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


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

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