Dirichlet Laplacian ($-{\Delta }_D$)

The nonnegative Dirichlet Laplacian is the self-adjoint realization of $-\mathrm{div}\mathrm{grad}$ with zero boundary values. On a bounded domain, its weak form is the Dirichlet energy on $H_0^1$. For polygonal domains the weak operator domain is preferable to a requirement of smoothness at corners.