The Resolvent of Closed Extensions of Cone Differential Operators

Schrohe, E. ; Seiler, J.

Canadian journal of mathematics, Tome 57 (2005) no. 4, pp. 771-811

Résumé

We study closed extensions $\underset{\scriptscriptstyle-}{A}$ 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 ${{\left( \lambda -\underset{\scriptscriptstyle-}{A} \right)}^{-1}}$ exists in a sector of the complex plane and decays like $1/\left| \lambda\right|$ as $\left| \lambda\right|\to \infty $ . Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of $\underset{\scriptscriptstyle-}{A}$ .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 $\dot{u}\,-\,\Delta u\,=\,f,$ $u\left( 0 \right)\,=\,0$ .

Détail

DOI : 10.4153/CJM-2005-031-1

Mots-clés : 35J70, 47A10, 58J40, Manifolds with conical singularities, resolvent, maximal regularity

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