I am a physicist interested in the use of the Heat Kernel for expressing the determinant of Laplace-type differential operators. I will use the Einstein summation convention and refer to "components of a tensor" when I say just "tensor".
I am working with $\det(-\Delta_4+m^2)$ where $\Delta_4$ is the four-dimensional Laplacian on a Riemannian manifold $\mathcal{M}$ with some boundaries $\partial \mathcal{M}$, and $m^2$ is a real, positive parameter. Using the Heat Kernel method it is known that (with Dirichlet boundary condition): \begin{align} \ln \det(-\Delta_4+m^2) =& \frac{1}{16\pi^2}\int_\mathcal{M} d^4x \sqrt{g} \left[\frac{1}{2}m^4 -\frac{1}{6}m^2 R +\frac{1}{72} R^2+\frac{1}{120}C^2\right] \\ &+B_R+B_{R^2}+B_{C^2}-\frac{1}{180}\chi(\mathcal{M}) \\ &+\frac{1}{16\pi^2}\int_{\partial \mathcal{M}}d^3x \sqrt{\gamma} \left[ \frac{1}{36}KR+\frac{2}{45}KR_{\textbf{n} \textbf{n}}-\frac{1}{90}K^{\mu \nu}R_{\textbf{n}\mu \textbf{n}\nu} \right. \\ &+\left.\frac{1}{15}K^{\mu \nu}R_{\mu \nu}+\frac{4}{315}K^3-\frac{2}{35}KK_{\mu \nu}K^{\mu \nu}+\frac{26}{945}K^\alpha_\beta K^\beta_\gamma K^\gamma_\alpha\right]. \tag{1} \end{align} With: $R$ the Ricci scalar, $C^2$ the Weyl quadratic, $B_X$ the boundary terms associated with $X$, $\chi(\mathcal{M})$ the Euler characteristic of $\mathcal{M}$, $\gamma$ the determinant of the induced metric on the boundary, $K$ the trace over the metric $g_{\mu \nu}$ of the second fundamental form $K_{\mu \nu}$, $R_{\textbf{n} \textbf{n}} = n^\mu n^\nu R_{\mu \nu}$ the projection of the Ricci tensor onto the normal of the boundary described by the unit 4-vector $n$, and finally $R_{\mu \nu \alpha \beta}$ the Riemann tensor.
In fact, in my work, I use $S=\ln \det(-\Delta_4+m^2)$ as an action functional, something we want to extremize: $\delta_g S=0$
My problem is, when I extremize this action, I find an equation of motion for the bulk part, but also another one on the boundary part. I use the Dirichlet boundary conditions, as mentioned above, and $\delta n^\mu|_{\partial \mathcal{M}} =0$ so that $\delta \gamma_{\mu \nu} |_{\partial \mathcal{M}}=0$. So my two questions are:
Is it really problematic to have an equation of motion on the boundary part for this type of system (an action of a Riemannian manifold)? If so, what type of boundary conditions should impose? I think I should try to find a mixed boundary condition modifying $(1)$ so that the Bulk part only remains after taking the variational, but I can't find such a boundary condition...