Resumen de Análisis de ecuaciones elípticas completamente no lineales, impropias respecto a términos de orden cero.
0.1. Historical precedents and aims The modern theory of Partial Differential Equations is constructed around the natural notion of solution associated to each precise kind of problem.
In the classical theory it is assumed that the solutions satisfy the equation pointwise since they are regular enough to allow all the required derivatives.
When trying to find solutions, it is sometimes better to enlarge the class of functions where we look for the solution, interpreting the equation in some generalized or weak sense. In a second step, it is studied if the generalized solution we have found is more regular, desirably classical, often as a consequence of the fact that it is a solution of the equation.
This strategy of dividing the problem has a long tradition; for example, it was necessary in the classical Dirichlet's work on his variational principle and it gave rise to Weirstrass' and then Hilbert's works and the theory of weak solutions with the notion of weak solution as is known nowadays, mainly based on integration by parts. We will not make much use of this kind of solutions along this work.
Another classical example is the eikonal equation |u|2 = 1 which appears in geometric optics. The structure of the equation itself makes it impossible to use integration by parts in a sensible way. For this example, it is well known the Monge's method of characteristics, which yields local solutions, and more recently, in the decade of 1980, the notion of viscosity solution introduced by P.
L. Lions and M. Crandall.
