Geodesic
Existence and uniqueness of classical solutions to second-order quasilinear elliptic equations
Electronic journal of differential equations, Tome 2010 (2010)
This article studies the existence of solutions to the second-order quasilinear elliptic equation

$ -\nabla \cdot(a(u) \nabla u) + v\cdot \nabla u=f $

with the condition $u(x_0)=u_0$ at a certain point in the domain, which is the 2 or the 3 dimensional torus. We prove that if the functions a, f, 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.
Classification : 35A05
Keywords: existence, uniqueness, quasilinear, elliptic 
@article{EJDE_2010__2010__a5,
     author = {Denny,  Diane L.},
     title = {Existence and uniqueness of classical solutions to second-order quasilinear elliptic equations},
     journal = {Electronic journal of differential equations},
     year = {2010},
     volume = {2010},
     zbl = {1198.35105},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2010__2010__a5/}
}
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Denny,  Diane L. Existence and uniqueness of classical solutions to second-order quasilinear elliptic equations. Electronic journal of differential equations, Tome 2010 (2010). http://geodesic.mathdoc.fr/item/EJDE_2010__2010__a5/