Homework 1. MA5016. Due September 13 in class

Exercise 1. Let L: \mathcal{C}^{\infty} (\mathbb{R}^n) \rightarrow \mathcal{C} (\mathbb{R}^n) be a linear operator such that:

  • L is a local operator, that is if f=g on a neighborhood of x then Lf(x)=Lg(x);
  • If f \in \mathcal{C}^{\infty} (\mathbb{R}^n) has a global maximum at x with f(x)\ge 0 then Lf (x) \le 0.

Show that for f \in \mathcal{C}^{\infty} (\mathbb{R}^n) and x \in \mathbb{R}^n,

Lf(x)=\sum_{i,j=1}^n \sigma_{ij} (x) \frac{\partial^2 f}{ \partial x_i \partial x_j} +\sum_{i=1}^n b_i (x)\frac{\partial f}{\partial x_i} -c(x)f(x),

where b_i, c and \sigma_{ij} are continuous functions on \mathbb{R}^n such that for every x \in \mathbb{R}^n, c(x) \ge 0 and the matrix (\sigma_{ij}(x))_{1\le i,j\le n} is symmetric and nonnegative.

 

Exercise 2.

  • Show that if f,g :\mathbb{R}^n \rightarrow \mathbb{R} are C^1 functions and \phi_1,\phi_2: \mathbb{R} \rightarrow \mathbb{R} are also C^1 then,
    \Gamma (\phi_1 (f), \phi_2 (g))=\phi'_1 (f) \phi_2'(g) \Gamma(f,g).
  • Show that if f:\mathbb{R}^n \rightarrow \mathbb{R} is a C^2 function and \phi: \mathbb{R}\rightarrow \mathbb{R} is also C^2,
    L \phi (f)=\phi'(f) Lf+\phi''(f) \Gamma(f,f).

 

Exercise 3 On \mathcal{C}_c (\mathbb{R}^n,\mathbb{R}), let us consider the diffusion operator L=\Delta +\langle \nabla U, \nabla \cdot \rangle, where U: \mathbb{R}^n \rightarrow \mathbb{R} is a C^1 function. Show that L is symmetric with respect to the measure \mu (dx)=e^{U(x)} dx.

Exercise 4 (Divergence form operator). On \mathcal{C}_c (\mathbb{R}^n,\mathbb{R}), let us consider the operator Lf=\mathbf{div} (\sigma \nabla f), where \mathbf{div} is the divergence operator defined on a C^1 function \phi: \mathbb{R}^n \rightarrow \mathbb{R}^n by
\mathbf{div} \text{ } \phi=\sum_{i=1}^n \frac{\partial \phi_i}{\partial x_i}
and where \sigma is a C^1 field of non negative and symmetric matrices. Show that L is a diffusion operator which is symmetric with respect to the Lebesgue measure.

Exercise 5: On \mathbb{R}^n, we consider the divergence form operator

Lf=\mathbf{div} (\sigma \nabla f),
where \sigma is a smooth field of positive and symmetric matrices that satisfies
a \|x \|^2 \le \langle x , \sigma x \rangle \le b \|x \|^2, \quad x \in \mathbb{R}^n,
for some constant 0 < a \le b. Show that with respect to the Lebesgue measure, the operator L is essentially self-adjoint on \mathcal{C}_c (\mathbb{R}^n,\mathbb{R})

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