Exercise 1. Let be a linear operator such that:
- is a local operator, that is if on a neighborhood of then ;
- If has a global maximum at with then .
Show that for and ,
where , and are continuous functions on such that for every , and the matrix is symmetric and nonnegative.
- Show that if are functions and are also then,
- Show that if is a function and is also ,
Exercise 3 On , let us consider the diffusion operator where is a function. Show that is symmetric with respect to the measure .
Exercise 4 (Divergence form operator). On , let us consider the operator where is the divergence operator defined on a function by
and where is a field of non negative and symmetric matrices. Show that is a diffusion operator which is symmetric with respect to the Lebesgue measure.
Exercise 5: On , we consider the divergence form operator
where is a smooth field of positive and symmetric matrices that satisfies
for some constant . Show that with respect to the Lebesgue measure, the operator is essentially self-adjoint on