Let A be a bounded domain in R^d, d>1, and {u_k} is the set of all L2-normalized Laplacian eigenfunctions on A with Dirichlet boundary condition (i.e., |u_k|_2 = 1). Is it true that these eigenfunctions are uniformly bounded, i.e., sup_k |u_k|_inf < infinity, where |.|_inf is the L-infinity norm (the maximum)? In other words, does there exist a constant C_A such that for any k and any x in A, |u_k(x)| < C_A?

If the answer is positive, please provide a reference or a proof.

If the answer is negative, please provide a counter-example. In that case, what are the conditions on the domain A to make this statement true?