There exists a planar domain with piecewise smooth boundary and one hole such that the second eigenfunction for the Laplacian with Neumann boundary conditions attains its maximum and minimum inside the domain.
Krzysztof Burdzy. "The hot spots problem in planar domains with one hole." Duke Math. J. 129 (3) 481 - 502, 15 September 2005. https://doi.org/10.1215/S0012-7094-05-12932-5