Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 82, pp. 1-21.

Title: Existence and uniqueness of classical solutions to
       second-order quasilinear elliptic equations

Author: Diane L. Denny (Texas A&M Univ., Corpus Christi, TX, USA)

Abstract:
 This article studies the existence of solutions to the
 second-order quasilinear elliptic equation
 $$
 -\nabla \cdot(a(u) \nabla u) +\mathbf{v}\cdot \nabla u=f
 $$
 with the condition $u(\mathbf{x}_0)=u_0$ at a certain point
 in the domain, which is the 2 or the 3 dimensional torus.
 We prove that if the functions a, f, $\mathbf{v}$ satisfy
 certain conditions, then there exists a unique classical solution.
 Applications of our results include stationary heat/diffusion
 problems with convection and with a source/sink, when
 the value of the solution is known at a certain location.
 	 
Submitted April 13, 2010. Published June 18, 2010.
Math Subject Classifications: 35A05.
Key Words: Existence; uniqueness; quasilinear; elliptic.