Coefficient identification of the regularized p-Stokes equations

The Antarctic and Greenland ice sheet simulation is challenging due to unknown parameters in the $p$-Stokes equations. In this work, we prove the existence of a solution to a parameter identification for the ice rheology and the friction coefficient. Additionally, we verify G\^ateaux differentiability of the coefficient-to-state operator by extending a similar result for distributed control. Moreover, we have more complicated boundary conditions. We only have to add a small diffusion term and assume the nonlinear exponent, which is given in applications, to be small enough to obtain the results. Finally, we state the adjoint equation and prove existence and uniqueness of a solution for this equation.