Integrability of very weak solution to the Dirichlet problem of nonlinear elliptic system
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Date
2019-01-02
Authors
Tong, Yuxia
Liang, Shuang
Zheng, Shenzhou
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
This article concerns the higher integrability of a very weak solution u ∈ θ + W1,r0(Ω) for max{1, p - 1} < r < p < n to the Dirichlet problem of the nonlinear elliptic system
-DαAαi(x, Du) = Bi(x, Du) in Ω,
u = θ on ∂Ω,
where A(x, Du) = (Aαi(x, Du)) for α = 1,..., n and i = 1,..., m, and each entry of B(x, Du) = Bi(x, Du)) for i = 1,..., m satisfies the monotonicity and controllable growth. If θ ∈ W1,q(Ω) for q > r, then we derive that the very weak solution u of above-mentioned problem is integrable with
u ∈ {θ + Lq*weak(Ω) for 1 ≤ q < n,
θ + Lτ(Ω) for q = n and 1 < τ < ∞,
θ + L∞(Ω) for q > n,
provided that r is sufficiently close to p, where q* = qn/(n - q).
Description
Keywords
Integrability, Very weak solution, Nonlinear elliptic system, Controllable growth
Citation
Tong, Y., Liang, S., & Zheng, S. (2019). Integrability of very weak solution to the Dirichlet problem of nonlinear elliptic system. <i>Electronic Journal of Differential Equations, 2019</i>(01), pp. 1-11.
Rights
Attribution 4.0 International