Applications of differential calculus to quasilinear elliptic boundary value problems with non-smooth data

Abstract.

This paper concerns boundary value problems for quasilinear second order elliptic systems which are, for example, of the type

$$ \begin{aligned} \partial _{j} {\left( {a^{{ij}}_{{\alpha \beta }} {\left( {u,\lambda } \right)}\partial _{i} u^{\alpha } + b^{j}_{\beta } {\left( {u,\lambda } \right)}} \right)} + c^{i}_{{\alpha \beta }} {\left( {u,\lambda } \right)}\partial _{i} u^{\alpha } & = d_{\beta } {\left( {u,\lambda } \right)}{\text{ in }}\Omega {\text{,}} \\ {\left( {a^{{ij}}_{{\alpha \beta }} {\left( {u,\lambda } \right)}\partial _{i} u^{\alpha } + b^{j}_{\beta } {\left( {u,\lambda } \right)}} \right)}\nu _{j} & = e_{\beta } {\left( {u,\lambda } \right)}{\text{ on }}\Gamma _{\beta } , \\ u^{\beta } & = \varphi ^{\beta } {\text{ on }}\partial \Omega \backslash \Gamma _{\beta } . \\ \end{aligned} $$

Here Ω is a Lipschitz domain in $\mathbb{R}^{N},$ ν _j are the components of the unit outward normal vector field on ∂Ω, the sets Γ_β are open in ∂Ω and their relative boundaries are Lipschitz hypersurfaces in ∂Ω. The coefficient functions are supposed to be bounded and measurable with respect to the space variable and smooth with respect to the unknown vector function u and to the control parameter λ. It is shown that, under natural conditions, such boundary value problems generate smooth Fredholm maps between appropriate Sobolev-Campanato spaces, that the weak solutions are Hölder continuous up to the boundary and that the Implicit Function Theorem and the Newton Iteration Procedure are applicable.