Degenerate elliptic operator

A degenerate elliptic operator has a positive semidefinite principal form that may lose positive definiteness at some points or jets. The p-Laplacian with $p>2$ degenerates where its solution has zero gradient.

For $u_{xx}+y^2{u}_{yy}=0$ with zero vertical-side values on $0<x<\pi$, the Fourier coefficient $b_n(y)$ obeys the Cauchy-Euler differential equation $y^2{b}_n^{\prime \prime }=n^2{b}_n$. Boundedness at $y=0$ removes the negative indicial root, leaving $b_n(y)=A_n{y}^{(1+\sqrt{1+4n^2})/2}$. Bounded top coefficients give absolute convergence of the Fourier sine series below the top boundary.

For the bounded continuous solutions in the sine-mode expansion for a quadratically degenerate elliptic equation, every surviving mode tends to zero at $y=0$. The estimate $p_n^{+}>n$ bounds the absolute sum by a constant times $y/(1-y)$, uniformly in $x$. The lower boundary trace must therefore be zero. Continuous boundary data prescribing a nonzero lower trace and zero vertical sides are incompatible with the equation.

For $P={\partial }_1^2+{\partial }_2^2$ in a three-dimensional domain, let $u_k=\varphi (x_1,x_2)\chi (x_3)\sin (k{x}_3)$ with fixed smooth compact support. Then $\Vert P{u}_k{\Vert }_2$ remains bounded whereas $\Vert u_k{\Vert }_{H^1}$ grows proportionally to $k$. Thus no uniform estimate $\Vert u{\Vert }_{H^1}\le C\Vert Pu{\Vert }_2$ holds. The missing third-direction coercivity, rather than boundary data, causes the failure.