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In Part I, we establish nonlinear ellipticity of the equation of contactinstantons with Legendrian boundary condition on punctured Riemann surfaces byproving the a priori elliptic coercive estimates for the contact instantonswith Legendrian boundary condition, and prove an asymptotic exponential$C^\infty$-convergence result at a puncture under the uniform $C^1$ bound. Weprove that the asymptotic charge of contact instantons at the punctures underthe Legendrian boundary condition vanishes. This eliminates the phenomenon ofthe appearance of spiraling cusp instanton along a Reeb core, which removes theonly remaining obstacle towards the compactification and the Fredholm theory ofthe moduli space of contact instantons in the open string case, which plaguesthe closed string case. In Part II, we derive an index formula which computesthe virtual dimension of the moduli space. These results are the analytic basisfor the sequels [Oh21a]-[Oh22b] and [OYb] containing applications to contacttopology and contact Hamiltonian dynamics.

Contact instantons with Legendrian boundary condition: a priori estimates, asymptotic convergence and index formula