Compactly-supported harmonic tensors

Question

Let $({M},g)$ be a connected and non-compact Riemannian manifold without boundary. If $L:\Gamma^{\infty}(E)\to \Gamma^{\infty}(E)$ is a linear second order elliptic operator on some smooth $\mathbb{R}$-bundle $E$ over ${M}$, is it then true that

$$Lu=0$$

for $u\in\Gamma^{\infty}_{c}(E)$ implies that $u=0$, or in other words, there are no compactly-supported smooth homogeneous solutions? I think such a result could be proven via some ``unique continuation property of elliptic system''. However, while looking in the literature, I didn't find a suitable version for this situation.

Examples of such operator $L$ I have in mind is for example the connection Laplacian $$\Delta_{C}:=g^{ij}\nabla_{i}\nabla_{j}:\Gamma^{\infty}(T^{\ast}{M}^{\otimes k})\to\Gamma^{\infty}(T^{\ast}{M}^{\otimes k})$$ with Levi-Civita connection $\nabla$ or closely related, the Hodge-de Rham Laplacian $$\Delta_{H}:=\mathrm{d}\delta+\delta\mathrm{d}:\Omega^{k}({M})\to\Omega^{k}({M})$$ with exterior derivative $\mathrm{d}$ and codifferential $\delta$.

Answer 1

The unique continuation is valid for generalized Laplacians. This follows from Hörmander's result in

Hörmander, Lars, Uniqueness theorems for second order elliptic differential equations, Commun. Partial Differ. Equations 8, 21-64 (1983). ZBL0546.35023 MR686819

The operators you mentioned are such generalized Laplacians. Here are a few details.

Suppose that $E$ is a metric vector bundle over the Riemann manifold $(M,g)$. Denote by $\Delta_M$ the scalar Laplacian determined by the metric $g$. Fix a connection on $E$ compatible with the metric on $E$ and set $\Delta_E=\nabla^*\nabla: C^\infty(E)\to C^\infty(E)$.

For any $u\in C^\infty(E)$ we have

$$\Delta_M |u(x)|_E^2=2\big\langle \Delta_E u(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$

A second order partial differential operator $L$ is called a generalized Laplacian if its principal symbol satisfies

$$ \sigma_L(\xi)=-|\xi|_g^2\cdot \mathbf{1}_{E_x},\;\;\forall x\in M,\;\;\forall \xi\in T^*_xM. $$

Any generalized Laplacian $L:C^\infty(E)\to C^\infty(E)$ has the form

$$ L=\nabla^*\nabla+ T=\Delta_E+T,$$

for some a connection $\nabla$ on $E$ compatible with the metric on $E$ and an endomorphism $T$ of the bundle $E$; see Proposition 10.1.34 here.

If $Lu=0$, then $\Delta_E u=-Tu$ and we deduce

$$\Delta_M |u(x)|_E^2=-2\big\langle Tu(x),u(x)\big\rangle_E-2\vert \nabla u(x)\vert_E^2.$$

At this point you can invoke the above results of Hörmander for the scalar function $|u(x)|_E^2$ to obtain the unique continuation.

Answer 2

H"ormander's result works perfectly well for any second order system with real diagonal principal part. The last few answers indicate steps to prove this, but it's more direct to just look at the proof and see that all the steps just work directly for systems of this type. It's very common that the hypothesis for such unique continuation results is simply some sort of differential inequality like this, with this sort of extension to bundles being straightforward.