Schwache Lösungen des Frankl-Morawetz-Problems im ℝ₃

Abstract

In a bounded simple connected region G ⊂ ℝ³ we consider the equation

$$L\left[ u \right]: = k\left( z \right)\left( {u_{xx} + u_{yy} } \right) + u_{zz} + d\left( {x,y,z} \right)u = f\left( {x,y,z} \right)$$

where k(z)⪋ 0 whenever z ⪋ 0.G is surrounded forz≥0 by a smooth surface Γ₀ with S:=Γ₀ ⋂ {(x,y,z)|=0} and forz<0 by the characteristic$\Gamma _2 :---(x^2 + y^2 )^{{\textstyle{1 \over 2}}} + \int\limits_z^0 {(---k(t))^{{\textstyle{1 \over 2}}} dt = 0}$ and a smooth surface Γ₁ which intersect the planez=0 inS and where the outer normal n=(n_x, n_y, n_z) fulfills$k(z)(n_x^2 + n_y^2 ) + n_z^2 |_{\Gamma _1 } > 0$. Under conditions on Γ₁ and the coefficientsk(z), d(x,y,z) we prove the existence of weak solutions for the boundary value problemL[u]=f inG with$u|_{\Gamma _0 \cup \Gamma _1 } = 0$. The uniqueness of the classical solution for this problem was proved in [1].