# Is the left regularizer for elliptic BVP a left inverse for the principal part?

Take a differential operator with elliptic symbol, consider just the principal part of the operator. Can one invert this principal part with some parametrix type construction (at least construct a left inverse)?

Background: the concern is on linear PDE systems $$\mathcal L(x, \frac{\partial}{\partial x})u=\mathcal S(x), \qquad \mathcal B(x,\frac{\partial}{\partial x})u|_{\partial\Omega}=\varphi(x)$$ whose symbol coincides with their principal symbol, $$\mathcal L_0(x,i\xi)=\mathcal L(x,i\xi),\qquad \mathcal B_0(x,i\xi)=\mathcal B(x,i\xi),$$ so there are no terms of lower order. The operator is considered on a bounded domain $\Omega\in\mathbb{R}^n$. There can be more equations in the systems than unknown functions $u=(u_1,\ldots, u_J)$. The symbol is assumed to be elliptic/full rank in $\Omega$ for $|\xi|\neq 0$.

The concern is not about existence, but only about regularity estimates given existence of the solution. To this purpose, some general theory asserts the existence of a left regularizer to $\mathcal A$, where $\mathcal Au=(\mathcal L\times\mathcal B)u=\left(\begin{array}\mathcal S\\\varphi\end{array}\right)$, that is $$\mathcal R\mathcal A=\mathcal{Id}-\mathcal T,$$ where $\mathcal T$ is compact.

The question is whether one can achieve that $\mathcal T=0$ because there are no lower order terms, as $\mathcal L_0(x,i\xi)=\mathcal L(x,i\xi)$. Is invertibility of the operator easy to analyze when the elliptic symbol has no lower order terms?

The aim is to get a regularity estimate for the solution $u$ just with $\mathcal S$ and $\varphi$ on the right hand side, without Lebesgue norm of $u$.

I recall that in the theory of $\Psi DO$ in $\mathbb{R}^n$, I saw a construction of a parametrix for an elliptic principal symbol which acted as inverse but could not find the source again.

A subquestion: is there literature on (redundant) PDE systems apart from that Russian article from the 70ies (link), which has its own peculariar notation and proofs which are at sometimes difficult to follow, relying on Russian articles not available in English. If the task is promising, I want to study the mentioned problem but lack other literature which construct regularizers with more perspicuous proofs.

• So this cannot be true, thank you for the reality check. The potential has to be gauged, yes. But this seems an easy case of non-uniqueness. Simpler to analyze than, say, the Helmholtz eqn. and eigenvalues of the Laplacian (which has a lower order term). When I was asking the question, I wanted to understand a particular remark in the literature that assured: analysis of invertibility is closely tied up with lower order terms. Do you know of a technique analyzing unicity of (linear) PDE by considering the terms in $\mathcal T$, where having just highest order is an advantage? – Tom Jul 27 '13 at 14:27