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Mathematics LibreTexts

1.E: Introduction (Exercises)


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


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.


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 \(\nabla u(x)=f'(r)\frac{x}{r}\).


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 \(\Omega\).


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


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
\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,
where \(\nu=(\nu_1,\ldots,\nu_n)\) is the exterior unit normal at the boundary \(\partial\Omega\).


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

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


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

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

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


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
\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.

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.


Find all radially symmetric solutions of
\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.

Hint: From an exercise above it follows that


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