I'm studying the regularity of time-harmonic linear elasticity with a traction free boundary condition and an appropriate radiation condition. I want to prove that the solutions that I have found have a certain degree of regularity. This is partly for its own sake, and partly because my uniqueness results only show that homogeneous solutions with $H^2$ smoothness are zero.
Letting $\sigma(\cdot)$ be the stress tensor and letting $\boldsymbol{g}_0$ and $\boldsymbol{f}_0$ be appropriate compactly supported forcing functions, the strong version of the problem is
$\rho\omega^2\boldsymbol{u}+\nabla\cdot\sigma(\boldsymbol{u})=\boldsymbol{g}_0$ for $\boldsymbol{x}\in\mathbb{R}^2\times\mathbb{R}$
$\sigma(\boldsymbol{u})\boldsymbol{e}_3=\boldsymbol{f}_0$ for $\boldsymbol{x}\in\mathbb{R}^2\times\{0\}$
$\boldsymbol{u}$ is outgoing as $|\boldsymbol{x}|\rightarrow\infty$
where "outgoing" is defined in terms of a radiation condition that looks something like
$\sigma(\boldsymbol{u})\hat{\boldsymbol{R}}-\boldsymbol{\Lambda}\boldsymbol{u}=o(R^{-1})$ as $R\rightarrow\infty$ for $x_3\ge CR^{1/3}$
$\sigma(\boldsymbol{u})\hat{\boldsymbol{R}}-\boldsymbol{M}\boldsymbol{u}=o(R^{-1})$ as $R\rightarrow\infty$ for $x_3<CR^{1/3}$
In the above, $R=|\boldsymbol{x}|$, $\hat{\boldsymbol{R}}$ is the unit vector corresponding to $R$ in spherical coordinates, $C$ is some constant chosen so that the dimensions match, and $\boldsymbol{\Lambda}$ and $\boldsymbol{M}$ are specific constant matrices (I can provide more details on them if it's important).
I've been looking up elliptic regularity results, but I haven't found any that directly apply to this problem. I have been working on a direct proof of regularity related to the specific Green's function involved, but it's very tedious and technical work, and I'll feel very silly if I try to publish that work, and then someone points out a reference that solves a more abstract problem and obviates it.
I know that in general, vectorial problems can fail to have boundary regularity. Results related only to the constant coefficient, isotropic case would be welcome, as would results that apply to a more general case.
Regularity away from the boundary is not difficult to demonstrate, at least for the constant coefficient case, as the problem can be reduced to scalar Helmholtz equations in potentials.
The main challenges with applying the results I've been able to find to this problem are
- It is a vectorial problem, not a scalar problem.
- The domain is unbounded, so the Neumann condition only holds on a plane, while the rest of the "boundary" has a radiation condition.
I would also like to use this result to show that the Green's function is unique in some sense, and that solutions with unpleasant forcings (could contain discontinuities and singularities as bad as delta functions) have a certain regularity; the challenge there is that the Green's function is only in $H^{1/2-\epsilon}$ while the forcing in that case is only in $H^{-3/2-\epsilon}$ for some $\epsilon>0$. I can already prove uniqueness of the Green's function, in fact, but I don't like my proof as I feel there should be some abstract results that make it easier.
