Asymptotics of Large Solutions of p-Laplace Equations on Cylinders Becoming Unbounded
N. N. Dattatreya
TL;DR
This work studies the asymptotics of large p-Laplacian solutions on cylinders S_ℓ that become unbounded in one direction, proving that they converge locally in Sobolev space to the cross-sectional problem on ω2 as ℓ → ∞. A quantitative convergence rate of order ℓ^{-1/p} for the gradient is established, and the analysis extends prior results to the range 1 < p < 2 while unifying finite boundary data and boundary blow-up (large) solutions. The authors show that, under mild uniqueness or nonuniqueness frameworks, the limit profile is independent of the axial variable and satisfies the cross-sectional PDE, thus enabling a clean reduction to a lower-dimensional problem. This has implications for understanding high-dimensional p-Laplacian behavior on elongated domains and provides explicit convergence rates applicable in both theoretical and applied contexts.
Abstract
In this article, we study the asymptotic behavior of large solutions for a quasi-linear equation involving the p-Laplacian, defined on a sequence of finite cylindrical domains converging to an infinite cylinder. We demonstrate that the sequence of solutions converges locally, in the Sobolev norm, to a solution of the corresponding cross-sectional problem. Moreover, we establish a convergence rate. As part of our analysis, we extend existing convergence results for the case $p\geq 2$, which previously lacked explicit convergence rates, to the range $1<p<2$. We additionally address solutions with finite Dirichlet boundary data within a unified framework and exhibit that this rate of convergence is independent of the boundary data.