On the Variable Exponent Riemann Boundary Value Problem for Liapunov Open Curve

In this paper, we first study the Riemann boundary value problem for Liapunov open curve in variable exponent space, we supplement the open curve as a closed one, and converted the problem to Riemann boundary value problem for closed curve in the variable exponent space, we solve the problem by discussing the singularity of the endpoints. Then we use these results to solve Hilbert boundary value problem for piecewise Liapunov closed curve in the variable exponent space, we also discuss the singularity of the discontinuity, we obtain the solvable conditions and explicit solutions of the Hilbert problem for piecewise Liapunov closed curve in the variable exponent space.