Partial Differential Equations by Jürgen Jost

Author:Jürgen Jost
Language: eng
Format: epub, pdf
Publisher: Springer New York, New York, NY

for all ν with (8.2.90). Thus we would have

(8.2.91)

On the other hand, by (8.2.88), we may assume

which in conjunction with (8.2.91) yields the contradiction

Consequently, (8.2.89) must hold. As in (8.2.87), we now obtain for each

The case of an infimum is treated analogously, and (8.2.85) follows.

We now want to carry out the corresponding analysis for the heat semigroup, again using the notation already established. In this case, the infinitesimal generator is the Laplace operator:

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