A conformal lower bound of weighted Dirac eigenvalues on manifolds with boundary

For the weighted Dirac eigenproblem on a compact spin manifold with the chiral boundary condition \begin{equation*} \left{ \begin{array}{ll} D\varphi = λf\varphi & $\text{in }$ M, \ $\mathbf{B}$\varphi = 0 & $\text{on }$ \partial M, \end{array} \right. \end{equation*} we first give a lower bound of the eigenvalue using the relative Yamabe constant \begin{equation*} λ^2 \geq \frac{n}{4(n-1)} Y(M,\partial M,[g]), \end{equation*} then prove that equality holds if and only if (up to a conformal transformation) $M$ is a hemisphere and $\phi$ is a Killing spinor. More generalizations are studied.