I would like to know for which choice of boundary conditions the title statement is true.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^n$, for which we regard the $L^2$-de-Rham complex.

We can impose on the domain of the differentials either (i) no boundary conditions (ii) tangential boundary conditions on all of $\partial\Omega$, or (iii) impose partial tangential boundary conditions on a 2reasonable" part $\Sigma_t$ of $\partial\Omega$. The last case has been investigated by e.g., in [1] and [2].

For closed smoothly bounded domains, it is known that is known that merely locally integrable differential forms $h$ with $dh = 0$ and $\star d \star = 0$ are already smooth. [3] As for the $L^2$-de-Rham complex for Lipschitz bounded domains, my impression is that the smoothness of the harmonic forms is expected or even taken for granted, but I have not found an explicit statement that clarifies this.

Unfortunately, I need comparable smoothness results only for application. Maybe it is even to simple for practioners of the field to write it down explicitly. Can provide me with some (available) resources which I can cite, or a combination of theorems?

[1] M. Mitrea: Mixed boundary-value problems for Maxwell's equation
[2] V. Gol'dshtein, I. Mitrea, M. Mitrea: Hodge decompositions with mixed boundary conditions and applications to partial differential equations on lipschitz manifolds
[3] T. Iwaniec, C. Scott, B. Stroffolini: Nonlinear Hodge Theory on Manifolds with Boundary.

1 Answer 1


Harmonic forms are smooth in the interior of the domain, not depending on how smooth the boundary is, no matter what boundary condition is imposed. Note that this is only about regularity, assuming existence. The reason is that the Hodge-Laplace operator is elliptic. Smoothness up to the boundary depends on the smoothness of the boundary, and if "correct" (i.e. complementing) boundary conditions are imposed. The latter requirement is also needed for existence and is automatic if you are considering one of the "usual" boundary conditions that arise from a variational formulation, e.g., the boundary conditions given in Taylor's PDE book, Volume 1, Chapter 5, Section 9.

  • $\begingroup$ @timur: Thanks for your reference to Taylor, which I learned about just yesterday. Still, in the refered section he assumes a smooth boundary from the beginning. Furthermore, how ellipticity of the Hodge Laplacian enforces regularity is not clear to me - in case you mean elliptic regularity, I am aware that the solutions for the Hodge-Laplacian with right-hand side in L^2 and, say, homogenous tangential boundary conditions may gain regularity between 1/2 and 1 for Lipschitz domains. Yet, I do not know how more regular data imply more regular solutions, as in the scalar case. $\endgroup$
    – shuhalo
    May 8, 2012 at 17:24
  • $\begingroup$ @Martin: The interior regularity can be shown by applying a general regularity theorem on elliptic operators, see e.g. Folland's PDE book. I don't know if it gives the best results, but I think McLean's book on strongly elliptic systems contains some discussions on regularity up to the boundary when the boundary is not smooth. $\endgroup$
    – timur
    May 8, 2012 at 19:08

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