Character and Dimension Formulae for Queer Lie Superalgebra

We study transmission problems with free interfaces from one random medium to another. Solutions are required to solve distinct partial differential equations, ({{rm L}_{+}}) and ({{rm L}_{-}}), within their positive and negative sets respectively. A corresponding flux balance from one phase to another is also imposed. We establish existence and ({L^{infty}}) bounds of solutions. We also prove that variational solutions are non-degenerate and develop the regularity theory for solutions of such free boundary problems.