(* verifying the Faber--Krahn inequality for regular n-gons, n=5,...,20 *)
evs=Table[NDEigenvalues[{-Laplacian[u[x,y],{x,y}],DirichletCondition[u[x,y]==0,True]},u[x,y], {x,y}∈RegularPolygon[n],1][[1]],{n,5,20}];
asympt=1+4Zeta[3]/n^3+(12-2BesselJZero[0,1]^2)Zeta[5]/n^5;
Show[ListPlot[Table[{n,evs[[n-4]] /(Pi/Area[RegularPolygon[n]] BesselJZero[0,1]^2)},{n,5,20}],PlotStyle->{Black,PointSize[Large]},AxesOrigin->{4,1}],Plot[asympt,{n,5,20},PlotRange->All]]