Von Kármán equations in L^p spaces

Abstract

We show the existence of solution in L ^p spaces for a generalized form of the classical Von Kármán equations where the coefficients of nonlinear terms are variable. We use Campanato's near operators theory.

Keywords:

• nonlinear elliptic equations
• Von Kármán equations
• near operators theory

AMS Subject Classifications::

• 35J60
• 35Q74
• 74H40
• 74B20
• 74K20

Acknowledgements

We thank Professor Piero Villaggio for his useful comments and remarks.

Notes

Notes

1. We set: , while are the exterior normal derivatives at the boundary.

2. d _Ω is the diameter of Ω, that is d _Ω = sup{|X − Y|, X, Y ∈ Ω}, and

where α = (α₁, … , α _n ). Moreover we set D ⁰ u = u.

3. We set: .

4. We denote by ℒ^2,λ(Ω), 0 < λ < n + 2, the vector space of the functions u ∈ L ²(Ω) such that

where Ω(x ₀, σ) = B(x ₀, σ) ∩ Ω, with B(x ₀, σ) = {x ∈ ℝ ⁿ : ‖x − x ₀‖ ≤ σ} andThe quantity (5.3) is a seminorm and ℒ^2,λ(Ω) is a Banach space equipped with the normIf the boundary of Ω is Lipschitz, if 0 < λ < n then L ^2,λ(Ω) and ℒ^2,λ(Ω) are isomorphic, while if n < λ ≤ n + 2 then ℒ^2,λ(Ω) is isomorphic to the space of Hölder continuous functions with 11.

5. H ^2,2,λ(Ω) is the space of the functions to which derivatives of second order belong to ℒ^2,λ(Ω).