Problems to Sections 10.3, 10.4


### Problems to Sections 10.3, 10.4

Problem 1. The area of the surface $z=z(x,y)$ is $$S=\iint_{D} \sqrt{1+z_x^2+z_y^2}\,dxdy \label{eq-10.4.P.1}$$ where $(x,y)\in D$ is an projection of the surface.

1. Write Euler-Lagrange PDE of the surface of the minimal area (with boundary conditions $z(x,y)=\phi(x,y)$ as $(x,y)\in \Gamma$ which is the boundary of $D$).
2. The potential energy is $$E= kS - \iint_D fz\,dxdy \label{eq-10.4.P.2}$$ with $S$ defined by (\ref{eq-10.4.P.1}) and $f$ areal density of external force.
3. Write Euler-Lagrange PDE of the surface of the minimal energy.
4. If the boundary condition for variational problem is $z(x,y)=\phi(x,y)$ as $(x,y)\in \Gamma'\subset \Gamma$ write the boundary condition for Euler-Lagrange PDE on $\Gamma''=\Gamma\setminus\Gamma'$.

Problem 2.

The energy of the small deformation of the plate $z=z(x,y)$ is $$E=\frac{1}{2}\iint_{D} \Bigl( A(\Delta z )^2 + 2B (z_{xy}^2-z_{xx}z_{yy})\Bigr)\,dxdy, \label{eq-10.4.P.3}$$ where $(x,y)\in D$ is an projection of the surface, and the total energy is $$E^*=E-\iint_{D} fz\,dxdy$$ with $f$ areal density of external force. One can rewrite (\ref{eq-10.4.P.3}) as $$E=\frac{A}{2}\iint_{D} (\Delta z )^2\,dxdy +\frac{B}{2}\int_\Gamma \bigl(-\Delta z\cdot \frac{\partial z}{\partial \nu} + \frac{1}{2}\frac{\partial\ }{\partial \nu}|\nabla z|^2\bigr)\,dxdy, \label{eq-10.4.P.5}$$ where $\Gamma=\partial D$ and $\nu$ is a unit internal unit normal to it.

In particular, for the circular plate with $D=\{x^2+y^2< a^2 \}$ the second term in (\ref{eq-10.4.P.5}) equals $$\frac{B}{2}\int_\Gamma \Bigl(z_{ss}z_r - z_{ss} z_r+ a^{-1}(z_r^2+z_s^2)\Bigr)\,ds \label{eq-10.4.P.6}$$ with $s=ad\theta$.

For variational problem write Euler-Lagrange PDE and also (for the circular plate only) write boundary-value problem for it, if in the variational problem boundary conditions are

1. $z|_\Gamma = \phi(\theta)$, $z_r|_\Gamma = \psi(\theta)$.
2. $z|_\Gamma = \phi(\theta)$.
3. none.