Laplacian eigenfunction

A Laplacian eigenfunction satisfies $-\Delta u=\lambda u$ together with specified boundary conditions.

A Dirichlet Laplacian eigenfunction vanishes on the boundary and satisfies $-\Delta u=\lambda u$.

For $n\ge 3$ and a bounded smooth domain, a Dirichlet Laplacian eigenfunction with eigenvalue $\lambda$ satisfies the displayed estimate. A signed power test for a Laplacian eigenfunction and the zero-boundary Sobolev inequality give a Moser iteration with exponents $p_j=2[n/(n-2){]}^j$. The geometric series ${\sum }_{j}1/p_j=n/4$ gives the exact power of $\lambda$, while ${\sum }_j(\log p_j)/p_j<\infty$ controls the remaining constant. The constant depends only on dimension, because the zero-boundary Sobolev inequality follows by extension by zero to the whole space.

A Dirichlet Laplacian eigenvalue is a number $\lambda$ for which $-\Delta u=\lambda u$ has a nonzero solution with zero boundary trace. At such a value, the shifted operator $\Delta +\lambda$ is not invertible.