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The Resolvent of Closed Extensions of Cone Differential Operators

Published:2005-08-01
Printed: Aug 2005
• E. Schrohe
• J. Seiler
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Abstract

We study closed extensions $\underline A$ of an elliptic differential operator $A$ on a manifold with conical singularities, acting as an unbounded operator on a weighted $L_p$-space. Under suitable conditions we show that the resolvent $(\lambda-\underline A)^{-1}$ exists in a sector of the complex plane and decays like $1/|\lambda|$ as $|\lambda|\to\infty$. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of $\underline A$. As an application we treat the Laplace--Beltrami operator for a metric with straight conical degeneracy and describe domains yielding maximal regularity for the Cauchy problem $\dot{u}-\Delta u=f$, $u(0)=0$.
 Keywords: Manifolds with conical singularities, resolvent, maximal regularity
 MSC Classifications: 35J70 - Degenerate elliptic equations 47A10 - Spectrum, resolvent 58J40 - Pseudodifferential and Fourier integral operators on manifolds [See also 35Sxx]

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