1.E: Introduction (Exercises)
( \newcommand{\kernel}{\mathrm{null}\,}\)
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 C2(R2) there are infinitely many linearly independent solutions of △u=0 in R2.
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
Rn∖{0} for n≥2. A function u is said to be radially symmetric if u(x)=f(r), where r=(∑nix2i)1/2.
Hint: Show that a radially symmetric u satisfies △u=r1−n(rn−1f′)′ by using ∇u(x)=f′(r)xr.
Q1.4
Prove the basic lemma in the calculus of variations:
Let Ω⊂Rn be a domain and f∈C(Ω) such that
∫Ω f(x)h(x) dx=0
for all h∈C20(Ω). Then f≡0 in Ω.
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
C1(¯Ω) of
E(v)=∫Ω F(x,v,∇v) dx−∫∂Ω g(v,v) ds,
is a solution of the boundary value problem
n∑i=1∂∂xiFuxi−Fu=0 in Ωn∑i=1Fuxiνi−gu=0 on ∂Ω,
where ν=(ν1,…,νn) is the exterior unit normal at the boundary ∂Ω.
Q1.7
Prove that ν⋅Tu=cosγ on ∂Ω, where γ is the angle between the container wall, which is here a cylinder, and the surface S, defined by z=u(x1,x2), at the boundary of S, ν is the exterior normal at
∂Ω.
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 Ω be bounded and assume u∈C2(¯Ω) is a solution of
div Tu=C in Ων⋅∇u√1+|∇u|2=cosγ on ∂Ω,
where C is a constant.
Prove that
$$
C={|\partial\Omega|\over|\Omega|}\cos\gamma\ .
\]
Hint: Integrate the differential equation over Ω.
Q1.9
Assume Ω=BR(0) is a disc with radius R and the center at the origin.
Show that radially symmetric solutions u(x)=w(r), r=√x21+x22, of the capillary boundary value problem are solutions of
(rw′√1+w′2)′=κrw in 0<r<Rw′√1+w′2=cosγ 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.
Q1.10
Find all radially symmetric solutions of
(rw′√1+w′2)′=Cr in 0<r<Rw′√1+w′2=cosγ if r=R.
Hint: From an exercise above it follows that
$$
C=\frac{2}{R}\cos\gamma.
\]
Contributors and Attributions
Integrated by Justin Marshall.