Abstract
The Meyer’s estimate of solutions to Zaremba problem for second-order elliptic equations in divergent form
Highlights
In this paper we estimate solutions to the Zaremba problem for elliptic equations in bounded Lipschitz domain D ∈ Rn, where n > 1, of the formL u := div(a(x)∇u) (1)with uniformly elliptic measurable and symmetric matrix a(x) = {ai j (x)}, i.e. ai j = a ji and n α−1|ξ|2 ≤ ai j (x)ξi ξj ≤ α|ξ|2 for almost all x ∈ D and for all ξ ∈ Rn . (2) i, j =1Below we assume that the set F ⊂ ∂D is closed and denote G = ∂D \ F
We assume that the set F ⊂ ∂D is closed and denote G = ∂D \ F
In the case of homogeneous Dirichlet problem for (3) with right-hand side f ∈ Lp (D), where p > 2, the increased integrability of the gradient of solutions to divergent uniformly elliptic equations with measurable coefficients on the plane follows from the results of [3]
Summary
Meyers estimates, Mixed problem, Embedding theorems, Capacity, Rapidly alternating type of boundary conditions. In this paper we estimate solutions to the Zaremba problem for elliptic equations in bounded Lipschitz domain D ∈ Rn, where n > 1, of the form
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