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Mathematics LibreTexts

1.E: Introduction (Exercises)

These are homework exercises to accompany Miersemann's "Partial Differential Equations" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Partial differential equations are differential equations that contains unknown multivariable functions and their partial derivatives. Prerequisite for the course is the basic calculus sequence.

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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