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Let $A$ be a bounded domain in $\mathbb R^d$, $d>1$, and $\{u_k\}$ is the set of all $L^2$-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., $sup_k \|u_k\|\infty |u_k\|_\infty < \infty$, infty$, where $\|.\|\infty$\|.\|_\infty$ is the $L^\infty$-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?

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Let A $A$ be a bounded domain in R^d, d>1$\mathbb R^d$, $d>1$, and {u_k} $\{u_k\}$ is the set of all L2-normalized $L^2$-normalized Laplacian eigenfunctions on A $A$ with Dirichlet boundary condition (i.e., |u_k|_2 $\|u_k\|_2 = 1)1$).

Is it true that these eigenfunctions are uniformly bounded, i.e., $sup_k |u_k|_inf \|u_k\|\infty < infinity, \infty$, where |.|_inf $\|.\|\infty$ is the L-infinity norm $L^\infty$-norm (the maximum)? In other words, does there exist a constant C_A $C_A$ such that for any k $k$ and any x in A$x\in A$, |u_k(x)| $|u_k(x)| < C_AC_A$?

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?

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# Boundness of Laplacian eigenfunctions

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?