Quasilinear second order elliptic equations with Venttsel boundary conditions
The aim of this paper is to study the existence and uniqueness of solutions of the quasilinear second order elliptic equations (1) \(a^{ij} (x,u, Du) D_{ij} u+ b(x,u, Du)=0\) in \(\Omega\), with quasilinear Venttsel boundary conditions (2) \(\alpha^{ij} (x,u, \partial u) \partial_{ij} u+ \beta(x,u, Du)=0\) on \(\partial \Omega\), where \(\Omega\) is a smooth domain in \(\mathbb{R}^ n\), \(\nu= (\nu^ 1, \dots, \nu^ n)\) is the inward normal vector field on the boundary manifold \(\partial \Omega\), \(\partial_ i= c^{ik} D_ k\), \(\partial_{ij}= \partial_ i \partial_ j\), \(i,j= 1,\dots, n\), where \(c^{ik}= \delta^{ik}- \nu^ i \nu^ k\), \(\delta^{ik}\) is the Kronecker symbol. Such boundary conditions arose from probability theory. The general assumptions, concerning coefficient matrices \(\{a^{ij}\}\) and \(\{\alpha^{ij}\}\) and vector fields \(b\) and \(\beta\), are motivated by the natural structure conditions of Ladyzhenskaya and Uraltseva. These conditions are sufficient to establish boundedness of the gradient and the second derivative. First, by locally flattening of the boundary, problem (1) (2) is reduced to the problem \[ a^{ij} (x,u,Du) D_{ij} u+ b(x,u, Du)=0 \quad \text{in } B^ +, \qquad \alpha^{st} (x) D_{st} u+ \beta(x,u, Du)=0 \quad \text{on } B^ 0, \] where \(B^ += \{x\in \mathbb{R}^ n\): \(| x|<1\), \(x_ n>0\}\), \(B^ 0= \{x\in \mathbb{R}^ n\): \(| x|<1\), \(x_ n=0\}\) and \(s,t= 1,\dots, n-1\). By the procedure of elliptic regularization, the proof of the existence of solution is reduced to a series of a priori estimates. As an application of the general theory, the authors solve the water wave problem. The last section deals with weak solutions of problem (1) (2) in the viscosity sense of P.-L. Lions and M. G. Crandall.
scientific article; zbMATH DE number 554327
scientific article