Geometric analysis of perturbed contact instantons with Legendrian boundary conditions

In the present article, we provide analytic foundation of the following nonlinear elliptic system, called the \emph{Hamiltonian-perturbed contact instanton equation}, $$ (du - X_H \otimes \gamma)^{\pi(0,1)} = 0, \quad d(e^{g_{H, u}}u^*(\lambda + H \otimes \gamma)\circ j) = 0 $$ associated to a contact triad $(M,\lambda,J)$ and contact Hamiltonian $H$ and its boundary value problem under the Legendrian boundary condition. (1) We identify the correct choice of the action functional for perturbed contact Hamiltonian trajectories which provides a gradient structure for the system and derive its first variation formula. (2) We identify the correct choice of the energy for the bubbling analysis for the finite energy solutions for the equation. (3) We develop elliptic regularity theory for the solution, called \emph{perturbed contact instantons}: We first establish a global $W^{2,2}$ bound by the Hamiltonian calculus and the harmonic theory of the vector-valued one form $d_Hu : = du - X_H(u)\otimes \gamma$ and its relevant Weitzenb\"ock formulae utilizing the contact triad connection of the contact triad $(M,\lambda, J)$. Then we establish $C^{k,\alpha}$-estimates by an alternating boot-strap argument between the $\pi$-component of $d_Hu$ and the Reeb-component of $d_Hu$. Along the way, we also establish the boundary regularity theorem of $W^{1,4}$-weak solutions of perturbed contact instanton equation under the weak Legendrian boundary condition. (4) Based on this regularity theory, we prove an asymptotic $C^\infty$ convergence result at a puncture under the hypothesis of finite energy.