Ω=RegionDifference[ImplicitRegion[0<x<Pi&& 0<y<Pi +x (Pi-x)/Pi,{x,y}], Disk[{Pi/3,Pi/2},Pi/4]];
(* numerical Neumann eigenvalues  for Ω *)
NDEigenvalues[-Laplacian[u[x,y],{x,y}], u[x,y],{x,y}∈Ω,10]
(* numerical Dirichlet eigenvalues for Ω *)
NDEigenvalues[{-Laplacian[u[x,y],{x,y}],DirichletCondition[u[x,y]==0,True]},u[x,y], {x,y}∈Ω,10]