Implicit Partial Differential Equations:
different approaching methods
Creators: name: Pisante, Giovanni
Publisher: name: Università degli Studi di Napoli Federico II
Year: 2004

description: Implicit Partial Differential Equations:
different approaching methods
PhD Thesis of Giovanni Pisante
Abstract
In the last few decades the interest of scientists in nonlinear analysis
has been constantly increasing and nonlinear partial differential
equations have become one of the main tools of modern mathematical
analysis.
In this thesis we deal with a family of nonlinear partial differential
equations, the so called ”implicit partial differential equations”.
The problem of solving this type of equations has been approached
looking at it from different points of view and consequently different
approaching methods were developed. The choice of the method to
apply depends on what properties we require on the solutions of the
equation.
The principal aim of this work is to present some different ways
to establish the existence of solutions of implicit partial differential
equations. In particular we focus our attention on two methods: the
Baire category method and the viscosity method.
The first one is a functional analytic method based essentially on
the Baire category theorem. It was introduced by Cellina in 1980 to
prove density properties of solutions of some differential inclusions and
it has been extensively studied and extended by many authors. In
particular Dacorogna and Marcellini in a series of papers extended the
method to the framework of implicit differential equations looking at
these as differential inclusions. We present this theory, making a survey
on the recent results that we can find in literature and we present new
proofs of general existence theorems. We would point out that the
Baire category method has the ”defect” to be purely existential, that
is it can establish only the existence of solutions (in fact it ensures the
existence of infinitely many solutions), but it does not give any other
information.
The viscosity approach for this type of equations is one of the oldest
methods applied in this field and it has received much attention
since its introduction by Crandall and Lions in 1982. It deals essentially
with scalar problems, i.e. the Hamilton-Jacobi equations, and it
is less general than the previous one. Nevertheless it has the advantage
that it gives much more information than existence of solutions;
for instance, uniqueness, stability, maximality, and, under suitable hypotheses,
explicit formulas.
i
Here we discuss about the existence of viscosity solutions for the
Hamilton-Jacobi equations looking at it from a ”geometrical” point of
view. The interest in finding geometrical conditions comes out from
the idea to compare the two above methods. Indeed, if we want to
use the viscosity method as a criterium to select, among the infinitely
many solutions given by the Baire category method, a preferred one, we
immediately deal with restrictive geometrical compatibility conditions.
In particular, generalizing these results, we present new geometrical
conditions sufficient and, in some cases, necessary for the existence of
viscosity solution of Hamilton-Jacobi equations with non necessarily
convex hamiltonian.
ii