## Example of non-uniformly elliptic equation

Consider solutions of the equation

$\displaystyle u_{xx}+y^2u_{yy}=0$ in $R=(0,\pi)\times(0,Y)$

$\displaystyle u(0,y)=u(\pi,y)=0$, $\forall\,y\in(0,Y)$

Any classical solution $u$ has a fourier expansion $\sum f_n(y)sinnx$. And $f_n$ satisfies

$y^2f''_n-n^2f_n=0$

This equation has two independent solutions $y^{\beta_n}, y^{\gamma_n}$, where $\beta_n=\frac{1}{2}(1+\sqrt{1+4n^2})>0$, $\gamma_n=\frac 12(1-\sqrt{1+4n^2})<0$. The fact $u$ is bounded at $y=0$ requires that $f_n(y)=y^{\beta_n}$, hence $f(0)=0$. It follows that $u(0)=0$, therefore $u$ can not take on prescribed no-zero boundary data.

$\text{Q.E.D}\hfill \square$

$\mathbf{Remark:}$ Gilbarg Trudinger, p116