# Is there a Hodge isomorphism theorem for part-tangential, part-normal, harmonic differential forms?

Let $M$ be an oriented compact Riemannian $n$-manifold with boundary $\partial M$. A differential $p$-form $\omega$ on $M$ is normal if $i^* \omega = 0$ holds, tangential if $i^* \star \omega = 0$ holds (where $i:\partial M \to M$ is the inclusion map), and harmonic if $\mathrm{d} \omega = 0 = \mathrm{d}^* \omega$ holds.

Here are two generalizations to the Hodge isomorphism theorem:

1. For each de Rham $p$-cohomology class $[\alpha] \in H^p(M)$ there exists a unique tangential harmonic $p$-form $\omega$.

2. For each relative de Rham $p$-cohomology class $[\alpha] \in H^p(M, \partial M)$ there exists a unique normal harmonic $p$-form $\omega$.

My question regards a situation where the boundary $\partial M$ is decomposed in two "non-overlapping" parts: $\partial M = A \cup B$ with $\partial A = \partial B = A \cap B$.

Is it true (possibly under under some additional conditions) that for each relative de Rham $p$-cohomology class $[\alpha] \in H^p(M, B)$ there is a unique harmonic $p$-form $\omega$ that is tangential to $A$ and normal to $B$?

This seems feasible with regard to the Lefschetz duality theorem and its generalization: $H^p(M) \simeq H^{n-p}(M, \partial M)$ and $H^p(M, A) \simeq H^{n-p}(M, B)$.

-