Existence of a unique solution to a quasilinear elliptic equation

The purpose of this paper is to prove the existence of a unique classical solution u ( x ) to the quasilinear elliptic equation − ∇ ⋅ ( a ( u ) ∇ u ) + v ⋅ ∇ u = f , where u ( x 0 ) = u 0 at x 0 ∈ Ω and where n ⋅ ∇ u = g on the boundary ∂Ω. We prove that if the functions a, f, v, g satisfy certain conditions, then a unique classical solution u ( x ) exists. Applications include stationary heat/diffusion problems with convection and with a source/sink, where the value of the solution is known at a spatial location x 0 ∈ Ω , and where n ⋅ ∇ u is known on the boundary.