/dcdsb.2009.11.935.

[21]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303. doi: 10.1016/0022-0396(92)90033-J.

[22]

A. A. Samarskii, Some problems of the theory of differential equations, Differentsial'nye Uravneniya, 16 (1980), 1925-1935; English transl.: Differ. Equ., 16 (1980).

[23]

A. A. Skovoroda, Magnetic Systems for Plasma Confinement, Fizmatlit, Moscow, 2009.

[24]

A. L. Skubachevskii, Nonclassical boundary-value problems. I, Sovrem. Mat. Fundam. Napravl., 26 (2007), 3-132; English transl.: J. Math. Sci. (N. Y.), 155 (2008), 199-334. doi: 10.1007/s10958-008-9218-9.

[25]

A. L. Skubachevskii, Nonclassical boundary-value problems. II, Sovrem. Mat. Fundam. Napravl., 33 (2009), 3-179; English transl.: J. Math. Sci. (N. Y.), 166 (2010), 377-561. doi: 10.1007/s10958-010-9873-5.

[26]

A. L. Skubachevskii, Vlasov-Poisson equations for a two-component plasma in a homogeneous magnetic field, Uspekhi Mat. Nauk, 69 (2014), 107-148; English transl.: Russian Math. Surveys, 69 (2014).

[27]

A. P. Soldatov, The The Bitsadze-Samarskii problem for functions analytic in the sense of Douglis, Differ. Uravn., 41 (2005), 396-407; English transl.: Differ. Equ., 41 (2005). doi: 10.1007/s10625-005-0173-7.

[28]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Math. Library, 18, North-Holland Publishing Co., Amsterdam-New York, 1978.

show all references

References:
[1]

M. S. Agranovich and M. I. Vishik, Elliptic problems with a parameter and parabolic problems of general type, Uspekhi Mat. Nauk, 19 (1964), 53-161; English transl.: Russian Math. Surveys, 19 (1964).

[2]

A. A. Arsen'ev, Existence in the large of a weak solution of Vlasov's system of equations, Zhurnal Vychisl. Matem. i Mat. Phys., 15 (1975), 136-147; English transl.: U.S.S.R. Comput. Math. Math. Phys., 15 (1975), 131-143.

[3]

J. Batt, Global symmetric solutions of the initial value problem of stellar dynamics, J. Differential Equations, 25 (1977), 342-364. doi: 10.1016/0022-0396(77)90049-3.

[4]

O. V. Besov, Some properties of the space $H_p^{(r_1, \ldots, r_m)}$, Izv. Vysch. Uchebn. Zaven. Mat., 1 (1960), 16-23.

[5]

A. V. Bitsadze and A. A. Samarskii, Some elementary generalizations of linear elliptic boundary value problems, Dokl. Akad. Nauk SSSR, 185 (1969), 739-740; English transl.: Soviet Math. Dokl., 10 (1969).

[6]

P. M. Blekher, Operators that depend meromorphically on a parameter, Vestnik Moscov. Univ., Ser. I Math. Mekh., 24 (1969), 30-36; English transl.: Moscow Univ. Math. Bull., 24 (1969).

[7]

R. J. Di Perna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757. doi: 10.1002/cpa.3160420603.

[8]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Second Edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 224 Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[9]

Y. Guo, Regularity for the Vlasov equations in a half-space, Indiana Univ. Math. J., 43 (1994), 255-320. doi: 10.1512/iumj.1994.43.43013.

[10]

P. L. Gurevich, Elliptic problems with nonlocal boundary conditions and Feller semigroups, Sovrem. Mat. Fundam. Napravl., 38 (2010), 3-173; English transl.: J. Math. Sci. (N. Y.), 182 (2012), 255-440. doi: 10.1007/s10958-012-0746-y.

[11]

A. K. Gushchin and V. P. Mikhailov, On the solvability of nonlocal problems for a second-order elliptic equation, Math. Sb., 185 (1994), 121-160; English transl.: Russian Acad. Sci. Sb. Math., 81 (1995), 101-136. doi: 10.1070/SM1995v081n01ABEH003617.

[12]

H. J. Hwang and J. J. L. Velázquez, On global existence for the Vlasov-Poisson system in a half-space, J. Differential equations, 247 (2009), 1915-1948. doi: 10.1016/j.jde.2009.06.004.

[13]

K. Yu. Kishkis, The index of the Bitsadze-Samarskii problem for harmonic functions, Differentsial'nye Uravneniya, 24 (1988), 105-110; English transl.: Differential Equations, 24 (1988).

[14]

V. A. Kondratiev, Boundary value problems for elliptic equations in domains with conical and angular points, Trudy Moskov. Mat. Obshch., 16 (1967), 209-292; English transl.: Trans. Moscow Math. Soc., 16 (1967).

[15]

V. A. Kondratiev and O. A. Oleinik, On asymptotics of solutions of nonlinear second order elliptic equations in cylindrical domains, in Partial differential equations and functional analysis, Progr. Nonlinear Differential Equations Appl., 22, Birkhäuser Boston, Boston, MA, 1996, 160-173.

[16]

V. V. Kozlov, The generalized Vlasov kinetic equation, Uspekhi Mat. Nauk 63 (2008), 93-130; English transl.: Russian Math. Surveys, 63 (2008), 691-726. doi: 10.1070/RM2008v063n04ABEH004549.

[17]

A. M. Krall, The development of general differential and general differential-boundary systems, Rocky Mountain J. Math., 5 (1975), 493-542. doi: 10.1216/RMJ-1975-5-4-493.

[18]

K. Miyamoto, Fundamentals of Plasma Physics and Controlled Fusion, Iwanami Book Service Centre, Tokyo, 1997.

[19]

S. A. Nazarov and B. A. Plamenevsky, Elliptic Problems in Domains with Piecewise Smooth Boundaries, de Guyter Exp. Math., 13, Walter de Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110848915.525.

[20]

I. Pankratova and A. Piatnitski, On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 935-970. doi: 10.3934/dcdsb.2009.11.935.

[21]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303. doi: 10.1016/0022-0396(92)90033-J.

[22]

A. A. Samarskii, Some problems of the theory of differential equations, Differentsial'nye Uravneniya, 16 (1980), 1925-1935; English transl.: Differ. Equ., 16 (1980).

[23]

A. A. Skovoroda, Magnetic Systems for Plasma Confinement, Fizmatlit, Moscow, 2009.

[24]

A. L. Skubachevskii, Nonclassical boundary-value problems. I, Sovrem. Mat. Fundam. Napravl., 26 (2007), 3-132; English transl.: J. Math. Sci. (N. Y.), 155 (2008), 199-334. doi: 10.1007/s10958-008-9218-9.

[25]

A. L. Skubachevskii, Nonclassical boundary-value problems. II, Sovrem. Mat. Fundam. Napravl., 33 (2009), 3-179; English transl.: J. Math. Sci. (N. Y.), 166 (2010), 377-561. doi: 10.1007/s10958-010-9873-5.

[26]

A. L. Skubachevskii, Vlasov-Poisson equations for a two-component plasma in a homogeneous magnetic field, Uspekhi Mat. Nauk, 69 (2014), 107-148; English transl.: Russian Math. Surveys, 69 (2014).

[27]

A. P. Soldatov, The The Bitsadze-Samarskii problem for functions analytic in the sense of Douglis, Differ. Uravn., 41 (2005), 396-407; English transl.: Differ. Equ., 41 (2005). doi: 10.1007/s10625-005-0173-7.

[28]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Math. Library, 18, North-Holland Publishing Co., Amsterdam-New York, 1978.

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