Laplacian eigenproblems on product regions and tensor products of Sobolev spaces

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• 作者：Giles Auchmuty^a ; ^{auchmuty@uh.edu" class="auth_mail" title="E-mail the corresponding author} ; M.A. Rivas^b ; ^{rivasma@wfu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Laplacian eigenproblems ; Tensor products ; Bases of Sobolev spaces ; Steklov eigenproblems
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 March 2016
• 年：2016
• 卷：435
• 期：1
• 页码：842-859
• 全文大小：433 K

文摘

Characterizations of eigenvalues and eigenfunctions of the Laplacian on a product domain Ω_p:=Ω₁×Ω₂${\mathrm{\Omega }}_p:={\mathrm{\Omega }}_1\times {\mathrm{\Omega }}_2$ are obtained. When zero Dirichlet, Robin or Neumann conditions are specified on each factor, then the eigenfunctions on Ω_p${\mathrm{\Omega }}_p$ are precisely the products of the eigenfunctions on the sets Ω₁${\mathrm{\Omega }}_1$, Ω₂${\mathrm{\Omega }}_2$ separately. There is a related result when Steklov boundary conditions are specified on Ω₂${\mathrm{\Omega }}_2$. These results enable the characterization of H¹(Ω_p)$H^1({\mathrm{\Omega }}_p)$ and $H_0^1({\mathrm{\Omega }}_p)$ as tensor products and descriptions of some orthogonal bases of the spaces. A different characterization of the trace space of H¹(Ω_p)$H^1({\mathrm{\Omega }}_p)$ is found.