# Boundary problem with elliptic 2nd order pde

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.

• My gut reaction: the key feature of the first problem that makes the trivial solution unique is its linearity. As the same is true of the second problem, I'm guessing it will also have to be unique. Jan 24, 2019 at 19:37