Parametric analyticity of semilinear eigenvalue problem and its uncertainty quantification

In this talk, we will discuss parametric semilinear elliptic eigenvalue problem. For last few decades, parametric elliptic problem has received tremendous attentions. Even if its ubiquity, the eigenvalue problem has only studied recently. Also, all the problems so far are only focused on linear case. In this study, to the best of our knowledge, we first attempt semilinear eigenvalue problem. A special case of our problem would be Gross Pitaevskii equation which describes super-fluidity and super-conductivity with random potential function. The main obstacle was showing analyticity of the ground state because the mixed derivative of them has no meaning without the analyticity. This obstacle is resolved by using implicit function theorem and multidimensional complex analysis. With careful analysis, we obtain the bound of mixed derivatives. With this bound, we suggest a method of uncertainty quantification for the ground state.