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I have read a paper of Z. Shen [1]. In the paper the author mentioned we can deal with two-dimensional elliptic systems by adding a dummy variable (the method of ascending) and use the results on the related three-dimensional problem. I guess it is a standard way to deal with elliptic systems with lower dimensions. However, I do not find any reference about it: can you give me some references or hints?

Reference

[1] Zhongwei Shen, "Convergence rates and Hölder estimates in almost-periodic homogenization of elliptic systems" (English), Analysis & PDE 8, No. 7, 1565-1601 (2015), DOI: 10.2140/apde.2015.8.1541, MR3399132, Zbl 1327.35025.

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    $\begingroup$ Just set the functions to be constant along the dummy variable? Simplest example: if $\Delta u = f$ on the two dimensional ball, set $\tilde{u}(x,y,z) = u(x,y)$ and $\tilde{f}(x,y,z) = f(x,y)$, then $\Delta \tilde{u} = \tilde{f}$ on the three dimensional ball, and then any estimates for solutions of the three dimensional Poisson problem gives an analogous estimate for the two dimensional problem. $\endgroup$
    – Willie Wong
    Commented Jan 1, 2022 at 14:12
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    $\begingroup$ (It is not so much "elliptic" specific: most students of PDE probably saw it first when learning about deriving the fundamental solution for the linear wave equation on $\mathbb{R}^{1+2}$ from that of $\mathbb{R}^{1+3}$.) $\endgroup$
    – Willie Wong
    Commented Jan 1, 2022 at 14:14

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