Regularity of solutions of the model Venttsel' problem for quasilinear parabolic systems with nonsmooth in time principal matrices

A. A. Arkhipova

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 3, pp. 470-490

Résumé

The Venttsel' problem in the model statement for quasilinear parabolic systems of equations with nondiagonal principal matrices is considered. It is only assumed that the principal matrices and the boundary condition are bounded with respect to the time variable. The partial smoothness of the weak solutions (Hölder continuity on a set of full measure up to the surface on which the Venttsel' condition is defined) is proved. The proof uses the $A(t)$-caloric approximation method, which was also used in [1] to investigate the regularity of the solution to the corresponding linear problem.

Bibliographie

[1] Arkhipova A. A., “Regularity of weak solutions to the model Venttsel problem for linear parabolic systems with nonsmooth in time principal matrix: $A(t)$-caloric approximation method”, Manuscripta Math., 151:3 (2016), 519–548 | DOI | MR | Zbl

[2] Ventsel A. D., “On the boundary conditions for multi-dimensional diffusion process”, Teor. Veroyat. Prim., 4:2 (1954), 172–185 (in Russian) | MR

[3] Luo Y., Trudinger N. S., “Quasilinear second order elliptic equations with Venttsel's boundary conditions”, Potential Analys., 3 (1994), 219–243 | DOI | MR | Zbl

[4] Apushkinskaya D. E., Nazarov A. I., “A Survey of results on nonlinear Venttsel problems”, Applications of Mathematics, 45:1 (2000), 69–80 | DOI | MR | Zbl

[5] Arkhipova A. A., Lukina A. A., “Otsenka reshenii modelnoi zadachi Venttselya v prostranstvakh Kampanato”, Probl. matem. analiza, 28 (2013), 47–56

[6] Arkhipova A. A., Lukina A. A., “Regulyarnost obobschennykh reshenii modelnoi zadachi Venttselya dlya lineinykh divergentnykh ellipticheskikh operatorov v prostranstvakh Kampanato”, Probl. matem. analiza, 72 (2013), 39–50

[7] Arkhipova A. A., “Chastichnaya regulyarnost obobschennykh reshenii modelnoi zadachi Venttselya dlya kvazilineinykh ellipticheskikh sistem uravnenii”, Probl. matem. analiza, 75 (2014), 3–20

[8] Giaquinta M., Giusti E., “Partial regularity for solutions to nonlinear parabolic systems”, Ann. Mat. Pura Appl., 97 (1973), 253–261 | DOI | MR

[9] Campanato S., “Partial Hölder continuity of solutions of quasilinear parabolic systems of second order with linear growth”, Rend. Sem. Mat. Univ. Padova, 64 (1981), 59–75 | MR | Zbl

[10] Campanato S., “On the nonlinear parabolic systems in divergence form. Hölder continuity and partial Hölder continuity of the solutions”, Ann. Mat. Pure and Appl. (IV), 137 (1984), 83–122 | DOI | MR | Zbl

[11] Giaquinta M., Martinazzi L., An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, Scuola Norm. Super., Pisa, 2012 | MR | Zbl

[12] Giaquinta M., “A counter-example to the boundary regularity of solutions to elliptic quasilinear systems”, Manuscripta Math., 26 (1978), 217–220 | DOI | MR

[13] Duzaar F., Grotowski J. F., “Optimal interior partial regularity for nonlinear elliptic systems; the method of A-harmonic approximation”, Manuscripta Math., 103 (2000), 267–298 | DOI | MR | Zbl

[14] Arkhipova A. A., “Chastichnaya regulyarnost reshenii modelnoi zadachi Venttselya dlya kvazilineinykh ellipticheskikh sistem s kvadratichnoi nelineinostyu po gradientu”, Probl. matem. analiza, 82 (2015), 15–30 | Zbl

[15] Duzaar F., Mingione G., “Second order parabolic systems, optimal regularity and singular sets of solutions”, Ann. Instit. H. Poincaré, Anal. Nonlinear., 22 (2005), 711–751 | MR

[16] Bögelein V., Duzaar F., Mingione G., “The boundary regularity of non-linear parabolic systems. I”, Ann. Inst. H. Poincaré, Anal. non Linearire, 27 (2010), 201–255 | DOI | MR | Zbl

[17] Arkhipova A. A., Stara J., John O., “Partial regularity for solutions of quasilinear parabolic systems with nonsmooth in time principal matrix”, Nonlinear Analysis, 95 (2014), 421–435 | DOI | MR | Zbl

[18] Arkhipova A. A., Stara J., “Boudary partial regulatiry for solutions of quasilinear parabolic systems with nonsmooth in time principal matrix”, Nonlinear Analysis, 120 (2015), 236–261 | DOI | MR | Zbl

[19] Krylov N. V., “Parabolic and elliptic equations with VMO coefficients”, Commun. Partial Differ. Equ., 32:1–3 (2007), 453–475 | DOI | MR | Zbl

[20] Krylov N. V., Lectures on elliptic and parabolic equations in sobolev space, Amer. Math. Soc., Providence, 2008 | MR

[21] Dong H., Krylov N. V., Li Xu, “On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains”, Algebra and Analyzis, 24:1 (2012), 53–93 | MR

[22] Dong H., Kim D., “Parabolic and elliptic systems with VMO coefficients”, Methods Appl. Math., 16:3 (2009), 365–388 | MR | Zbl

[23] Dong H., Kom D., “On the $L_p$-solvability of higher order parabolic and elliptic systems with BMO coefficients”, Arch. Rat. Mech. Anal., 199 (2011), 889–941 | DOI | MR | Zbl

[24] Dong H., Kim D., “Global regularity of weak solutions to quasilinear elliptic and parabolic equations with controlled growth”, Communic. in PDE, 36 (2011), 1750–1777 | DOI | MR | Zbl

[25] Campanato S., “Equazioni paraboliche del second ordine e spazi $L^{2,\theta}(\Omega,\delta)$”, Ann. Mat. Pura Appl., 137:4 (1966), 55–102 | DOI | MR

[26] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967