First, let's start with the assumption/definition that $mathbf{B} = mu mathbf{H}$, where $mathbf{H}$ is called the magnetic field, $mathbf{B}$ is called the magnetic induction, and $mu$ is the permeability of the substance/medium. As discussed in this answer, there can be a finite surface current density, $mathbf{K}$, on the boundary of a conductor. This is illustrated using the usual boundary condition of: $$ oint_{C} mathbf{H} cdot dmathbf{l} = int_{S'} da left[ mathbf{j} + partial_{t} mathbf{D} right] cdot mathbf{n}' tag{0} $$ where $mathbf{j}$ is the current density (specifically the macroscopic average current density, [see pages 248--258 in Jackson, [1999] book for definition and derivation]), $S'$ is a closed surface with an outward unit normal $mathbf{n}'$, and $partial_{t} = tfrac{ partial }{ partial t }$.

It turns out that in the limit where there is a finite $mathbf{K}$ the right-hand side of Equation 0 goes to $mathbf{K} cdot mathbf{t} Delta l$, where $mathbf{t}$ is the unit vector transverse to the surface $S'$ and $Delta l$ is the scale length of the pill box transverse to the surface $S'$. In this limit, the boundary conditions between two media of different permeabilities go to: $$ begin{align} mathbf{n} cdot left( mathbf{B}_{2} - mathbf{B}_{1} right) & = 0 tag{1a} mathbf{H}_{1} cdot mathbf{n} & = frac{ mu_{1} }{ mu_{2} } left( mathbf{H}_{2} cdot mathbf{n} right) tag{1b} mathbf{n} times left( mathbf{H}_{2} - mathbf{H}_{1} right) & = mathbf{K} tag{1c} end{align} $$ where subscripts $1$ and $2$ refer to the two different regions. Note this assumes that $mu$ is not a function of $mathbf{H}$, i.e., the linear limit [e.g., see pages 193--194 in Jackson, [1999]].

For discussion of induced eddy currents and magnetic diffusion, see pages 219--223 in Jackson [1999].

the question is what is the governing differential equation on a point that is exactly on the boundary?

As I mention above, there can be abrupt changes right at the boundary for various reasons (e.g., induced currents). However, these are often confined to regions one or less skin depths from the surface. For copper at room temperature, the skin depth is $sim0.0652 left( nuleft[ Hz right] right)^{-1/2}$ m. That is, for incident electromagnetic fluctuations at frequencies above ~100 MHz the skin depth is on the order of microns. For extremely thin materials, this could pose a problem for such low frequencies but would be negligible for any sufficiently thick material.

So in the limit that we care about the exact conditions across the boundary, then we need to use Equation 0 with the modification allowing for a finite $mathbf{K}$ flowing along the surface.

References

• J.D. Jackson, Classical Electrodynamics, Third Edition, John Wiley & Sons, Inc., New York, NY, 1999.