Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
Venttsel boundary conditions; elliptic equations; parabolic equations; a priori estimates; existence theorems; boundary value problems
Venttsel boundary conditions; elliptic equations; parabolic equations; a priori estimates; existence theorems; boundary value problems
Summary:
We review the recent results for boundary value problems with boundary conditions given by second-order integral-differential operators. Particular attention has been paid to nonlinear problems (without integral terms in the boundary conditions) for elliptic and parabolic equations. For these problems we formulate some statements concerning a priori estimates and the existence theorems in Sobolev and Hölder spaces.
References:
[1] D. E. Apushkinskaya: An estimate for the maximum of solutions of parabolic equations with the Venttsel condition. Vestnik Leningrad Univ. Mat. Mekh. Astronom. (1991), 3–12. MR 1166371
[2] D. E. Apushkinskaya, A. I. Nazarov: Gradient estimates for solutions of stationary degenerate Venttsel problems I. Research rep. no. MRR 058–97, Austral. Nat. Univ., Centre for Math. Anal. (1997).
[3] D. E. Apushkinskaya, A. I. Nazarov: Gradient estimates for solutions of stationary degenerate Venttsel problems II. Research rep. no. MRR 019–98, Austral. Nat. Univ., Centre for Math. Anal. (1998).
[4] D. E. Apushkinskaya, A. I. Nazarov: Gradient estimates for solutions of stationary degenerate Venttsel problems. Probl. Mat. Anal. 18 (1998), 43–69.
[5] D. E. Apushkinskaya, A. I. Nazarov: Hölder estimates of solutions to degenerate Venttsel boundary value problems for parabolic and elliptic equations of nondivergent form. Probl. Mat. Anal. 17 (1997), 3–18.
[6] D. E. Apushkinskaya, A. I. Nazarov: Hölder estimates of solutions to initial-boundary value problems for parabolic equations of nondivergent form with Wentzel (Venttsel) boundary condition. Amer. Math. Soc. Transl. (2) 64 (1995), 1–13. MR 1334136
[7] D. E. Apushkinskaya, A. I. Nazarov: On the quasilinear stationary Venttsel problem. Zap. Nauchn. Sem. S.-Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 221 (1995), 20–29. MR 1359746
[8] D. E. Apushkinskaya, A. I. Nazarov: The initial-boundary value problem for nondivergent parabolic equation with Venttsel boundary condition. Algebra Anal. 6 (1994), 1–29. MR 1322117
[9] D. E. Apushkinskaya, A. I. Nazarov: The nonstationary Venttsel problem with quadratic growth with respect to the gradient. Probl. Mat. Anal. 15 (1995), 33–46. MR 1420673
[10] J. R. Cannon, G. H. Meyer: On diffusion in a fractured medium. SIAM J. Appl. Math. 3 (1971), 434–448. DOI 10.1137/0120047
[11] E. I. Galakhov, A. L. Skubachevskii: On shrinking nonnegative semigroups with nonlocal conditions. Mat. Sb. 165(207) (1998), 45–75. MR 1616432
[12] N. Ikeda, S. Watanabe: Stochastic Differential Equations and Diffusion Processes. (1981), North-Holland, Amsterdam-New York, Kodansha, Tokyo. MR 0637061
[13] P. Korman: Existence of periodic solutions for a class of nonlinear problems. Nonlinear Anal. 7 (1983), 873–879. DOI 10.1016/0362-546X(83)90063-9 | MR 0709040 | Zbl 0523.35006
[14] P. Korman: Existence of solutions for a class of semilinear noncoercive problems. Nonlinear Anal. 10 (1986), 1471–1476. DOI 10.1016/0362-546X(86)90116-1 | MR 0869554 | Zbl 0621.35008
[15] O. A. Ladyzhenskaya, N. N. Uraltseva: A survey of results on the solvability of boundary value problems for uniformly elliptic and parabolic second order quasilinear equations having unbounded singularities. Uspekhi Mat. Nauk 41 (1986), 59–83. MR 0878325
[16] K. Lemrabet: Ventcel’ boundary value problem in a non-smooth domain. C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), 531–534. MR 0792383
[17] G. M. Lieberman: Second Order Parabolic Differential Equations. World Scientific, Singapore-New Jersey-London-Hong Kong, 1996. MR 1465184 | Zbl 0884.35001
[18] G. M. Lieberman, N. S. Trudinger: Nonlinear oblique boundary value problems for nonlinear elliptic equations. Trans. Am. Math. Soc. 295 (1986), 509–545. DOI 10.1090/S0002-9947-1986-0833695-6 | MR 0833695
[19] V. V. Lukyanov, A. I. Nazarov: A solution of Venttsel problems for the Laplacian and the Helmholtz equation by iterated potentials. Zap. Nauchn. Sem. S.-Petersburg. Otdel. Mat. Inst. Steklov. (POMI) 250 (1998).
[20] Y. Luo: An Aleksandrov-Bakel’man type maximum principle and applications. J. Diff. Eq. 101 (2) (1993), 213–231. DOI 10.1006/jdeq.1993.1011 | MR 1204327
[21] Y. Luo: Quasilinear second order elliptic equations with elliptic Venttsel boundary conditions. Nonlinear Anal. 16 (1991), 761–769.
[22] Y. Luo, N. S. Trudinger: Linear second order elliptic equations with Venttsel boundary conditions. Proc. Roy. Soc. Edinburgh Sect. A 118 (1991), 193–207. MR 1121663
[23] A. I. Nazarov: On the estimates of the Hölder constants for solutions of the oblique derivative problem for a parabolic equation. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 163 (1987), 130–131. MR 0918945
[24] A. I. Nazarov: Interpolation of linear spaces and maximum estimates for solutions of parabolic equations. Part. Diff. Eq., Acad. Nauk SSSR, Siberian Dep. Math. Inst., Novosibirsk, 1987, pp. 50–72. MR 0994027
[25] M. Shinbrot: Water waves over periodic bottoms in three dimensions. J. Inst. Math. Applic. 25 (1980), 367–385. DOI 10.1093/imamat/25.4.367 | MR 0578084 | Zbl 0441.76009
[26] K. S. Tulenbaev, N. N. Uraltseva: Nonlinear boundary value problem for elliptic equations in general form. Part. Diff. Eq., Acad. Nauk SSSR, Siberian Dep. Math. Inst., Novosibirsk, 1987, pp. 95–112. MR 0994029
[27] A. D. Venttsel: On boundary conditions for multidimensional diffusion processes. Teor. Veroyatnost. i Primenen. 4 (1959), 172–185. Zbl 0089.13404
[28] S. Watanabe: Construction of diffusion processes with Venttsel boundary conditions by means of Poisson point processes of Brownian excursions. Probability theory, Banach center Publ., vol. 5, PWN, Warsaw, 1979, pp. 255–271. MR 0561485
Search
Partner of
