ISSN:
1436-5081
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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=(nx, ny, nz) 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].
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01538034
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