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UBC Theses and Dissertations

A sharp inequality for Poisson's equation in arbitrary domains and its applications to Burgers' equation Xie, Wenzheng


Let Ω be an arbitrary open set in IR³. Let || • || denote the L²(Ω) norm, and let [formula omitted] denote the completion of [formula omitted] in the Dirichlet norm || ∇•||. The pointwise bound [forumula omitted] is established for all functions [formula omitted] with Δ u є L² (Ω). The constant [formula omitted] is shown to be the best possible. Previously, inequalities of this type were proven only for bounded smooth domains or convex domains, with constants depending on the regularity of the boundary. A new method is employed to obtain this sharp inequality. The key idea is to estimate the maximum value of the quotient ⃒u(x)⃒/ || ∇u || ½ || Δ u || ½, where the point x is fixed, and the function u varies in the span of a finite number of eigenfunctions of the Laplacian. This method admits generalizations to other elliptic operators and other domains. The inequality is applied to study the initial-boundary value problem for Burgers' equation: [formula omitted] in arbitrary domains, with initial data in [formula omitted]. New a priori estimates are obtained. Adapting and refining known theory for Navier-Stokes equations, the existence and uniqueness of bounded smooth solutions are established. As corollaries of the inequality and its proof, pointwise bounds are given for eigenfunctions of the Laplacian in terms of the corresponding eigenvalues in two- and three-dimensional domains.

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