For the eigenvalue problem: \begin{align*} \nabla^2 \phi + \lambda \phi &= 0 \\ \end{align*} with the boundary condition along the entire boundary: \begin{align*} a(x,y) \phi + b(x,y) \nabla \phi \cdot \tilde{n} &= 0 \\ \end{align*} Use the following Rayleigh Quotient to demonstrate that all eigevalues are non-negative ($\lambda \ge 0$): \begin{align*} \lambda &= \frac{-\oint \phi \nabla \phi \cdot \tilde{n} \, ds + \iint_R \left| \nabla \phi \right|^2 \, dx \, dy}{\iint_R \phi^2 \, dx \, dy} \\ \end{align*}
Clearly, the denominator is positive. Any eigenfunction can't be zero everywhere, or it wouldn't be an eigenfunction, and it's real valued so it's square is positive.
The numerator term on the right must be non-negative since the integrand is non-negative since it is the square of a real number.
The numerator terms on the left: I see no obvious reasons why this can't be negative and why this can't be larger than the term on the right and I'm not sure how to proceed in demonstrating that.