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Nonlinear oblique boundary value problems for nonlinear elliptic equations. (English) Zbl 0619.35047
The paper is concerned with nonlinear oblique elliptic boundary value problems of the general form $(1)\quad F[u]=F(x,u,Du,D^ 2u)=0\quad in\quad \Omega;\quad G[u]=G(x,u,Du)=0\quad on\quad \partial \Omega$ ($$\Omega\subset {\mathbb{R}}^ n$$ bounded smooth domain). The operator F is elliptic, i.e. the derivative matrix $$[\partial F/\partial r_{ij}]$$ of F(x,z,p,r) (F defined on $$\Omega \times {\mathbb{R}}\times {\mathbb{R}}^ n\times S^ n$$; $$S^ n = set$$ of symmetric real matrices $$r=(r_{ij})_{n,n})$$ is positive at (x,z,p,r). Furthermore G is called oblique at (x,z,p), if $$G_ p\cdot \gamma$$ is positive at (x,z,p), where $$\gamma$$ is the unit inner normal to $$\partial \Omega$$, and $$G_ p$$ is the partial derivative with respect to p.
Adding some ”natural” structure conditions we get the main Theorem (Theorem 1.1): If $$0<\alpha <1$$ and $$u\in C^ 2({\bar \Omega})$$ is a solution of (1), then there exists a constant C satisfying $$| u|_{2,\alpha;\Omega}\leq C$$ $$(C^{2,\alpha}({\bar \Omega})$$- estimate). (C depends on the structure conditions and the bounds of u, Du and $$D^ 2u.)$$
Corollary 1.2: Assuming additionally sup $$F_ z<0$$ and sup $$G_ z\leq 0$$ there exists a unique classical solution $$u\in C^{2,\alpha}({\bar \Omega})$$ for (1) for all $$\alpha <1.$$
The theorem and the corollary are new, and the $$C^{2,\alpha}({\bar \Omega})$$-estimate is the culmination of several estimates of lower order established separately in the following six sections of the paper. (§ 2: global Hölder estimates, § 3, § 4: gradient bounds, § 5, § 6: global bounds for $$D^ 2u$$, § 7: existence and uniqueness of solutions.)
In the very detailed paper the resultant theorems include previous work such as that of the first author himself (on quasilinear equations) and Lions and the second author (on boundary conditions.)
Reviewer: F.Wille

##### MSC:
 35J60 Nonlinear elliptic equations 35J65 Nonlinear boundary value problems for linear elliptic equations 35B45 A priori estimates in context of PDEs 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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