The Resolvent of Closed Extensions of Cone Differential Operators

By Elmar Schrohe, J. Seiler. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2002

53 citations recorded by the source corpus.

Abstract 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 $\dot{u}\,-\,\Delta u\,=\,f,$ $u\left( 0 \right)\,=\,0$ .

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d5ea9bd5cfd0a17341f7e97d1b75a75e8b483640

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Elmar Schrohe
2. T. Krainer
3. N. Roidos
4. Juan B. Gil
5. G. Mendoza
6. J. Seiler
7. Gerd Grubb

In Computational Theory and Mathematics

1. Elmar Schrohe
2. T. Krainer
3. G. Mendoza
4. Juan B. Gil
5. N. Roidos
6. J. Seiler

In Mathematical Physics

1. Elmar Schrohe
2. T. Krainer
3. G. Mendoza
4. N. Roidos
5. Juan B. Gil
6. J. Seiler
7. Gerd Grubb

Citations in the corpus

• Existence and maximal Lp-regularity of solutions for the porous medium equation on manifolds with conical singularities (2015, background): “[5], [19], [27], [28]; the present version is due to Gil, Krainer and Mendoza [13].”
• Resolvents of elliptic cone operators (2004, background): “[16], [17], or the technically convenient but rather restrictive dilation invariance of the domain, cf.”