UMD PDE Qualifying Exams/Jan2005PDE

Problem 1Edit

Let   be a harmonic function on   and suppose that

 

Show that   is a constant function.

SolutionEdit

Let  

If   is harmonic (i.e.  ) then so must   (surely,  ). Then since the absolute value as an operator is convex, we have that   is a subharmonic function on  .

Then by the mean value property of subharmonic functions, for any  we have

 

where the second inequality is due to Cauchy-Schwarz (Hölder) inequality.

This estimate hold for all  . Therefore if we send   we see that for all    which gives us that   is constant.

Problem 2Edit

Let   be a piecewise smooth weak solution of the conservation law  

a) Derive the Rankine-Hugoniot conditions at a discontinuity of the solution.

b)Find a piecewise smooth solution to the IVP

 

 


SolutionEdit

a)Edit

When we solve the PDE by methods of characteristics, the characteristic curves can cross, causing a shock, or discontinuity. The task at hand, is to find the curve of discontinuity, call it  . Multiply the PDE by  , a smooth test function with compact support in  . Then by an integration by parts:

 

Let   denote the open region in   to the left of   and similarly   denotes the region to the right of  . If the support of   lies entirely in either of these two regions, then all of the above boundary terms vanish and we get  

Now suppose the support of   intersects the discontinuity  .

b)Edit

We can calculate  . Therefore, the shock wave extends vertically from the origin. That is,

 

Problem 3Edit

Consider the evolution equation with initial data

 

 

a) What energy quantity is appropriate for this equation? Is it conserved or dissipated?

b) Show that   solutions of this problem are unique.

SolutionEdit

3aEdit

Consider the energy  . Then  . Integrate by parts to get  . The boundary terms vanish since   implies   (similarly at  ). Then by the original PDE we get

 

where the last equality is another integration by parts. The boundary terms vanish again by the same argument. Therefore,   for all  ; that is, energy is dissipated.

3bEdit

Suppose   are two distinct solutions to the system. Then   is a solution to

 

 

This tells us that at  ,  . Therefore,  . Since   then   for all  . This implies  . That is,  .

Problem 4Edit

Let   be a bounded open set with smooth boundary  . Consider the initial boundary value problem for  :

 

where   is the exterior normal derivative. Assume that   and that   for  . Show that smooth solutions of this problem are unique.

SolutionEdit

Suppose   are two distinct solutions. Then   is a smooth solution to

 

Consider the energy  . It is easy to verify that  . Then

 

Therefore   implies   for all  . Thus,   for all   which implies  


Problem 5Edit

Let   be a bounded open set with smooth boundary. Let  . Let   and define the functional

 .

Show that   is a minimizer of   over   if and only if   satisfies the variational inequality

  for all  .

SolutionEdit

  Suppose   minimizes  , i.e.  . Then for any fixed  , if we let   then  . Let  ; then we can say that  . Now we must compute  . We have

 


 

 

Since we know   then

  as desired.


  Conversely suppose

 

Then  

Therefore,   for all  . That is,   for all  , as desired.