7.E: Elliptic Equations of Second Order (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.
Q7.1
Let γ(x,y) be a fundamental solution to △, y∈Ω. Show that
$$
-\int_\Omega\gamma(x,y)\ \triangle\Phi(x)\ dx=\Phi(y)\quad\hbox{for all}\ \
\Phi\in C_0^2(\Omega)\ .
\]
Hint: See the proof of the representation formula.
Q7.2
Show that |x|−1sin(k|x|) is a solution of the Helmholtz equation
$$
\triangle u+k^2u=0 \ \mbox{in}\ \mathbb{R}^n\setminus\{0\}.
\]
Q7.3
Assume u∈C2(¯Ω), Ω bounded and sufficiently regular, is a solution of
△u=u3 in Ωu=0 on ∂Ω.
Show that u=0 in Ω.
Q7.4
Let Ωα={x∈R2: x1>0,0<x2<x1tanα}, 0<α≤π. Show that
u(x)=rπαksin(παkθ)
is a harmonic function in Ωα satisfying u=0 on ∂Ωα, provided k is an integer. Here (r,θ) are polar coordinates with the center at (0,0).
Q7.5
Let u∈C2(¯Ω) be a solution of △u=0 on the quadrangle Ω=(0,1)×(0,1) satisfying the boundary conditions u(0,y)=u(1,y)=0 for all y∈[0,1] and uy(x,0)=uy(x,1)=0 for all x∈[0,1]. Prove that u≡0 in ¯Ω.
Q7.6
Let u∈C2(Rn) be a solution of △u=0 in Rn satisfying u∈L2(Rn), i. e., ∫Rn u2(x) dx<∞. Show that u≡0 in Rn.
Hint: Prove
∫BR(0) |∇u|2 dx≤const.R2∫B2R(0) |u|2 dx,
where c is a constant independent of R.
To show this inequality, multiply the differential equation by ζ:=η2u, where
η∈C1 is a cut-off function with properties: η≡1 in BR(0), η≡0 in the exterior of B2R(0),
0≤η≤1, |∇η|≤C/R. Integrate the product, apply integration by parts and use
the formula 2ab≤ϵa2+1ϵb2, ϵ>0.
Q7.7
Show that a bounded harmonic function defined on Rn must be a constant (a theorem of Liouville).
Q7.8
Assume u∈C2(B1(0))∩C(¯B1(0)∖{(1,0)}) is a solution of
△u=0 in B1(0)u=0 on ∂B1(0)∖{(1,0)}.
Show that there are at least two solutions.
Hint: Consider
$$
u(x,y)=\frac{1-(x^2+y^2)}{(1-x)^2+y^2}.
\]
Q7.9
Assume Ω⊂Rn is bounded and u, v∈C2(Ω)∩C(¯Ω) satisfy △u=△v and max∂Ω|u−v|≤ϵ for given ϵ>0. Show that max¯Ω|u−v|≤ϵ.
Q7.10
Set Ω=Rn∖¯B1(0) and let u∈C2(¯Ω) be a harmonic function in Ω satisfying lim|x|→∞u(x)=0. Prove that
$$
\max_{\overline{\Omega}}|u|=\max_{\partial\Omega}|u|\ .
\]
Hint: Apply the maximum principle to Ω∩BR(0), R large.
Q7.11
Let Ωα={x∈R2: x1>0, 0<x2<x1tanα}, 0<α≤π, Ωα,R=Ωα∩BR(0), and assume f is given and bounded on ¯Ωα,R.
Show that for each solution u∈C1(¯Ωα,R)∩C2(Ωα,R) of △u=f in
Ωα,R satisfying u=0 on ∂Ωα,R∩BR(0), holds:
For given ϵ>0 there is a constant C(ϵ) such that
$$
|u(x)|\le C(\epsilon)\ |x|^{{\pi\over\alpha}-\epsilon}\qquad\hbox{in}\
\Omega_{\alpha,R}.
\]
Hint: (a) Comparison principle (a consequence from the maximum principle): Assume Ω is bounded, u,v∈C2(¯Ω)∩C(¯Ω) satisfying −△u≤−△v in Ω and u≤v on ∂Ω. Then u≤v in Ω.
(b) An appropriate comparison function is
v=Arπα−ϵsin(B(θ+η)) ,
A, B, η appropriate constants, B, η positive.
Q7.12
Let Ω be the quadrangle (−1,1)×(−1,1) and u∈C2(Ω)∩C(¯Ω) a solution of the boundary value problem −△u=1 in Ω, u=0 on ∂Ω. Find a lower and an upper bound for u(0,0).
Hint: Consider the comparison function v=A(x2+y2), A=const.
Q7.13
Let u∈C2(Ba(0))∩C(¯Ba(0)) satisfying u≥0, △u=0 in Ba(0). Prove (Harnack's inequality):
$$
{a^{n-2}(a-|\zeta|)\over (a+|\zeta|)^{n-1}}u(0)\le u(\zeta)\le
{a^{n-2}(a+|\zeta|)\over (a-|\zeta|)^{n-1}}u(0)\ .
\]
Hint: Use the formula (see Theorem 7.2)
u(y)=a2−|y|2aωn∫|x|=a u(x)|x−y|n dSx
for y=ζ and y=0.
Q7.14
Let ϕ(θ) be a 2π-periodic C4-function with the Fourier series
ϕ(θ)=∞∑n=0(an cos(nθ)+bnsin(nθ)) .
Show that
u=∞∑n=0(an cos(nθ)+bnsin(nθ))rn
solves the Dirichlet problem in B1(0).
Q7.15
Assume u∈C2(Ω) satisfies △u=0 in Ω. Let Ba(ζ) be a ball such that its closure is in Ω.
Show that
|Dαu(ζ)|≤M(|α|γna)|α|,
where M=supx∈Ba(ζ)|u(x)| and γn=2nωn−1/((n−1)ωn).
Hint: Use the formula of Theorem 7.2, successively to the k th derivatives in balls with radius a(|α|−k)/m, k=o,1,…,m−1.
Q7.16
Use the result of the previous exercise to show that u∈C2(Ω) satisfying △u=0 in Ω is real analytic in Ω.
Hint: Use Stirling's formula
n!=nne−n(√2πn+O(1√n))
as n→∞, to show that u is in the class CK,r(ζ), where K=cM and r=a/(eγn). The constant c is the constant in the estimate nn≤cenn! which follows from Stirling's formula. See Section 3.5 for the definition of a real analytic function.
Q7.17
Assume Ω is connected and u∈C2(Ω) is a solution of △u=0 in Ω. Prove that u≡0 in Ω if Dαu(ζ)=0 for all α, for a point ζ∈Ω. In particular, u≡0 in Ω if u≡0 in an open subset of Ω.
Q7.18
Let Ω={(x1,x2,x3)∈R3: x3>0}, which is a half-space of R3. Show that
G(x,y)=14π|x−y|−14π|x−¯y|,
where ¯y=(y1,y2,−y3), is the Green function to Ω.
Q7.19
Let Ω={(x1,x2,x3)∈R3: x21+x22+x23<R2, x3>0}, which is half of a ball in R3. Show that
G(x,y)=14π|x−y|−R4π|y||x−y⋆|−14π|x−¯y|+R4π|y||x−¯y⋆|,
where ¯y=(y1,y2,−y3), y⋆=R2y/(|y|2) and ¯y⋆=R2¯y/(|y|2),
is the Green function to Ω.
Q7.20
Let Ω={(x1,x2,x3)∈R3: x2>0, x3>0}, which is a wedge in R3. Show that
G(x,y)=14π|x−y|−14π|x−¯y|−14π|x−y′|+14π|x−¯y′|,
where ¯y=(y1,y2,−y3), y′=(y1,−y2,y3) and ¯y′=(y1,−y2,−y3),
is the Green function to Ω.
Q7.21
Find Green's function for the exterior of a disk, i. e., of the domain Ω={x∈R2: |x|>R}.
Q7.22
Find Green's function for the angle domain Ω={z∈C: 0<argz<απ}, 0<α<π.
Q7.23
Find Green's function for the slit domain Ω={z∈C: 0<argz<2π}.
Q7.24
Let for a sufficiently regular domain Ω∈Rn, a ball or a quadrangle for example,
F(x)=∫Ω K(x,y) dy,
where K(x,y) is continuous in ¯Ω×¯Ω where x≠y, and which satisfies
|K(x,y)|≤c|x−y|α
with a constants c and α, α<n.
Show that F(x) is continuous on ¯Ω.
Q7.25
Prove (i) of the lemma of Section 7.5.
Hint: Consider the case n≥3. Fix a function η∈C1(R1) satisfying 0≤η≤1, 0≤η′≤2, η(t)=0 for t≤1, η(t)=1 for t≥2 and consider for ϵ>0 the regularized integral
Vϵ(x):=∫Ω f(y)ηϵdy|x−y|n−2,
where ηϵ=η(|x−y|/ϵ). Show that Vϵ converges uniformly to V on compact subsets of Rn as ϵ→0, and that ∂Vϵ(x)/∂xi converges uniformly on compact subsets of Rn to
∫Ω f(y)∂∂xi(1|x−y|n−2) dy
as ϵ→0.
Contributors and Attributions
Integrated by Justin Marshall.