Abstract
We establish Holder continuity of generalized solutions of the Dirichlet problem, associated to a degenerate nonlinear fourth-order equation in an open bounded set , with data, on the subsets of where the behavior of weights and of the data is regular enough.
Highlights
IntroductionWhere Ω is a bounded open set of Rn, n > 4, 2 < p < n/2, max(2p,√n) < q < n, ν and μ are positive functions in
In this paper, we will deal ◦ (W (ν, μ, Ω))of the form with equations involving an operator A : W Ω) → Au =(−1)|α|DαAα x, ∇2u, |α|=1,2
We will assume that the right-hand sides of our equations, depending on unknown function, belong to L1(Ω)
Summary
Where Ω is a bounded open set of Rn, n > 4, 2 < p < n/2, max(2p,√n) < q < n, ν and μ are positive functions in. Ω with properties precised later, is the Banach space of all functions u : Ω → R with the properties |u|q, ν|Dαu|q, μ|Dβu|p ∈ L1(Ω), |α| = 1,. |β| = 2, and “zero” boundary values; ∇2u = {Dαu : |α| ≤ 2}. The functions Aα satisfy growth and monotonicity conditions, and in particular, the following strengthened ellipticity condition (for a.e. x ∈ Ω and ξ = {ξα : |α| = 1, 2}): Aα(x, ξ)ξα ≥ c2 ν(x) ξα q + μ(x) ξα p − g2(x),. |α|=2 where c2 > 0, g2(x) ∈ L1(Ω)
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