# Self-dual differential on $4$-manifold with boundary

Let $$(M,g)$$ be an oriented compact Riemannian $$4$$-manifold with boundary $$\partial M$$. Let $$a\in \Omega^1$$ such that $$*a\big|_{\partial M}=0$$, i.e. $$a(\nu)=0$$ on $$\partial M$$, where $$\nu$$ denotes the unit normal field.

Question. If $$d^+a=0$$, can we get $$da=0$$?

PS:

• If $$M$$ is closed, we know it is true, by $$d(a\wedge da)=|d^+a|^2-|d^-a|^2$$ and Stokes formula.

• If $$a\big|_{\partial M}=0$$, by the same method, we know it is still true.

The reason to ask such a question: Under the same condition, we have that $$\|a\|_{L^{p}_1}\leq C(\|da\|_{L^p}+\|d^*a\|_{L^p}+\|a\|_{L^p}),$$ by Theorem 5.1 of Katrin Wehrheim, Uhlenbeck Compactness, Princeton University, Princeton, NJ. I wonder can we have a similar estimate.

No. For example, take $$M$$ to be the Euclidean ball with $$a=-x^1 dx^0+x^0 dx^1+x^3 dx^2-x^2 dx^3.$$