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Consider the elliptic second-order PDE on some bounded domain $D$ (in any dimension)

$-\Delta_D u + \alpha u = 0$

subject to the constraint

$\gamma^i \nabla_i u + \beta u = 0$,

where $\Delta_D$ is the Laplacian on $D$ (with respect to some metric), and $\alpha$, $\beta$, and $\gamma^i$ are real smooth functions on $D$.

A zeroth question, to address user254433's comment below, is what compatibility conditions on $\alpha$, $\beta$, $\gamma^i$, and presumably the metric on $D$ are required to ensure that the system above has solutions? user254433 derived such a condition in one dimension.

Now my main question. Assume the system satisfies the above compatibility conditions so that it admits solutions, and say if any (real) solution to the above system is positive on the boundary $\partial D$, it must be positive everywhere in $D$. What does this imply on the coefficients $\alpha$, $\beta$, and $\gamma^i$?

Note that here I'm asking for necessary conditions on the coefficients. Indeed, I do know of a sufficient condition: if $\alpha \geq 0$ everywhere, this ensures by standard minimum principles that $u$ cannot have any negative local minima (since otherwise at the minimum we'd have $-\Delta_D u + \alpha u < 0$), and so if it's positive on $\partial D$ it must be positive everywhere. However, if necessary conditions on the coefficients are unknown, I'm also interested in learning about sufficient conditions that are weaker than $\alpha \geq 0$ everywhere.

Note that these conditions (necessary or sufficient) need not be local; for instance I'm happy to have conditions on integrals of these coefficients over $D$, etc.

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This is a long comment, not an answer. I just want to mention a technicality that suggests your problem might need to be rephrased. You will want to assume that the system itself is compatible. Let me illustrate with $n=1$ dimensions.

If $n=1$, we can write your system as \begin{align} u''+\Gamma(x)u'+\alpha(x)u=0,\\ u'+\beta(x)u=0, \end{align} where, using the low dimensional structure, we set some coefficients equal to 1. The problem then becomes: if $u|_{\partial D}>0\to u|_D>0$, does this imply anything on $\Gamma,\alpha,\beta$?

Here's my observation: if no positive solutions exist, then the answer must be negative. So the problem is only interesting in the case when the system is compatible. In higher dimensions, the precise conditions are unclear to me, but for $n=1$, it is clear:

To test for compatibility, we differentiate the second equation and substitute $u'$ terms: \begin{align} 0=u''+\beta u'+\beta' u=u''+(\beta'-\beta^2)u. \end{align} Let us also eliminate $u'$ from the "main" equation: \begin{align} u''+(\alpha-\Gamma\beta)u=0. \end{align} Solving both equations for $u''$, we get \begin{align} [\alpha-\Gamma\beta-(\beta'-\beta^2)]u=0. \end{align} For nontrivial solutions, we thus need \begin{align} \alpha=\Gamma\beta+\beta'-\beta^2. \end{align} So before even addressing your question about positive solutions, we need conditions on $\alpha,\beta,\Gamma$.

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    $\begingroup$ This is a very good point; indeed, for the particular case I have in mind I'm assuming that the coefficients satisfy whatever compatibility conditions they must for the problem to have solutions. I'll add this to the question. $\endgroup$ – Sebastian Nov 10 '18 at 2:40

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