Abstract
We show existence theorems of global strong solutions of Dirichlet problem for second-order fully nonlinear systems that satisfy the Campanato's condition of ellipticity. We use the Campanato's near operators theory.
Acknowledgments
Part of the special issue, “Variational Analysis and Applications.”
Notes
1Indeed it is enough to assume
We observe that by these assumptions we have that a. e. in Ω it results F(x, u, Du, 0) = 0 and g(x, 0, 0) = 0.
2In the case of linear operator this condition is equivalent to Cordes condition (see [Citation15]). The connection between this and others ellipticity conditions are studied in [Citation16].
3
p = (p
1,…, p
n
), p
i
∈ ℝ
N
, if p ∈ ℝ
nN
. (|)
N
and ‖ ‖
N
are, respectively, the scalar product and the norm in ℝ
N
. is the vector space of N-ples
of n × n matrices with
i, j = 1,…, n, k = 1,…, N, equipped with the scalar product:
.
4These conditions are necessary also in the case of linear equations since, for example, if λ > 0 is a eigenvalue of Δ, then as everybody knows, the problem
5Here we use: for any a, b ∈ ℝ+ we have (γa + δb)2 ≤ γ(γ + δ)a 2 + δ(γ + δ)b 2.
6Where t 0 ≤ t 1, so that (−t 0, t 0) is the neighborhood V(x 0) of Theorem 2.2.
7We don't make any restriction if we set, in Condition A
x
, a(x) = 1. It is enough to assume ,
,
.
,
,
verify in the same way the required conditions.
8Indeed
The last estimate is obtained by Lemma 3.1 setting k = (γ + δ)2.
9It easy to show that ϵ0 ∈ (0, 1), because we can consider following system
10By Hypothesis (Equation3), we have F(x, u, p, 0) = 0 and moreover we can assume a(x) = 1 without loss of generality.