Energy distribution for Dirichlet eigenfunctions on right triangles (Q6601844)

This article is concerned with various properties of the eigenfunctions related to the Laplace operator with homogeneous Dirichlet boundary condition on traingular domains in \(\mathbb{R}^2\). In their first result the authors show that on the triangle \(T=\big\{(x,y): 0\leq x\leq 1,\, 0\leq y\leq 1-x\big\}\) there exists an orthonormal basis for \(L^2(T)\) of eigenfunctions of \(-h^2 \Delta u=u\) such that \(\|hu_x\|_{L^2(T)}=\|hu_y\|_{L^2(T)}=1/2\). In their second result, the authors consider the triangle \(\Omega=\big\{(x, y):0\leq x\leq a,\, 0\leq y\leq 1-x/a\big\}\). It is shown that if \(\{u_j\}\) is a sequence of orthonormal eigenfunctions on \(\Omega\) , then there exists a density one subsequence \(\{u_{j_k}\}\) such that \(\limsup_{k\to \infty} \int_\Omega |h (u_{j_k})_x|^2 dV\leq 7/8\).