Δ[u_]:=Laplacian[u[x,y],{x,y}];
(* Neumann eigenvalues for the unit square *)
DEigenvalues[{-Δ[u]},u[x,y],{x,y}∈Rectangle[{0,0},{1,1}],10]
(* Dirichlet eigenvalues for the unit disk *)
DEigenvalues[{-Δ[u],DirichletCondition[u[x,y]==0,True]},u[x,y],{x,y}∈Disk[],10]
(* Dirichlet eigenvalues for  the isosceles right triangle with sides π *)
DEigenvalues[{-Δ[u],DirichletCondition[u[x,y]==0,True]},u[x,y],{x,y}∈Triangle[{{0,0},{Pi,0},{0,Pi}}],10]