ASSIGNED EXERCISES OF MATH544: PDE set 7 April, 2001

THE LAPLACE EQUATION Ronald B Guenther and John W Lee, Partial Differential Equations of Mathematical Physics and Integral Equations

In all types of equations we studied the following:

1. Exact Solutions

2. Uniqueness of solutions

3. Maximum-Minimum Principles

Here, in the elliptic case, I will include into our study the subject Green’s Function Technique for the Laplace operator.

1. Green’s function for the Laplace operator: Let D be a domain and B its boundary. Let the closure of D be D̄ = D ∩ B. Let u, v ∈ C^{2}(D) ∩ C^{1}(D̄). Then the following is an identity

∫_{D} [v∇^{2} u − u∇^{2} v] dV = ∫_{B} [v \frac{∂u}{∂n} − u\frac{∂v}{∂n}] dS  (1)

where \frac{∂u}{∂n} =< N, grad u >. Here N is the unit normal vector field defined at each point of the surface B. Now consider the following Dirichlet problem

∇^{2} u = h, in D,  (2)
u = f, on B.  (3)

where h and f some continuous functions in D and on B respectively. Let v = G(x, y) with

∇_{y}^{2} G(x, y) = δ(x − y), in D  (4)
G(x, y)|_{y∈B} = 0.  (5)

Then we find that

u(x) = ∫_{D} G(x, y)h(y) dv_{y} + ∫_{B} f(y) \frac{∂G(x, y)}{∂n_{y}} dS_{y}  (6)

Hence the solution of a Dirichlet problem reduces to the determination of Green’s function G(x, y) satisfying (4)-(5).

2. Find the Green function when D is a ball and B is the sphere with radius a. Since \frac{1}{|x−y|} satisfies

∇^{2} \frac{1}{|x − y|} = 4πδ(x − y)

then the following form for G is more suitable

G(x, y) = \frac{1}{4π} [\frac{1}{|x − y|} − \frac{q}{|x^{∗} − y|}]

where q and x^{∗} are to be determined thorough the boundary condition. Let y ∈ B then

G|_{y∈B} = 0

or

\frac{1}{|x − y|} = \frac{q}{|x^{∗} − y|}

This gives

q = \frac{a}{r}, x^{∗} = q^{2} x.  (7)

where r^{2} = |x|^{2}. We find also

\frac{∂G(x, y)}{∂n_{y}} =< \frac{y}{a}, grad_{y}G(x, y) > |_{y∈B} = \frac{r^{2} − a^{2}}{4π |x − y|^{3}}

Hence the solution of the Dirichlet problem when h = 0 is

u(x) = ∫_{B} \frac{a^{2} − r^{2}}{4π |x − y|^{3}} f(y) dS_{y}

3. Using the Green’s function technique solve the following Dirichlet problem.

∇^{2} u = 0, − ∞ < x, y < ∞, z > 0,  (8)
u(x, y, 0) = f(x, y), − ∞ < x, y < ∞.  (9)

where f(x, y) is a continuous function in R^{2}.

4. Solve the above problem by use of separation of variables

5. Prove the following theorem:

Theorem. Let D be a bounded normal domain, and suppose that the Green’s function G(x, y) for the Dirichlet problem exists. Then

G(x, y) = G(y, x)

for x, y ∈ D. Consequently , in addition to ∇_{y}^{2} G(x, y) = 0 for y ≠ x, we also have ∇_{x}^{2} G(x, y) = 0 for x ≠ y.

Solution. First way. Let u = G(z, x) and v = G(z, y) in (1). Using (4) and (5) and using the property

∫_{D} f(y) δ(y − x) d^{3}y = f(x)

of the delta-distribution one obtains G(x, y) = G(y, x). Second way. Let K_{a}(c) represent a ball with radius a centered at c and S_{a}(c) is the sphere bounding this ball.

K_{a}(a) = {x ∈ R^{3}||x − c| < a}, S_{a}(c) = {x ∈ R^{3}||x − c| = a}

Fix x, x′ ∈ D. Fix ε > 0 so small that K_{ε}(x), K_{ε}(x′) ⊂ D and |x − x′| > 2ε. Apply (1) to the domain

D_{ε} = D − (K_{ε}(x) ∪ K_{ε}(x′))

for u(y) = G(x, y), and v(y) = G(x′, y)

6. In two dimensions let D be a disc with radius a and B is the circle bounding D with radius a. Consider

∇^{2} u = 0, r < a,  (10)
u = f on B.  (11)

a. Find the formal solution. b. Find reasonable restrictions on f so that the formal solution is a solution.

7. Let D = {(x, y, z) ∈ R^{3}|z > 0} and B = {(x, y, z)|z = 0}. Solve the following Dirichlet problem

∇^{2} u = 0, in D, u = f(x, y) on B.  (12)

Remark 1. The last two problems were solved in class. They respectively represent discrete and continuous spectrum in the method of separation of variables.

8. In two dimensions consider

∇^{2} u = 0, 0 < x < a, 0 < y < b,  (13)
u(x, 0) = f(x), u(x, b) = 0, 0 ≤ x ≤ a,  (14)
u(0, y) = u(a, y) = 0, 0 ≤ y ≤ b.  (15)

a. Find the formal solution, b. Find reasonable restrictions on f(x) so that the formal solution is a solution.

9. Consider

∇^{2} u(r, θ) = 0, 0 ≤ r < a,  (16)
\frac{∂u(a, θ)}{∂r} + α u(a, θ) = f(θ),  (17)

with f is 2π periodic continuous function and α > 0 a constant. a. Solve formally by separation of variables. b. Find reasonable conditions on f so that the formal solution is a solution.

10. Suppose that k > 0 is a constant and ∇^{2}u−k^{2}u = 0. Formulate Dirichlet, Neumann , and Robin problems of this operator L = ∇^{2} − k^{2} and prove that each problem has at most one solution.

11. Prove the following theorems.

Theorem 1. Let u ∈ C^{2}(D) ∩ C^{1}(D̄), where D is abounded domain.

(i) If ∇^{2} u ≤ 0 in D and u ≥ 0 on B. then u ≥ 0 in D, ii) If ∇^{2} u ≥ 0 in D and u ≤ 0 on , B then u ≤ 0 in D.

Theorem 2. Let u ∈ C^{2}(D) ∩ C^{1}(D̄), where D is a bounded domain.

(i) If ∇^{2} u ≥ 0 in D, then u(x) ≤ M = max_{y∈B} u(y) for all x ∈ D. (ii) If ∇^{2} u ≤ 0 in D, then u(x) ≥ m = min_{y∈B} u(y) for all x ∈ D. (iii) If∇^{2} u = 0 in D, then m ≤ u(x) ≤ M for all x ∈ D.

Theorem 3. Let D be a bounded domain and B its boundary. Let u ∈ C^{2}(D) ∩ C^{1}(D̄). Then there is at most one solution of the problem

∇^{2} u = h, in D,  (18)
u = f, on B.  (19)

Theorem 4. Suppose that u ∈ C^{2}(D) ∩ C^{1}(D̄) and satisfies the Dirichlet problem as in the previous theorem (Thm. 3). Suppose that h(x) is bounded by a constant H in D and f is bounded by F on B. Then there is a constant C depending only on the domain D such that |u(x)| ≤ F + CH

Proof. Find a function w such that

∇^{2} w ≤ −H in D and w ≥ ±u on B.  (20)

Given such a function we have ∇^{2} (w ± u) ≤ −H ± h ≤ 0 in D and w ± u ≤ 0 on B Using Thm. 1 above we must have

w ± u ≥ 0

or w ≥ |u| in D. Hence we have to find an w satisfying (20). Let

w(x) = F + H[e^{d} − ex_{1} − c_{c}], x ∈ D̄  (21)

where D lies in between the planes x_{1} = c_{1} and x_{1} = c_{1} + d and C = e^{d}. Prove that this w satisfies all requirements of (20).

Theorem 5. Suppose that the functions u_{i} ∈ C^{2}(D) ∩ C^{1}(D̄), i = 1, 2, satisfies ∇^{2} u_{i} = h_{i} in D and u_{i} = f_{i} on B, where h_{i} and f_{i} are given. If

|h_{1}(x) − h_{2}(x)| ≤ ε

for all x ∈ D and

|f_{1}(x) − f_{2}(x)| ≤ ε

for all x ∈ B. then

|u_{1}(x) − u_{2}(x)| ≤ (1 + C)ε

for all x ∈ D, where C is a constant determined by the domain D.

Remark 2. First three theorems were proved in the class.

12. Prove the following theorem. Theorem 6. Let u ∈ C^{2}(D) ∩ C^{1}(D̄) for some bounded domain D.

(1) If ∇^{2} u < 0 in D and u ≥ 0 on B , then u ≥ 0 in D. (ii) If ∇^{2} u > 0 in D and u ≤ 0 on B, then u ≤ 0 in D.