Global Solvability of Dirichlet Problem for Fully Nonlinear Elliptic Systems

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.

Keywords:

• Fully nonlinear elliptic systems
• Near operators theory

Mathematics Subject Classification:

• 35J47
• 35J60

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 [15]). The connection between this and others ellipticity conditions are studied in [16].

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

is not well posed.

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

and we observe that it has solutions ν such that 0 < ν₁ < ν < ν₂ < 1, where , .

¹⁰By Hypothesis (3), we have F(x, u, p, 0) = 0 and moreover we can assume a(x) = 1 without loss of generality.