Homogenization of Elliptic Problems with Neumann Boundary Conditions in Non-smooth Domains

Uniform regularity estimates in homogenization theory of elliptic system with lower order terms

“…Let u ε be the weak solution to L ε (u ε ) = F in Ω and u ε = 0 on ∂Ω, where F ∈ L 2 (Ω; R m ). Then we have 16) where u satisfies L 0 (u) = F in Ω and u = 0 on ∂Ω. Moreover, if the coefficients of L ε satisfy (1.…”

Weighted gradient estimates for elliptic problems with Neumann boundary conditions in Lipschitz and (semi-)convex domains

“…The weighted Robin boundary value problem (R) p, ω is said to be uniquely solvable if, for any f ∈ L p ω (Ω; R n ) and F ∈ L p * ω p * /p (Ω), there exists a unique u ∈ W 1,p ω (Ω) such that (1.7) holds true with g ≡ 0. In particular, if α ≡ 0 in (1.6) and (1.8), then the Robin problem (R) p and the weighted Robin problem (R) p, ω are, respectively, the Neumann problem (N) p and the weighted Neumann problem (N) p, ω (see, for instance, [33,34,74]). The Neumann problem (N) p is said to be uniquely solvable if, for any f ∈ L p (Ω, R n ), F ∈ L p * (Ω) and g ∈ W −1/p,p (∂Ω) satisfying the compatibility condition Ω F(x) dx = g, 1 ∂Ω , there exists a u ∈ W 1,p (Ω), unique up to constants, such that (1.7) holds true.…”

“…Moreover, when A has the small BMO coefficients and Ω is a bounded Reifenberg flat domain, or A has partial small BMO coefficients and Ω is a bounded Lipschitz domain with small Lipschitz constant, the estimate (1.10) with g ≡ 0 and p ∈ (1, ∞) was established, respectively, in [16] and [29] for the Neumann problem (N) p . Furthermore, for the Neumann problem (N) p on a general Lipschitz domain, it was proved in [33,34] that, if A is symmetric and A ∈ VMO(R n ), then (1.10) holds true for any p ∈ ( 3 2 − ε, 3 + ε) when n ≥ 3, or p ∈ ( 4 3 − ε, 4 + ε) when n = 2, where ε ∈ (0, ∞) is a positive constant depending only on the Lipschitz constant of Ω and n. We point out that, when A := I in the Neumann problem (N) p , the range of p obtained in [33,34] is even sharp for general Lipschitz domains (see, for instance, [32]). In particular, if A is symmetric, A ∈ VMO(R n ) and Ω is convex, it was proved in [34] that (1.10) with F ≡ 0 holds true for the Neumann problem (N) p with any given p ∈ (1, ∞).…”

“…Furthermore, for the Neumann problem (N) p on a general Lipschitz domain, it was proved in [33,34] that, if A is symmetric and A ∈ VMO(R n ), then (1.10) holds true for any p ∈ ( 3 2 − ε, 3 + ε) when n ≥ 3, or p ∈ ( 4 3 − ε, 4 + ε) when n = 2, where ε ∈ (0, ∞) is a positive constant depending only on the Lipschitz constant of Ω and n. We point out that, when A := I in the Neumann problem (N) p , the range of p obtained in [33,34] is even sharp for general Lipschitz domains (see, for instance, [32]). In particular, if A is symmetric, A ∈ VMO(R n ) and Ω is convex, it was proved in [34] that (1.10) with F ≡ 0 holds true for the Neumann problem (N) p with any given p ∈ (1, ∞). Moreover, for any given p ∈ (1, ∞) and ω ∈ A p (R n ), the weighted estimate (1.11), with F L p * ω p * /p (Ω) replaced by F L p ω (Ω) , was established in [74] for the weighted Neumann problem (N) p, ω when Ω is a bounded (semi-)convex domain.…”

“…Remark 1.12. In the special case of Theorem 1.11 that Ω ⊂ R 3 is a bounded C 1 domain and A ∈ VMO(R 3 ), the global estimate (1.23) was established in [4, Theorem 1.1] via a different way, which is similar to that used in [33,34]. However, two inequalities used in [4, p. 8, line 10 and p. 11, line 12], which are essential for the proof of [4, Theorem 1.1], are not correct.…”