In the first part of the talk we will compare three different notions of weak solutions of the -Laplace equation: Sobolev weak solutions based on distributional derivatives, semi-continuous weak solutions based on the comparison principle, and viscosity solutions based on generalized point-wise derivatives or jets. For it is easy to show that Sobolev weak solutions are semi-continuous weak solutions and that these are viscosity solutions. In fact, viscosity solutions are semi-continuous weak solutions. A sketch of the proof, ultimately based on Jensen's comparison principle will be presented.
In the second part, we concentrate on the case ,
where we are forced to use viscosity solutions.
We will present applications to -harmonic
functions, and discuss the -eigenvalue problem.