An existence result for a nonlinear transmission problems

Original · EN

Let Ωᵒ and Ωⁱ be open bounded subsets of Rⁿ of class C¹,α such that the closure of Ωⁱ is contained in Ωᵒ. Let fᵒ be a function in C¹,α(∂Ωᵒ) and let F and G be continuous functions from ∂Ωⁱ to R. By exploiting an argument based on potential theory and on the Leray-Schauder principle we show that under suitable and completely explicit conditions on F and G there exists at least one pair of continuous functions (uᵒ, uⁱ) such that { arrayll Δuᵒ=0&in ΩᵒΩⁱ, Δuⁱ=0&in Ωⁱ, uᵒ(x)=fᵒ(x)&for all x∈∂Ωᵒ, uᵒ(x)=F(x,uⁱ(x))&for all x∈∂Ωⁱ, νΩᵢ·∇ uᵒ(x)-νΩᵢ·∇ uⁱ(x)=G(x,uⁱ(x))&for all x∈∂Ωⁱ, array. where the last equality is attained in certain weak sense. In a simple example we show that such a pair of functions (uᵒ, uⁱ) is in general neither unique nor local unique. If instead the fourth condition of the problem is obtained by a small nonlinear perturbation of a homogeneous linear condition, then we can prove the existence of at least one classical solution which is in addition locally unique.