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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 , yΩ. 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 uC2(¯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={xR2: x1>0,0<x2<x1tanα}, 0<απ. 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 uC2(¯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 u0 in ¯Ω.

Q7.6

Let uC2(Rn) be a solution of u=0 in Rn satisfying uL2(Rn), i. e., Rn u2(x) dx<. Show that u0 in Rn.

Hint: Prove
BR(0) |u|2 dxconst.R2B2R(0) |u|2 dx,
where c is a constant independent of R.
To show this inequality, multiply the differential equation by ζ:=η2u, where
ηC1 is a cut-off function with properties: η1 in BR(0), η0 in the exterior of B2R(0),
0η1, |η|C/R. Integrate the product, apply integration by parts and use
the formula 2abϵa2+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 uC2(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 ΩRn is bounded and u, vC2(Ω)C(¯Ω) satisfy u=v and maxΩ|uv|ϵ for given ϵ>0. Show that max¯Ω|uv|ϵ.

Q7.10

Set Ω=Rn¯B1(0) and let uC2(¯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 ΩBR(0), R large.

Q7.11

Let Ωα={xR2: x1>0, 0<x2<x1tanα}, 0<απ, Ωα,R=ΩαBR(0), and assume f is given and bounded on ¯Ωα,R.

Show that for each solution uC1(¯Ωα,R)C2(Ωα,R) of u=f in
Ωα,R satisfying u=0 on Ωα,RBR(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,vC2(¯Ω)C(¯Ω) satisfying uv in Ω and uv on Ω. Then uv in Ω.

(b) An appropriate comparison function is
v=Arπαϵsin(B(θ+η)) ,
A, B, η appropriate constants, B, η positive.

Q7.12

Let Ω be the quadrangle (1,1)×(1,1) and uC2(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 uC2(Ba(0))C(¯Ba(0)) satisfying u0, 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)|xy|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 uC2(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=supxBa(ζ)|u(x)| and γn=2nωn1/((n1)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,,m1.

Q7.16

Use the result of the previous exercise to show that uC2(Ω) satisfying u=0 in Ω is real analytic in Ω.

Hint: Use Stirling's formula
n!=nnen(2πn+O(1n))
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 nncenn! 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 uC2(Ω) is a solution of u=0 in Ω. Prove that u0 in Ω if Dαu(ζ)=0 for all α, for a point ζΩ. In particular, u0 in Ω if u0 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|xy|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|xy|R4π|y||xy|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|xy|14π|x¯y|14π|xy|+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={xR2: |x|>R}.

Q7.22

Find Green's function for the angle domain Ω={zC: 0<argz<απ}, 0<α<π.

Q7.23

Find Green's function for the slit domain Ω={zC: 0<argz<2π}.

Q7.24

Let for a sufficiently regular domain ΩRn, a ball or a quadrangle for example,
F(x)=Ω K(x,y) dy,
where K(x,y) is continuous in ¯Ω×¯Ω where xy, and which satisfies
|K(x,y)|c|xy|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 n3. Fix a function ηC1(R1) satisfying 0η1, 0η2, η(t)=0 for t1, η(t)=1 for t2 and consider for ϵ>0 the regularized integral
Vϵ(x):=Ω f(y)ηϵdy|xy|n2,
where ηϵ=η(|xy|/ϵ). Show that Vϵ converges uniformly to V on compact subsets of Rn as ϵ0, and that Vϵ(x)/xi converges uniformly on compact subsets of Rn to
Ω f(y)xi(1|xy|n2) dy
as ϵ0.

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7.E: Elliptic Equations of Second Order (Exercises) is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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