An inverse eigenvalue problem for a general convex domain: an extension to higher dimensions
Let \(D\subset R^ 3\) be a simply connected bounded domain with the bounding surface S. Let us suppose that the \(\lambda_ n\) are the eigenvalues of the problem \(\nabla^ 2 u+\lambda u=0\) in D, \(\partial u/\partial n+\gamma u=0\) on S, where \(\partial /\partial n\) denotes differentiation along the inward pointing normal to the bounding surface S and \(\gamma\) is a positive constant. The objective of this paper is to determine the geometry of D and the impedance \(\gamma\) from the asymptotic form of the trace function \(\theta (t)=tr(e^{-\nabla^ 2 t})=\sum_{1\leq n\leq \infty}\ell^{-\lambda_ nt}\) for small positive t. The author proves that if \(0<\gamma \ll 1\) then \[ \theta (t)\sim V(4\pi t)^{-3/2}+S(16\pi t)^{-1}+(12\pi^{3/2} t^{1/2})^{- 1}\int_{S}(H-3\gamma)ds+O(t^{1/2})\quad as\quad t\to 0, \] and if \(\gamma \gg 1\) then \[ \theta (t)\sim V(4\pi t)^{-3/2}-(16t\pi)^{- 1}(S-2\gamma^{-1}\int_{S}Hds)+(12\pi^{3/2} t^{1/2})^{- 1}\int_{S}Hds+O(t^{1/2})\quad as\quad t\to 0. \] In these formulae V and S are respectively, the volume and the surface area of D; \(H=2^{- 1}(R_ 1^{-1}+R_ 2^{-1})\) where \(R_ 1\) and \(R_ 2\) are the principal radii of curvature. For the Dirichlet and Neumann problems similar results were obtained by \textit{A. Pleijel} [Ark. Mat. 2, 553-569 (1954; Zbl 0055.088); 12. Scand. Math. Kongr., Lund 1953, 222-240 (1954; Zbl 0056.097)].
scientific article; zbMATH DE number 4000569
scientific article