Abstract
A majorant of the Hölder constant is established for solutions of the two-phase quasilinear parabolic or elliptic boundary-value problems with degenerate and nondegenerate Venttsel conditions on an interface. In addition, gradient estimates for solutions of the two-phase quasilinear nondegenerate Venttsel problems are obtained, and the solvability in Sobolev and Hölder spaces is proved. Bibliography: 14 titles.
Similar content being viewed by others
REFERENCES
J. R. Cannon and G. H. Meyer, On diffusion in a fractured medium," SIAM J. Appl. Math., 3, 434–448 (1971).
P. Korman, Existence of periodic solutions for a class of nonlinear problems," Nonlinear Anal., 7, 873–879 (1983).
P. Korman, Existence of solutions for a class of semilinear noncoercive problems," Nonlinear Anal., 10, 1471–1476 (1986).
M. Shinbrot, Water waves over periodic bottoms in three dimensions," J. Inst. Math. Applic., 25, 367–385 (1980).
D. E. Apushkinskaya and A. I. Nazarov, A survey of results on nonlinear Venttsel problems," Appl. Math. (toappear).
D. E. Apushkinskaya and A. I. Nazarov, Linear two-phase Venttsel problems," Report No. 8, 1999/2000, Institut Mittag-Leffler (1999).
D. E. Apushkinskaya and A. I. Nazarov, The initial boundary-value problem for nondivergent parabolic equations with Venttsel boundary condition," Algebra Analiz, 6, 1–29 (1994).
D. E. Apushkinskaya and A. I. Nazarov, H¨older estimates of solutions to initial boundary-value problems for parabolic equations of the nondivergent form with Venttsel boundary condition," Am. Math. Soc. Transl., 164, 1–13 (1995).
D. E. Apushkinskaya and A. I. Nazarov, The nonstationary Venttsel problem with quadratic growth with respect to the gradient," Probl. Mat. Anal., 15, 33–46 (1995).
D. E. Apushkinskaya, An estimate for the maximum of solutions of parabolic equations with Venttsel condition," Vestnik Leningrad. Univ., Ser. Mat. Mekh. Astronom., No. 2, 3–12 (1991).
A. I. Nazarov and N. N. Uraltseva, Convex-monotone hulls and an estimate of the maximum of the solution of a parabolic equation," Zap. Nauchn. Semin. LOMI, 147, 95–109 (1985).
O. A. Ladyzhenskaya and N. N. Uraltseva, Estimates of the H¨older constant for functions satisfying a uniformly elliptic or uniformly parabolic quasilinear inequality with unbounded coefficients," Zap. Nauchn. Semin. LOMI, 147, 72–94 (1985).
D. E. Apushkinskaya and A. I. Nazarov, H¨older estimates of solutions to degenerate Venttsel boundary-value problems for parabolic and elliptic equations of the nondivergent form," Probl. Mat. Anal., 17, 3–19 (1997).
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Apushkinskaya, D.E., Nazarov, A.I. Quasilinear Two-Phase Venttsel Problems. Journal of Mathematical Sciences 115, 2704–2719 (2003). https://doi.org/10.1023/A:1023305432516
Issue Date:
DOI: https://doi.org/10.1023/A:1023305432516