Helsinki University of Technology Somersalo/Bingham Mathematics

Mat-1.198 Scattering Theory

3^{rd} set of exercises, 12.2.2003

N.B. There is no lecture on Monday, the 10^{th} of February.

1. Let ∂D be a bounded C^{2} surface. Using the estimate

|n(y) · (z − y)| ≤ L|z − y|^{2} for all z,y ∈ ∂D,

show that there is an h_{0} > 0 such that in

B = {x = z + hn(z)|z ∈ ∂D, |h| ≤ h_{0}}

the boundary projection z + hn(z) ↦ z is unique. Further, show that it is possible to choose h_{0} so small that

|x_{1}− x_{2}|^{2} ≥ \frac{1}{2} |z_{1}− z_{2}|^{2} for all x_{1},x_{2} ∈ B,

where z_{j} is the boundary projection of x_{j}.

2. Show that for 0 < β < 1, we have the estimate

|t ln \frac{1}{t}| ≤ \frac{1}{1 − β} t^{β} when 0 < t < 1.

3. (Jump relation, special case) Let

Φ_{0}(x) = \frac{1}{4π|x|}

and define

w(x) = ∫_{∂D} \frac{∂Φ_{0}}{∂n(y)}(x − y)dS(y), x ∉ ∂D

where ∂D is a bounded C^{2} surface. Prove that

w(x) = \{ \begin{matrix}−1, & x ∈ D \\ 0, & x ∈ ℝ^{3}\ \overline{D}. \end{matrix}

4. (Jump relation, continued) Show that

w(x) = −\frac{1}{2} when x ∈ ∂D

and w is interpreted as an improper integral.