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Proof of Theorem 1.3 - Part (i)

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1499)

Abstract

This Chapter 8 and the next Chapter 9 are devoted to the proof of Theorem 1.3 and Theorem 1.4. In this chapter we prove part (i) of Theorem 1.3. In the proof we make use of Sobolev’s imbedding theorems (Theorems 8.1 and 8.2) and a λ-dependent localization argument due to Masuda [Ma] (cf. Lemma 8.4) in order to adjust estimate

$$\parallel (A_p - \lambda {\rm I})^{ - 1} \parallel \, \le \frac{{c_p (\varepsilon )}}{|\lambda| }\,\,for\,all\,\lambda \in \,\Sigma p(\varepsilon ) $$
(4)

Keywords

  • Positive Constant
  • Sobolev Space
  • Bounded Domain
  • Unique Function
  • Closed Subspace

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Correspondence to Kazuaki Taira .

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© 2009 Springer-Verlag Berlin Heidelberg

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Taira, K. (2009). Proof of Theorem 1.3 - Part (i). In: Boundary Value Problems and Markov Processes. Lecture Notes in Mathematics(), vol 1499. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01677-6_8

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