The solvability of the fully nonlinear stationary Venttsel' problem is established. The equation and the boundary condition are assumed to be uniformly elliptic. Bibliography: 12 titles.
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D. E. Apushkinskaya and A. I. Nazarov, “A survey of results on nonlinea Venttsel' problems”,Appl. Math., to appear.
D. E. Apushkinsakaya and A. I. Nazarov, “On the quasilinear stationary Venttsel' problem,”Zap. Nauchn. Semin. POMI,221, 20–29 (1995).
D. E. Apushkinsakaya and A. I. Nazarov, “Estimates for the gradient of solutions to stationary degenerate Venttsel' problems,”J. Math. Sci.,98, No. 6, 654–673 (2000).
N. S. Trudinger, “Fully nonlinear, uniformly elliptic equations under natural structure conditions,”Trans. Amer. Math. Soc.,278, 751–769 (1983).
G. M. Lieberman and N. S. Trudinger, “Nonlinear oblique bondary-value problems for nonlinear elliptic equations,”Trans. Amer. Math. Soc.,295, 509–546 (1986).
D. E. Apushkinsakaya and A. I. Nazarov, “Hölder estimates of solutions to initial-boundary value problems for parabolic equations of nondivergence form with Wentzel (Venttsel') boundary condition”,Amer. Math. Soc. Transl. Ser. II.,164, 1–13 (1995).
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D. E. Apushkinskaya and A. I. Nazarov, “The initial-boundary value problem for nondivergence parabolic equations with Venttsel' boundary condition,”St. Petersburg Math. J.,6, No. 6, 1127–1149 (1995).
G. M. Lieberman, “The Dirichlet problem for quasilinear elliptic equations with continuously differentiable boundary data,”Commun. Partial Differ. Equations,11, No. 2, 167–229 (1986).
D. E. Apushkinskaya and A. I. Nazarov, “Boundary estimates for the first-order derivatives of a solution to a nondivergence parabolic equation with composite right-hand side and coefficients of lower-order derivatives,”J. Math. Sci.,77, No. 4, 3257–3276 (1995).
O. V. Besov, V. P. Il'in, and S. M. Nikol'skii,Integral Represntations of Functions and Embedding Theorems [in Russian], Nauka, Moscow (1996). [English Translation of the 1st ed.:Integral Representations of Functions and Embedding Theorems, Vol. I–II, Wiley, New York (1978, 1979).]
D. Gilbarg and N. S. Trudinger,Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin (1983).
Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 3–26.
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Apushkinskaya, D.E., Nazarov, A.I. The venttse’ problem for nonlinear elliptic equations. J Math Sci 101, 2861–2880 (2000). https://doi.org/10.1007/BF02672175
- Elliptic Equation
- Nonlinear Elliptic Equation
- Quasilinear Elliptic Equation
- Require Estimate
- Boundary Estimate