Consider a long cylinder $C = D \times (-L,L) \subset \mathbf{R}^3$, with heat applied to its horizontal boundary according to $\varphi$ and perfectly insulated ends. The steady state $u: C \to \mathbf{R}$ representing the temperature is the solution of the system \begin{equation} \begin{cases} \Delta u = 0 \quad \text{in $C$} \\ u = \varphi \quad \text{on $\partial D \times (-L,L)$} \\ \frac{\partial u}{\partial \nu} = 0 \quad \text{ on $D \times \{-L,L\}$}. \end{cases} \end{equation}

In heuristic terms, 'most important' for the values of $u$ on $D \times \{ 0 \}$ are the boundary values with similar height. Heat applied further away, 'nearer to the ends' of the cylinder is still 'felt' by $u$ on $D \times \{ 0 \}$, but 'much less'.

Question. How does one establish estimates that quantify the 'waning influence' of $\varphi$ on $u$ with the height? Maybe this would be in terms of a weight function $w: (-L,L) \to \mathbf{R}$, say $w(t) = \lvert L - t \rvert$ or something stronger?

Edit. Apparently there are estimates with exponentially decaying weight $w(t) = \mathrm{e}^{-C \lvert t \rvert}$, perhaps something like \begin{equation} \lvert u(\cdot,0) \rvert_{L^2(D)} \leq C \lvert \mathrm{e}^{-C \lvert t \rvert} \varphi\rvert_{L^2(\partial D \times (-L,L))}. \end{equation} Although I am inclined to believe these bounds by fiat, I am ultimately most interested in the arguments used to derive them (or indeed a reference to such arguments). This wasn't explicit in the wording of the earlier version of the question, so I've amended it to clarify this.