I am reading some notes about harmonic functions and associated boundary problems and I came through a proposition stating that the problem
$u_{xx}+u_{yy}=0$
$u(0,y)=u(1,y)=u(x,1)=u(x,0)=0$ , where $x,y\in[0,1]$
has only the 0 solution. I would like kindly to ask you if the following problem
$u_{xx}+u_{yy}+au_{y}+bu_{x}=0$
$u(0,y)=u(1,y)=u(x,1)=u(x,0)=0$ , where $x,y\in[0,1]$
has also only the $0$ solution. Any hints or refferences are welcome. Thanks in advance.
@dmitri Yes, the solution to your problem is unique. This follows from the (strong) maximum principle. The basic idea is that if you had a non-zero solution you would have a point in the square where it attains a positive maximum or a negative minimum. At that point your first derivatives would be zero and the second ones would have a definite sign. The degenerate case can be dealt with using a barrier type of argument. You can find more info here:
http://www.mi.uni-koeln.de/~gsweers/pdf/maxprinc.pdf
Check Corollary 7.
Hope this helps.