**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.

**Exercise 2.**

- 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