Isoperimetry for semilinear torsion problems in Riemannian two-manifolds

Given (M2,g) – a smooth, complete Riemannian 2-manifold. This note discusses three types of isoperimetric properties arising from , the non-negative solution to the semilinear (0⩽γ<1) torsion problem Δgu=−uγ in O subject to u|∂O=0 for a relatively compact domain O⊂M2 with smooth boundary ∂O.