General Mixed Boundary Problems for Elliptic Differential Equations

Abstract

Let G be a bounded domain in Rⁿ with a smooth n-1-dimensional boundary Γ. Assume that Γ is divided by a smooth n-2 dimensional manifold Γ₀ on two parts Γ_l and Γ₂, so that Γ̄₁ ∪ Γ̄₂ = Γ , Γ̄₁ ∪ Γ̄₂ = Γ₀. Consider in G a second order elliptic equation

$$ {\text{L}}\left( {{\text{x,D}}} \right){\text{u = }}\,\,\sum\limits_{{\text{i,k = 1}}}^{\text{n}} {\,\,{\text{a}}_{{\text{ik}}} \left( {\text{x}} \right)} \,\,\frac{{\partial ^2 {\text{u}}}} {{\partial {\text{x}}_{\text{i}} \partial {\text{x}}_{\text{k}} }}\,\,\,\, + \,\,\,\sum\limits_{{\text{k = 1}}}^{\text{n}} {{\text{b}}_{\text{k}} \left( {\text{x}} \right)\frac{{\partial {\text{u}}}} {{\partial {\text{x}}_{\text{k}} }} + {\text{c}}\left( {\text{x}} \right){\text{u = f}}\left( {\text{x}} \right)} $$

(0.1)

with a mixed boundary conditions

$$ {\text{B}}_1 \left( {{\text{x,D}}} \right){\text{u}}\left| {_{\Gamma _1 } } \right.\,\, = {\text{g}}_1 $$

(0.2)

$$ {\text{B}}_2 \left( {{\text{x,D}}} \right){\text{u}}\left| {_{\Gamma _1 } } \right.\,\, = {\text{g}}_2 $$

(0.3)