Geodesic
On initial boundary value problem for parabolic differential operator with non-coercive boundary conditions
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 5, pp. 547-558.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider initial boundary value problem for uniformly $2$-parabolic differential operator of second order in cylinder domain in ${\mathbb R}^n $ with non-coercive boundary conditions. In this case there is a loss of smoothness of the solution in Sobolev type spaces compared with the coercive situation. Using by Faedo–Galerkin method we prove that problem has unique solution in special Bochner space.
Keywords: non-coercive problem, parabolic problem, Faedo–Galerkin method. 
@article{JSFU_2020_13_5_a2,
     author = {Alexander N. Polkovnikov},
     title = {On initial boundary value problem for parabolic differential operator with non-coercive boundary conditions},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {547--558},
     publisher = {mathdoc},
     volume = {13},
     number = {5},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2020_13_5_a2/}
}
TY  - JOUR
AU  - Alexander N. Polkovnikov
TI  - On initial boundary value problem for parabolic differential operator with non-coercive boundary conditions
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2020
SP  - 547
EP  - 558
VL  - 13
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2020_13_5_a2/
LA  - en
ID  - JSFU_2020_13_5_a2
ER  - 
%0 Journal Article
%A Alexander N. Polkovnikov
%T On initial boundary value problem for parabolic differential operator with non-coercive boundary conditions
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2020
%P 547-558
%V 13
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2020_13_5_a2/
%G en
%F JSFU_2020_13_5_a2
Alexander N. Polkovnikov. On initial boundary value problem for parabolic differential operator with non-coercive boundary conditions. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 13 (2020) no. 5, pp. 547-558. http://geodesic.mathdoc.fr/item/JSFU_2020_13_5_a2/

[1] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, R.I., 1968 | MR

[2] J.L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin et al., 1972 | MR

[3] V.P. Mikhailov, Partial Differential Equations, Mir, M., 1978 | MR

[4] R. Temam, Navier-Stokes equations: Theory and Numerical Analysis, Studies in Math. and its Appl., 2, 1979 | MR | Zbl

[5] S. Agmon, A. Douglis, L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, Part 1”, Comm. Pure Appl. Math., 12 (1959), 623–727 | DOI | MR | Zbl

[6] S. Campanato, “Sui problemi al contorno per sistemi di equazioni differenziale lineari del tipo dell'elasticitá”, Ann. della Scuola Norm. Superiore, Cl. di Sci, Ser. III, 13:2 (1959), 223–258 | MR | Zbl

[7] S. Campanato, “Proprietá di taluni spazi di distribuzioni e loro applicazione”, Ann. della Scuola Norm. Superiore, Cl. di Sci., Ser. III, 14:4 (1960), 363–376 | MR

[8] A.S. Peicheva, A.A. Shlapunov, “On the completeness of root functions of Sturm-Liouville problems for the Lame system in weighted spaces”, Z. Angew. Math. Mech., 95:11 (2015), 1202–1214 | DOI | MR | Zbl

[9] A.N. Polkovnikov, A.A. Shlapunov, “On spectral properties of a non-coercive mixed problem associated with the $\overline \partial$-operator”, Journal of Siberian Federal University. Math. and Phys., 6:2 (2013), 247–261

[10] A. Polkovnikov, A. Shapunov, “On non-coercive mixed problems for parameter-dependent elliptic operators”, Math. Commun., 15 (2015), 1–20 | MR

[11] A. Polkovnikov, A.A. Shlapunov, Sib. Math. J., 58:4 (2017), 676–686 | DOI | MR | Zbl

[12] A. Shlapunov, N. Tarkhanov, “On completeness of root functions of Sturm-Liouville problems with discontinuous boundary operators”, J. of Differential Equations, 10 (2013), 3305–3337 | DOI | MR | Zbl

[13] J.J. Kohn, “Subellipticity of the $\overline \partial~$-Neumann problem on pseudoconvex domains: sufficient conditions”, Acta Math., 142:1–2 (1979), 79–12 | DOI | MR

[14] A. Shlapunov, N. Tarkhanov, Sib. Adv. Math., 26:1 (2016), 30–76 | DOI | MR

[15] M. Schechter, “Negative norms and boundary problems”, Ann. Math., 72:3 (1960), 581–593 | DOI | MR | Zbl

[16] H. Gaevsky, K. Greger, K. Zaharias, Nonlinear Operator Equations and Operator Differential Equations, Mir, M., 1978 | MR

[17] A.F. Filippov, Differential equations with discontinuous right-hand side, Nauka, M., 1985 (in Russian) | MR | Zbl

[18] T.H. Gronwall, “Note on the derivatives with respect to a parameter of the solutions of a system of differential equations”, Ann. of Math., 20:2 (1919), 292–296 | DOI | MR | Zbl

[19] D.S. Mitrinović, J.E. Pečarić, A.M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Mathematics and its Applications (East European Series), 53, Kluwer Ac. Publ., Dordrecht, 1991 | MR | Zbl

[20] A.A. Shlapunov, “Spectral decomposition of Green's integrals and existence of $W^{s,2}~$-solutions of matrix factorizations of the Laplace operator in a ball”, Rend. Sem. Mat. Univ. Padova, 96 (1996), 237–256 | MR | Zbl