Abstract:
Let Ω, Ω1, Ω2, . . . ⊂ R d be open sets, λ > 0 sufficiently large and f, f1, f2, . . . ∈ L ∞ R d with d ≥ 2. Consider the elliptic operator A on R d formally given by Au = − X d i,j=1 ∂j (aij∂iu) − X d i=1 ∂i(biu) +X d i=1 ci∂iu + a0u. We study conditions on the sequence of open sets (Ωn)n∈N such that solutions of ( Aun + λun = fn in D′ (Ωn), un = 0 on R d \ Ωn, converge uniformly to the solution of the corresponding limit problem ( Au + λu = f in D′ (Ω), u = 0 on R d \ Ω. The resolvent convergence in L 2 for general second-order elliptic operators in divergence form was studied extensively in [15]. The notion of ‘regular convergence’ of domains was introduced in [3] as a sufficient condition for the uniform convergence of the resolvents of the Laplacian operator (i.e., for A = −∆). Little is known about the uniform convergence on varying domains for general elliptic operators. We find necessary and also sufficient conditions on the sequence of open sets (Ωn)n∈N and on the coefficient functions of A such that whenever limn→∞ fn = f weakly∗ in L ∞(R d ), it follows that the sequence of solutions (un)n∈N converges uniformly to the solution u on Ω under two different types of convergence of open sets .
