Elliptische Differentialgleichungen mit variablen Koeffizienten in Gebieten mit unbeschränktem Rand

Abstract

We discuss the spectrum of a symmetric elliptic differential operator A with domain$\mathop {H^m }\limits^o (\Omega ) \cap H^{2m} (\Omega )$ in regions Ω with unbounded boundary$\dot \Omega$, where are$\bar \Omega$uniformely of class C^2m and on$\dot \Omega$ the normal condition x·ν(x)≦μ for sufficient small positiveμ. We prove the A-priori-estimate$\parallel u\parallel _{m,\Omega } \leqq c\parallel (l + r) (A - k)u\parallel _{o,\Omega }$ and show for all k>k, k≧0 suitable, there are no eigenvalues of A and by characterizing weighted Sobolev spaces with negative norm the existence of solutions$(l + r)_2 ^{ - 1} u \in \mathop H\limits^0{^m} (\Omega ) \cap H^{2m} (\Omega )$ of the equation (A−k)u=f, (1+r)f∈L²(Ω).