## Appendix A General Results for the Laplacian Operator in Bounded Domains

In Chapter 2, Section 2.9, we used the result that a function u(x) with ux = 0 on x = 0,1 satisfies i u2xxdx > n2\ u2xdx (A4.1)

Jo Jo and the more general result

/ |V2u |2 dr > n I " "2 jb js i|2 dr > n y Vu y2 dr, (A4.2)

h b where B is a finite domain enclosed by the simply connected surface d B on which zero-flux (Neumann) conditions hold; namely, n Vu = 0 where n is the unit outward normal to d B. In (A4.2), x is the least positive eigenvalue of V2 + x for B with Neumann conditions on d B and where || ■ y denotes a Euclidean norm. By the Euclidean norm here we mean, for example,

We prove these standard results in this section: (A4.1) is a special case of (A4.2) in which u is a single scalar and r a single space variable.

By way of illustration we first derive the one-dimensional result (A4.1) in detail and then prove the general result (A4.2).

Consider the equation for the scalar function w(x), a function of the single space variable x, given by wxx + fiw = 0, (A4.4)

where \x represents the general eigenvalue for solutions of this equation satisfying Neumann conditions on the boundaries; namely, wx (x) = 0 on x = 0,1.

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