Global Hölder regularity for nonhomogeneous elliptic systems with non linearity greater than two and with controlled growth

Giovanna Idone D.I.M.E.T. Faculty of Engineering, University of Reggio Calabria, Italy

                       e-mail idone@ing.unirc.it

2000 Mathematics Subject Classification: 35J65, 35J55. Keywords: Nonlinear Elliptic Systems, Global Hölder Regularity, Higher Gradient Summability.

The aim of this paper is to study the global Hölder continuity in Ω̄ of a solution u ∈ H^{1,q}(Ω) to the following Dirichlet problem

\{ \begin{matrix}u − g ∈ H_{0}^{1,q}(Ω) \\ ∑_{i=1}^{n} D_{i}a^{i}(x, Du) = ∑_{i=1}^{n} D_{i}F^{i}(x, u) − F^{0}(x, u, Du) in Ω \end{matrix}  (1)

where Ω is a bounded open set in R^{n}, n ≥ 2, and q is a real number > 2.

We will prove the following regularity theorem:

Theorem. There exists a number λ = λ(ν, M, n), 2 < λ < n, such that if u ∈ H^{1,q}(Ω) is a solution of the Dirichlet problem (1), ∂Ω is of class C^{2}, g ∈ H^{1, \frac{nq}{n−q} ,(\frac{μ}{q−1})}(Ω), 0 < \frac{μ}{q−1} < λ, F^{0}(x, u, Du), F^{i}(x, u), i = 1, 2, …, n, have controlled growth, then

Du ∈ L^{q, \frac{μ}{q−1}} (Ω)

and if n − q < \frac{μ}{q−1} < λ

u ∈ C^{0,α}(Ω̄, R^{N}) with α = 1 − \frac{n}{q} + \frac{μ}{q − 1} \frac{1}{q}.

An essential tool for proving the global Hölder continuity of u is the global higher sommability of the gradient.

The results above generalize previous results by S. Campanato, G. Idone and K.Widman and ensure the global Hölder continuity of the solutions for n ≤ q + 2 (in particular for n ≤ 4).