We consider the following degenerate elliptic PDE on the unit ball $B\subset \mathbb{R}^n$: $$ L(u):= -\mathrm{div}(a\nabla u) = 0, $$ where $a\in C^{\infty}(B;[0,+\infty))$ satisfies $a(0)=0$ and $a(x)>0$ for $x\neq 0$.

If $a$ was positive everywhere, then the operator $T:u\mapsto (L(u),tr_{\partial B}(u))$ would be a Fredholm operator from $W^{2,2}(B)\to L^2(B)\times L^2(\partial B)$. In the case where $a$ vanishes at the origin, can we still say something on $T$? How far is it from a Fredholm operator?

If yes, can we generalize this to the following operators? I am interested in elliptic operators of order $2m$ of the form $$ Au = \sum_{|\alpha|,|\beta|\leq m} D^{\alpha}(a_{\alpha\beta} D^{\beta}u), $$ where the coefficients $a_{\alpha \beta}$, for $|\alpha|=|\beta|=m$, vanishes only at a given point $x_0$ (or more generally on some "small" set), and define a strongly elliptic operator $A$ for functions supported outside of $x_0$.

We consider the following degenerate elliptic PDE on the unit ball $B\subset \mathbb{R}^n$: $$ L(u):= -\operatorname{div}(a \, \nabla u) = 0, $$ where $a\in C^\infty(B;[0,+\infty))$ satisfies $a(0)=0$ and $a(x)>0$ for $x\neq 0$.

If $a$ was positive everywhere, then the operator $T:u\mapsto (L(u),\operatorname{tr}_{\partial B}(u))$ would be a Fredholm operator from $W^{2,2}(B) \to L^2(B)\times L^2(\partial B)$. In the case where $a$ vanishes at the origin, can we still say something on $T$? How far is it from a Fredholm operator?

If yes, can we generalize this to the following operators? I am interested in elliptic operators of order $2m$ of the form $$ Au = \sum_{|\alpha|,|\beta|\leq m} D^{\alpha}(a_{\alpha\beta} D^{\beta}u), $$ where the coefficients $a_{\alpha \beta}$, for $|\alpha|=|\beta|=m$, vanishes only at a given point $x_0$ (or more generally on some "small" set), and define a strongly elliptic operator $A$ for functions supported outside of $x_0$.