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In this paper we generalise the results proved in [N. Katzourakis, An$L^\infty$ regularisation strategy to the inverse source identification problemfor elliptic equations, SIAM J. Math. Anal. 51:2, 1349-1370 (2019)] by studyingthe ill-posed problem of identifying the source of a fully nonlinear ellipticequation. We assume Dirichlet data and some partial noisy information for thesolution on a compact set through a fully nonlinear observation operator. Wedeal with the highly nonlinear nonconvex nature of the problem and the lack ofweak continuity by introducing a two-parameter Tykhonov regularisation with ahigher order $L^2$ "viscosity term" for the $L^\infty$ minimisation problemwhich allows to approximate by weakly lower semicontinuous cost functionals.

On the inverse source identification problem in $L^\infty$ for fully nonlinear elliptic PDE