We know that for a direct problem with Dirichlet Boundary Condition (with Laplacian operator) that if two domains $M_1$ and $M_2$ are such that $M_1 \subset M_2$, then $\lambda(M_2) \leq \lambda(M_1)$, and hence, $N_{M_1} (\lambda) \leq N_{M_2} (\lambda)$. Why doesn't exist a similar result for a direct problem with Neumann Boundary Condition, i.e. $M_1 \subset M_2 \not \Rightarrow N_{M_1} (\lambda) \leq N_{M_2} (\lambda)$? Is there anyone could give me a **clever counterexample?** I think this is related by the fact that $H_0^1(M_1) \subset H_0^1(M_2)$, but $H^1(M_1) \not \subset H^1(M_2)$.

**Precision :** $N(\lambda) \equiv \text{the number of eigenvalues less than } \lambda$.