Math 6410 • ODE

\textbf{Fundamental Elliptic Estimate.}
Let \(\T^n=S^1\times\cdots\times S^1\) be the \(n\)-dimensional torus. Let
\(u\in C^\infty(\T^n)\), which is the same as saying \(u\) is a complex valued smooth
function on \(\R^n\) which is \(2\pi\)-periodic in each coordinate. Then for \(k\in\R\), the
Sobolev norm is given by
\[
  \lVert u\rVert_k^2=\sum_\alpha (1+\lvert\alpha\rvert^2)^k\lvert u_\alpha\rvert^2
\]
where \(u\sim\sum_\alpha u_\alpha e^{i\alpha\cdot x}\) is the Fourier series expansion of
\(u\) such that the sum runs through all multiindices \(\alpha\in\mathbb{Z}^n\). Let
\(H^k(\T^n)\) be the \(\lVert\cdot\rVert_k\)-completion of the trigonometric polynomials on
\(\T^n\) (finite sums of the form
\(p(x)=\sum_{\lvert\alpha\rvert\le N}c_\alpha e^{i\alpha\cdot x}\).)
\begin{enumerate}
\item[(a.)] Show that for \(k=2\), there is a constant \(c(k,n)<\infty\) so that whenever
\(u\in H^{k+2}(\T^n)\) then
\[
  \lVert u\rVert_{k+2}\le c\bigl(\lVert\Delta u\rVert_k+\lVert u\rVert_0\bigr).
  \tag{\dagger}
\]
\item[(b.)] Supposing that \((\dagger)\) holds for all \(k\), prove that if
\(f\in C^\infty(\T^n)\) such that \(\int_{\T^n} f\,dx=0\) then there is
\(u\in C^\infty(\T^n)\) such that
\[
  \Delta u=f\qquad\text{on }\T^n.
\]
\end{enumerate}