Assume $\boldsymbol{g}:\mathbb{R}_{+}^{n}\to [0,1]^{n}$ and $\boldsymbol{h}:\mathbb{R}_{+}^{n}\to \mathbb{R}^{n}$ continuous functions, and that if I know $\boldsymbol{g}$ to be decreasing* then I can deduce $\boldsymbol{h}$ strictly increasing. Now let $V=\left\{\boldsymbol{y}\in \mathbb{R}_{+}^{n}\mid \forall i\quad 0<h_{i}(\boldsymbol{y})\right\}$ and solve for each $f_{i}$ with \begin{equation*} \begin{cases} Lf_{i}=0&\text{ in }V\\ f_{i} = g_{i}&\text{ on }\partial V. \end{cases} \end{equation*} Assume $S$ a constant, positive definite $n\times n$ matrix and \begin{equation*} Lf=-\frac{1}{2}\sum_{jk}S_{jk}\frac{\partial^{2} f}{\partial y_{j}\partial y_{k}} +\frac{1}{2}\sum_{j}S_{jj}\frac{\partial f}{\partial y_{j}}+f. \end{equation*} My intuition is that if $\boldsymbol{g}$ is in fact decreasing then so is $\boldsymbol{f}$. Is it true?

*If $x_{i}\leq y_{i}$ for all $i$ then $g_{i}(\boldsymbol{x})\geq g_{i}(\boldsymbol{y})$ for all $i$.

Background I am trying to build a system of functions inductively. Each successive function is determined by solving the PDE using boundary conditions prescribed by previous solutions. If I can show by induction that all these functions are decreasing, then I can claim something about boundaries being uniquely determined and the system of functions being uniquely determined. This SE question is relevant.

What have I done? I have been reading in Evans about various maximum principles. But even if I could show that $f_{i}$ has no minimum in the interior of $V$, I don't think it would be enough for the induction... Edit: just realised that Evans assumes bounded domains everywhere :(

Even just a good book recommendation would be welcome!



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