Prescribing the Gaussian Curvature in a Subdomain of \mathbb {S}^2 with Neumann Boundary Condition

The problem of prescribing the Gaussian curvature under a conformal change of the metric leads to the equation: −u+2=2K(x)eu. Here we are concerned with the problem posed on a subdomain ⊂ S2 under Neumann boundary condition. By using min-max techniques we give a new existence result that generalizes and unifies previous work on the argument. For sign-changing K, compactness of solutions is not known in full generality, and this difficulty is bypassed via an energy comparison argument.

prescribed gaussian curvature problem; neumann boundary condition; variational methods

Prescribing the Gaussian Curvature in a Subdomain of \mathbb {S}^2 with Neumann Boundary Condition Articles