# Can one show that the solution to this elliptic PDE is a decreasing function?

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 $$V=\left\{\boldsymbol{y}\in \mathbb{R}_{+}^{n}\mid \forall i\quad 0 and solve for each $$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!