On Laplacian eigenvalue equation with constant Neumann boundary data

Let & be a bounded Lipschitz domain in R" and we study boundary behaviors of solutions to the following Laplacian eigenvalue equation with constant Neumann data:<br /> {-Delta u = cu in Omega partial derivative u/partial derivative v = -1 on a partial derivative Omega. (0.1)<br /> First, by using properties of Bessel functions and proving new inequalities on elemen- tary symmetric polynomials, we obtain the following inequality for rectangular boxes, balls and equilateral triangles:<br /> lim(c ->mu 2)c integral(partial derivative Omega)u(c) d sigma >= n -1/n P2(Omega)/Iota Omega Iota, (0.2)<br /> with equality achieved only at cubes and balls. In the above, u, is the solution to (0.1) and mu 2 is the second Neumann Laplacian eigenvalue. Second, let Ky be the best constant for the Poincare inequality with the vanishing mean condition over a, and we prove that k <= mu(2) and that the equality holds if and only if fague do > 0 for any c <euro> (0,mu 2). As a consequence, k = mu 2 on balls, rectangular boxes and equilateral triangles, and balls maximize k over all Lipschitz domains with fixed volume. As an application, we extend the symmetry breaking results from ball domains obtained in Bucur-Buttazzo-Nitsch (J Math Pures Appl (9) 107(4): 451-463, 2017), to wider class of domains, and give quantitative estimates for the precise breaking threshold at balls and rectangular boxes. It is a direct consequence that for domains with K < mu(2), (0.2) is never true, while whether it is valid for domains on which k] = mu(2) remains open.