Some estimates for the symmetrized first eigenfunction of the Laplacian (Q1275358)

Consider the first eigenfunction \(u> 0\) of the Laplace operator with Dirichlet boundary condition on a bounded domain \(D\) of \({\mathbb R}^2\) which is associated to the eigenvalue \(\lambda_1\). Denote \(u^*\) the radially nonincreasing rearrangement of \(u\) and \(D^*\) the disc centered at the origin which has same area as \(D\), then the authors prove the estimate \[ \|u^* -U\|_{L^\infty (D^*)}\leq C\sqrt{\lambda_1 -\lambda_1^*}, \] where \(U\) is the first eigenfunction (conveniently normalized) of the Laplacian on \(D^*\) and \(\lambda_1^*\) is the associated eigenvalue. Moreover, denoting \(B\) the disc centered at the origin such that the Laplace operator with Dirichlet boundary condition on \(B\) admits also \(\lambda_1\) as its first eigenvalue, they get an estimate on \(u^*-v\) on \(B\), where \(v\) is the first eigenfunction of the Laplacian on \(B\).