Wellposedness of variable-coefficient conservative fractional elliptic differential equations (Q2840390)

This paper deals with the Dirichlet boundary value problem of a one-sided variable-coefficient conservative fractional elliptic differential equation in one space dimension. The authors recall the existing results for the Galerkin weak formulation with a constant diffusivity coefficient which were proved by \textit{V. J. Ervin} and \textit{J. P. Roop} [Numer. Methods Partial Differ. Equations 22, No. 3, 558--576 (2006; Zbl 1095.65118)] and present a counterexample to show that the bilinear form of the weak formulation is indefinite. Also, a Petrov-Galerkin weak formulation for the variable-coefficient conservative fractional elliptic differential equation is derived.