Math 6410 • ODE

In Kirchhoff-Love Plate Theory, the vertical displacement \(w(x)\) of an isotropic static
plate under pure bending satisfies
\[
  Lw:=D\left(\frac{\partial^4w}{\partial x_1^4}
  +2\frac{\partial^4w}{\partial x_1^2\partial x_2^2}
  +\frac{\partial^4w}{\partial x_2^4}\right)=-f,
\]
where \(D>0\) is a known constant bending stiffness, and \(-f(x)\) is a given applied
transverse load. Assume that the plate lies within a simply connected bounded region
\(U\subset\R^2\) with smooth boundary \(\partial U\). Define the bilinear form
\[
  B(w,v)=\int_U (Lw)v\dd x.
\]
\begin{enumerate}
\item[(a)] Suppose that the boundary of the plate is \textit{clamped}, that is, the
displacement \(w\) satisfies
\[
\begin{aligned}
  w &= 0, && \text{on }\partial U,\\
  \frac{\partial w}{\partial\eta} &= 0, && \text{on }\partial U,
\end{aligned}
\]
where \(\eta\) is the unit outward normal. Let \(X\) be the space of smooth functions
satisfying the above boundary conditions. Show that \(B\) is symmetric and positive
semidefinite over \(X\times X\).
\item[(b)] Using an elementary estimate for solutions of Poisson's equation, show that
\(B\) is bounded and coercive over \(X\times X\) in a well-chosen norm.
\item[(c)] By extending \(X\) appropriately, use the Lax-Milgram Theorem to prove that
the boundary value problem above admits a unique weak solution for all sufficiently
smooth \(f\).
\end{enumerate}