Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases - Grubb, Gerd (MATHEMATICA SCANDINAVICA, 2023)
Let P be a symmetric 2a-order classical strongly elliptic pseudodifferential operator with even symbol p(x,ξ) on R^n (0<a<1), for example a perturbation of (-Δ )^a. Let Ω R^n be bounded, and let P_D be the Dirichlet realization in L_2(Ω) defined under the exterior condition u=0 in R^nΩ. When p(x,ξ) and Ω are C^∞ , it is known that the eigenvalues λ_j (ordered in a nondecreasing sequence for j→ ∞) satisfy a Weyl asymptotic formula equation* λ _j(P_D)=C(P,Ω )j^2a/n+o(j^2a/n) for j→ ∞ , equation* with C(P,Ω) determined from the principal symbol of P. We now show that this result is valid for more general operators with a possibly nonsmooth x-dependence, over Lipschitz domains, and that it extends to P=P+P’+P”, where P’ is an operator of order <{2a, a+12} with certain mapping properties, and P” is bounded in L_2(Ω ) (e.g. P”=V(x)L_∞(Ω)). Also the regularity of eigenfunctions of P_D is discussed.