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Boundary Operators and Boundary Maximum Principles

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Boundary Value Problems and Markov Processes

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

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Abstract

Let D be a bounded domain of Euclidean space R N, with smooth boundary ∂D; its closure \(\overline {D} = D \cup {\partial D}\) is an N-dimensional, compact smooth manifold with boundary (see Figure 11.1 below). In this chapter, following Bony–Courrège–Priouret ( [15, Chapter II]) we characterize Ventcel’–Lévy boundary operators T (Theorem 11.3) and Ventcel’ boundary operators Γ = Λ + T (Theorem 11.4) defined on the compact smooth manifold \(\overline {D}\) with boundary ∂D in terms of the positive boundary maximum principle:

$$\displaystyle \,\,x^{\prime }_{0} \in {\partial D}, \,\, u \in C^{2}(\overline {D}) \hspace{.5em} \mbox{and} \hspace{.5em} u(x^{\prime }_{0}) = \max _{x \in \overline {D}} u(x) \geq 0 \Longrightarrow (\varGamma u)(x^{\prime }_{0}) \leq 0. $$

This chapter will be very useful in the study of Markov processes with general Ventcel’ boundary conditions in the last Chapter 16.

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  1. Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki, Japan

    Kazuaki Taira

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Taira, K. (2020). Boundary Operators and Boundary Maximum Principles. In: Boundary Value Problems and Markov Processes. Lecture Notes in Mathematics, vol 1499. Springer, Cham. https://doi.org/10.1007/978-3-030-48788-1_11

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