An Aleksandrov-Bakelman type maximum principle and applications
This paper is concerned with various local estimates of solutions of second order linear elliptic equations with Venttsel boundary conditions. Venttsel boundary conditions arose from probability theory [\textit{A. D. Venttsel}, Theor. Probab. Appl. 4, 164-177 (1960), transl. from Teor. Verojatn. Primen. 4, 172-185 (1959; Zbl 0089.134)] and three-dimensional water wave theory [\textit{M. Shinbrot}, J. Inst. Math. App. 25, 367-385 (1980; Zbl 0441.76009)]. The existence and uniqueness of classical solutions to linear Venttsel boundary value problems is treated by \textit{Y. Luo} and \textit{N. S. Trudinger} [Proc. R. Soc. Edinb., Sect. A 118, No. 3/4, 193-207 (1991; Zbl 0771.35014)]. The work described in this paper is preliminary for the study of the corresponding nonlinear Venttsel boundary value problems [\textit{Y. Luo}, Nonlinear Anal., Theory Methods Appl. 16, No. 9, 761-769 (1991; Zbl 0731.35035), \textit{Y. Luo} and \textit{N. S. Trudinger}, Potential Anal. 3, No. 2, 219-243 (1994)]. The existence result for nonlinear problems usually depends strongly upon a priori estimates. In Venttsel boundary value problems the boundary Hölder estimates are desired, since the interior estimates are known. This paper gives a boundary version of the Aleksandrov-Bakelman maximum principle and a boundary version of the weak Harnack inequality which are fundamental for the boundary Hölder estimates for solutions to nonlinear Venttsel boundary value problems.
scientific article; zbMATH DE number 166899