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Mathematics LibreTexts

1.E: Introduction (Exercises)

Q1.1

Find nontrivial solutions \(u\) of
$$
u_xy-u_yx=0 \ .
$$

Q1.2

Prove: In the linear space \(C^2(\mathbb{R}^2)\) there are infinitely many linearly independent solutions of \(\triangle u=0\) in \(\mathbb{R}^2\).

Hint: Real and imaginary part of holomorphic functions are solutions of the Laplace equation.

Q1.3

Find all radially symmetric functions which satisfy the Laplace equation in
\(\mathbb{R}^n\setminus\{0\}\) for \(n\ge2\). A function \(u\) is said to be radially symmetric if \(u(x)=f(r)\), where \(r=(\sum_i^nx_i^2)^{1/2}\).

Hint: Show that a radially symmetric \(u\) satisfies \(\triangle u=r^{1-n}\left(r^{n-1}f'\right)'\) by using \(\nabla u(x)=f'(r)\frac{x}{r}\).

Q1.4

Prove the basic lemma in the calculus of variations:
Let \(\Omega\subset\mathbb{R}^n\) be a domain and \(f\in C(\Omega)\) such that
$$
\int_\Omega\ f(x)h(x)\ dx=0
$$
for all \(h\in C^2_0(\Omega)\). Then \(f\equiv0\) in \(\Omega\).

Q1.5

Write the minimal surface equation (1.2.2.1) as a quasilinear equation of second order.

Q1.6

Prove that a sufficiently regular minimizer in
\(C^1(\overline{\Omega})\) of
$$
E(v)=\int_\Omega\ F(x,v,\nabla v)\ dx-\int_{\partial\Omega}\ g(v,v)\ ds,
$$
is a solution of the boundary value problem
\begin{eqnarray*}
\sum_{i=1}^n\frac{\partial}{\partial x_i}F_{u_{x_i}}-F_u&=&0\ \ \mbox{in}\ \Omega\\
\sum_{i=1}^nF_{u_{x_i}}\nu_i-g_u&=&0\ \ \mbox{on}\ \partial\Omega,
\end{eqnarray*}
where \(\nu=(\nu_1,\ldots,\nu_n)\) is the exterior unit normal at the boundary \(\partial\Omega\).

Q1.7

Prove that \(\nu\cdot Tu=\cos\gamma\) on \(\partial\Omega\), where \(\gamma\) is the angle between the container wall, which is here a cylinder, and the surface \(S\), defined by \(z=u(x_1,x_2)\), at the boundary of \(S\), \(\nu\) is the exterior normal at
\(\partial\Omega\).

Hint: The angle between two surfaces is by definition the angle between the two associated normals at the intersection of the surfaces.

Q1.8

Let \(\Omega\) be bounded and assume \(u\in C^2(\overline{\Omega})\) is a solution of
\begin{eqnarray*}
\text{div}\ Tu&=&C\  \mbox{in}\ \Omega\\
\nu\cdot\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}&=&\cos\gamma\ \mbox{on}\
\partial\Omega,
\end{eqnarray*}
where \(C\) is a constant.

Prove that
$$
C={|\partial\Omega|\over|\Omega|}\cos\gamma\ .
$$

Hint: Integrate the differential equation over \(\Omega\).

Q1.9

Assume \(\Omega=B_R(0)\) is a disc with radius \(R\) and the center at the origin.
Show that radially symmetric solutions \(u(x)=w(r)\), \(r=\sqrt{x_1^2+x_2^2}\), of the capillary boundary value problem are solutions of
\begin{eqnarray*}
\left(\frac{rw'}{\sqrt{1+w'^2}}\right)'&=&\kappa r w\ \ \mbox{in}\ 0<r<R\\
\frac{w'}{\sqrt{1+w'^2}}&=&\cos\gamma\ \ \mbox{if}\ r=R.
\end{eqnarray*}

Remark. It follows from a maximum principle of Concus and Finn [7] that a solution of the capillary equation over a disc must be radially symmetric.

Q1.10

Find all radially symmetric solutions of
\begin{eqnarray*}
\left(\frac{rw'}{\sqrt{1+w'^2}}\right)'&=&C r \ \ \mbox{in}\ 0<r<R\\
\frac{w'}{\sqrt{1+w'^2}}&=&\cos\gamma\ \ \mbox{if}\ r=R.
\end{eqnarray*}

Hint: From an exercise above it follows that
$$
C=\frac{2}{R}\cos\gamma.
$$

Q1.11

Show that \(\text{div}\ Tu\) is twice the mean curvature of the surface defined by \(z=u(x_1,x_2)\).

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