Differential equations with degeneration and resurgent analysis (Q633568)

The paper is devoted to the study of asymptotic expansions for degenerate elliptic equations of the form \(H(-r^{2}(d/dr), r)u=f\), where \(H(p,r):B_{1} \rightarrow B_{2}\) is an operator valued symbol, \(B_{1}, B_{2}\) are Banach spaces. The symbol \(H\) is polynomial in \(p\) and analytic in \(r\) in some neighborhood of \(r=0\). Endless continuability of the solutions is proven and their asymptotic expansions are obtained, firstly in the scalar case, \(B_{1}=B_{2}=\mathbb{C}\), and next in the general case of Fredholm equations with endless continuable right hand side. A main tool in this study is the resurgent analysis developed by J. Ecalle in the eighties of the last century.