Time and Date: 2-5 PM, October 31, 2015

國立台灣師範大學數學系

104 學年度上學期博士班資格考試題 科目：偏微分方程

Math/NTNU Qualifying Exam of PDEs in Oct. 2015

Time and Date: 2-5 PM, October 31, 2015

1. (15 pt.)

Let u : [0, π] × (R

+

∪ {0}) → R fulfill

∂

²

u

∂t

²

− ∂

²

u

∂x

²

= 0, ∀ (x,t) ∈ (0, π) × R

+

with the initial data u(x, 0) =

∑∞ n=1

α

n

sin nx, ∂u

∂t (x, 0) =

∑∞ n=1

β

n

sin nx, and boundary conditions

u(0,t) = 0 = u( π,t), ∀ t > 0.

Represent the solution u as a Fourier series u(x,t) =

∑∞ n=1

γ

n

(t) sin nx, and compute the coefficients γ

n

(t).

2. (25 pt.) Let

K(x,t) = 1

√ 4 πt e

^{−|x|24 t}

, ∀ (x,t) ∈ R × R

+

.

1

Assume that there exist constant numbers, M > 0 and α ∈ (0,1), such that the real-valued function f ∈ C(R × R) ∩ L

^∞

( R × R) fulfills

| f (x

2

,t

₂

) − f (x

1

,t

₁

) | ≤ M · (|x

2

− x

1

|

^α

+ |t

2

−t

1

|

^α/2

) for all (x

₁

,t

₁

), (x

₂

,t

₂

) ∈ R

²

. Let

z(x,t) =

∫ _t

0

∫ _∞

−∞

K(x − y,t − τ) · f (y,τ) dy dτ.

Show with details that

(a) (20 pt.) z,

^∂z_∂x

,

^∂z_∂t

,

^∂_∂x^2z₂

are continuous in R × (R

+

∪ {0}).

(b) (5 pt.) The equation

∂z

∂t = ∂

²

z

∂x

²

+ f holds in the domain R × R

+

.

Hint: Consider the family of functions, z

_h

(x,t) =

∫ _t−h

0

∫ _∞

−∞

K(x − y,t − τ) · f (y,τ) dy dτ, where h ∈ (0,t/2).

3. (30 pt.) The following is a standard procedure to establish regularity of weak solutions of elliptic PDEs.

Let Ω ⊂ R

^d

be an open, bounded, and simply-connected subset. Assume that Γ : R → R is smooth and bounded (i.e., |Γ| ≤ M for some constant M > 0).

Suppose that v : Ω → R is a weak solution of

∆u + Γ(u)|∇u|

²

= 0 (1)

in the Sobolev space W

^1,2

( Ω). If v is a weak solution of Eq.(1) in W

^1,2

( Ω) ∩ W

^1,p

( Ω), where p > d, then the L

^p

-theory of elliptic PDEs implies

v ∈ W

^2,q

( Ω

^′

)

for some q ∈ (1,∞) and any proper open set Ω

^′

⊂ Ω. The so-called boot- strapping argument is to proceed this procedure until one derives the interior smoothness of v, i.e. v ∈ C

^∞

( Ω).

2

(a) (10 pt.) Give the definitions of weak derivatives and weak solutions of Eq.(1) in W

^1,2

( Ω).

(b) (20 pt.) Explain how to apply the boot-strapping argument to derive in- terior smoothness of v, i.e. prove that v ∈ C

^∞

( Ω).

Hints: You should first figure out `q =?' in each step stated above.

In other words, in the L

^p

-theory, ∆v = f ∈ L

^r

for some r > 1 implies that v ∈ W

^2,s

( Ω

^′

), where s =?

4. (30 pt.) Denote by B

R

:= {x ∈ R

^d

: |x| < R} the open ball of radius R > 0 with center at the origin of R

^d

. Let u : R

^d

→ R

^d

be defined by

u(x) = x

|x| .

(a) As d = 1, is it true that u ∈ W

^1,p

( B

1

) for some p ∈ [1,∞)? If it is yes, what is the range of p? If it is not, explain why.

(b) As d = 2, is it true that u ∈ W

^1,p

( B

1

) for some p ∈ [1,∞)? If it is yes, what is the range of p? If it is not, explain why.

(c) As d = 3, is it true that u ∈ W

^1,p

( B

1

) for some p ∈ [1,∞)? If it is yes, what is the range of p? If it is not, explain why.

3

1  

PDE Qualify Exam

2016/10/31 1. Solve following problems. (10 points for each problem)

(1).

2 2

2 2 2

1 0 for 0 1, 0;

(1 ) (0, ) 0;

(1, ) 0;

( , 0) 0;

( , 0) ( ).

u u

x t

x t x

u t u t u x

u x g x

t

       

   

 

 

 



 

 

(2).

2 2

2 2 2

2

1 1

0 for 1;

(1, ) sin .

u u u

r r r r r

u



 

       

   

 



(3).

0 for 0 , 0 2, 0 1;

0 for 0, 0, 1;

0 for ;

0 for ; 2 ( , , 0) 2 .

u u x y z

u x y z

u x

x

u y

x

u x y x

z

 









        

    

   

 

   



   

 



2. (a). Show that if

2

2 0 for 0 ; (0, ) 0.

u u

k x l

t x

u t

x

      

  



 

 

the maximum of u for 0 x l and

0  must occur at t t1 t0 or at xl. (10 points)

(b). Show that there is no maximal principle for the wave equation. (10 points) (c). Let u x( )C²( ) C( ) be a solution of

1

( ) ( ) 0

n k

k k

u a x u c x u

 x

    



 , where ( )c x  in  . 0 Show that u0 on   implies u0 in  . (10 points)

2  

3. Show that the modified Green’s function for the boundrary value problem u f

  , 0 x 1, u(0)u(1), u(0)u(1), where f L²( ) , and satifies ¹

0 f x dx( ) 0



is

1 ( )2 1

( , )

12 2 2

g x    x^  x . (15 points) 

4. Suppose that L is strongly elliptic of order 2m on a bounded domain  and

satisfies ²

2

( 1) Re^m ( ) ^m

m

a x_ ^ C



 







 ^{for all Rn , x} , and that

L . L^

(a). Show that there is an orthonormal basis {u_j} for H₀( ) consisting of

eigenfunctions for L such that u_jC^( ) for all j and u satisfies boundrary _j

conditions _ⁱu_j 0 on  for i1, 2,m1. The eigenvalues are real and only accumulate only at  . (15 points)

(b) Show that there is an orthonormal basis {u_j} for L²( ) consisting of

eigenfunctions for the Laplacian such that u_jC^( ) and u_j  on 0  for all j. The eigenvalues are all negative. (10 points)

109 學年度上學期博士班資格考試題 科目: 偏微分方程

2020 年 10 月 30 日

1. Solve the following initial boundary value problem u_t= u_xx + 5, 0 < x < π, t > 0 u(0, t) = 1, u(π, t) = 6, t > 0

u(x, 0) = 1 + ⁵_πx + 2 sin 3x, 0 < x < π.

2. (a) State any version of maximum principle for heat equation in a bounded domain.

(b) Let Ω denote an open bounded set of Rⁿand T > 0 be a fix number. Prove a uniqueness theorem for the following initial boundary value problem

ut− ∆u = f, in Ω× (0, T ) u(x, 0) = g(x), in Ω

u = 0, on ∂Ω× (0, T ) where f and g are continuous such that g = 0 on ∂Ω.

3. Let Ω be a a region in Rⁿand u∈ C²(Ω). Show that△u ≥ 0 in Ω if and only if for each ξ ∈ Ω :

u(ξ)≤ 1 ω_nρⁿ⁻¹

∫

|x−ξ|=ρ

u(x) dS_x

for all ρ sufficiently small, where ω_nis the surface area of the unit sphere in Rⁿ.

4. (a) Define the notion of distribution.

(b) Let u be a distribution on R and suppose that u^′ = 0 on R.

Show that u = constant; i.e. show that there is a number a such that u(ϕ) =

∫

R

aϕ dx for all ϕ ∈ C0^∞(R).

5. (a) Let u∈ W0^1,2satisfy

∫

Ω

∇u · ∇ϕ dx ≥ 0 ∀ϕ ∈ W0^1,2, ϕ ≥ 0.

Show that u≥ 0 a.e. in Ω.

(b) Let u∈ W₀^1,2satisfy the inequality in (a), show that infΩ u≥ inf

∂Ωu (essinf)

1

109 學年度下學期博士班資格考試題 科目: 偏微分方程

2021 年 04 月 26 日

1. Solve the following initial boundary value problem

4u_t= u_xx, 0 < x < 1, t > 0,

u(x, 0) = a₁sin(πx) +· · · + amsin(mπx), 0≤ x ≤ 1,

u(0, t) = u(1, t) = 0, t > 0

where a₁,· · · , am are constants and m is a positive integer.

2. State and prove the mean value property for harmonic functions.

3. (a) Let Ω be a bounded domain and Γ be a nonempty open subset of ∂Ω such that Γ is real analysis. Suppose that ∆u = 0 in Ω, u = 0,∇u = 0 on Γ.

Show that u is identically equal to zero in Ω.

(b) State (without proof) carefully the theorems you used in (a)

4. Let K be a compact subset in Rⁿ. Define f to be uniform Hölder continuous with exponent α∈ (0, 1] in K, (denoted by f ∈ C^α(K)), if

sup

x,y∈K,x̸=y

{|f(x) − f(y)|

|x − y|^α }

<∞.

Show that f g∈ C^γ(K) if f ∈ C^α(K) and g ∈ C^β(K), where γ = min(α, β).

5. Let

Φ(x, t) = 1

(4πt)^n/2e^−|x|24t for all x∈ Rⁿand t > 0.

(a) Show that ∫

Rⁿ

Φ(x, t) dx = 1, ∀t > 0.

(b) Let f ∈ C(Rⁿ)∩

L^∞(Rⁿ) be a real-valued function, and u(x, t) =

∫

Rⁿ

Φ(x− y, t)f(y) dy.

Show that, for any fixed x ∈ Rⁿ, lim

t→0⁺u(x, t) = f (x).

1