For this opportunity of giving my first webinar in the COVID era, which we hope will finish

soon, not the seminar, the era. And so this is intended for a large audience. It's a sort

of historical approach to a subject which is, well, quite technical singularities of

solutions to Hamilton-Jacobi equation. We PDE people, we study nonlinear equations and

usually what we study at first is the basic theory, namely existence, uniqueness, well-posedness,

stability, and so on. But then there are a lot of other questions that need to be addressed

as the research goes on. One of these is the study of the, well, of the regularity of solution

and of the sets of points where solutions are not regular. And to introduce such a topic,

let me quote this paper by Kanning and Sobolewski from 2016 about Hamilton-Jacobi equation.

So they say that the evolutionary Hamilton-Jacobi equation appears that we know very well, this

is the time derivative plus fully nonlinear function of the space gradient, appears in

diverse mathematical models ranging from analytical mechanics to combinatorics, condensed matter,

turbulence, and cosmology. And in many of these applications, the objects of interest are

described by singularities of solutions, which inevitably appear for generic initial data

after a finite time due to the nonlinearity of hj of the equation. Therefore, one of the

central issues, both for theory and applications, is to understand the behavior of the system

after singularities form. And this is the overview of the talk. So we will, well, consider

a fully nonlinear first order PDE of Hamilton-Jacobi type, which means essentially that the Hamiltonian

h is convex in the momentum variable in the gradient. We take a viscosity solution u in

some domain. And then we want to study, we know that viscosity solutions, as we saw just

before, they are in general non-smooth. So the equation has to be interpreted in a suitable

generalized sense. And the object of our study is exactly the set of points at which the

gradient does not exist. Since we are assuming Lipschitz continuity, this is a small set,

a negligible set with respect to Lebesgue measure. Nevertheless, sets of measure zero

can be very nasty. And you may want to, well, to control sides, the form, the evolution

of such sets. The typical examples we have in mind are the Hamilton-Jacobi equation from

mechanics, time dependent. So set on the whole space Rn with an initial condition, the Cauchy

problem, but also stationary. Let's say Iconal type equations, which means, I mean, some

norm of the gradient of the solution squared equals some potential. And this is the typical,

one of the most general case. A is of course a matrix, symmetric positive definite matrix

in omega. And also we can study the so-called weak-can solutions, solutions of other stationary

problems on Rn on manifolds, which are defined either through the viscosity equation or with

the approach due to fatigue using calibrated curves and then, and then Lux-Elinik semigroups.

We will get back to this point later, as you will see. But let me start with an example

coming from nature of the singularities of a solution of Hamilton-Jacobi equation. Let

me take the distance function from Sfs in Rn, which is a well-known object. Well, this

function is, well, is non-smooth in general. In fact, always, if you take it, if you take

like in these two pictures, the distance from the boundary of a bounded domain, even though

if the boundary is very smooth, like in this example, when you picture the distance, and

here you picture the distance using the transport theory, optimal transport theory. Sandpiles

at the equilibrium regime show a graph that actually, well, graphs the distance function

times a certain parameter, which depends on the material, distance from the boundary of

the table. And here you can see the singularities appear.

Okay. Let me connect this notion of singularities with other characteristic notions for solutions

of nonlinear first-order PDEs. So let's start from the Hamilton-Jacobi equation of mechanics.

And let's recall that the basic tool in the analysis of first-order PDEs is the method

of characteristics. That means you want to solve a first-order PDE of this form. When

then you can look at a system of ODEs, the so-called characteristics, starting from the

initial manifold, which in this case is Rn t equal to zero, even initial condition for

the state, and an initial condition for the momentum, which is exactly the gradient of