The variable exponent periodic boundary value problems on the Lyapunov open curve

Abstract This study investigates the periodic Riemann boundary value problem on a Lyapunov open curve within the framework of variable exponent spaces. By extending the open curve to a closed one, we demonstrate that this extension preserves the class of the problem. Periodicity is eliminated through a tangent transformation, and a weight function is introduced to handle discontinuous coefficients. The equivalence of the problem is established using weighted variable indices. By analyzing the singularities at the endpoints, we construct the corresponding canonical function, which yields an explicit solution. Additionally, the problem is extended from the single open curve case to the more general scenario involving multiple open curves.