We consider a nonlinear integral operator which involves a Nemytskij type operator and which appears in the applications as a pull-back of layer potential operators. We prove an analyticity result in Schauder spaces by splitting the operator into the composition of a nonlinear operator acting into Roumieu classes and a composition operator.Peer reviewe

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Since 1984, he has worked on boundary value problems in complex analysis and numerical methods to singular integral equations and has become the leading expert in this area in P. R. China. His research interests include boundary value problems in hyper-complex analysis and asymptotic analysis of orthogonal polynomials using Riemann-Hilbert technique. Professor Jinyuan Du has not only devoted his scientific activities to fundamental theoretical research, but in his early years also to research mathematical applications to agriculture.

We investigate the effective thermal conductivity of a two-phase composite with thermal resistance at the interface. The composite is obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material. The diameter of each inclusion is assumed to be proportional to a positive real parameter ǫ. Under suitable assumptions, we show that the effective conductivity can be continued real analytically in the parameter ǫ around the degenerate value ǫ = 0, in correspondence of which the inclusions collapse to points.

We consider a sufficiently regular bounded open connected subset Ω of R n such that 0 ∈ Ω and such that R n \ cl Ω is connected. Then we choose a point w ∈]0, 1[ n . If ǫ is a small positive real number, then we define the periodically perforated domain T (ǫ) ≡ R n \ ∪ z∈Z n cl(w + ǫΩ + z). For each small positive ǫ, we introduce a particular Dirichlet problem for the Laplace operator in the set T (ǫ). More precisely, we consider a Dirichlet condition on the boundary of the set w + ǫΩ, and we denote the unique periodic solution of this problem by u[ǫ]. Then we show that (suitable restrictions of) u[ǫ] can be continued real analytically in the parameter ǫ around ǫ = 0.