A maximum principle for the $p$-Laplacian, an eigenvalue estimate and a stabilization phenomenon for the large-$p$ regime
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arXiv
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| Format: | Preprint |
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2026
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| author | Carrillo-Reina, Kevin Cortissoz, Jean C. |
| author_facet | Carrillo-Reina, Kevin Cortissoz, Jean C. |
| contents | We establish an explicit maximum principle for the Dirichlet problem associated with the $p$-Laplacian ($p>1$), where the constant depends on both $p$ and the geometry of the domain. From this result we derive two main applications. First, we obtain a new lower bound for the first nontrivial eigenvalue of the $p$-Laplacian, which improves upon existing estimates in certain parameter regimes and for thin domains. Second, we prove an existence theorem for nonlinear boundary value problems of the form \[ -Δ_p u = λf(u) \quad \text{in } Ω, \qquad u=0 \quad \text{on } \partial Ω, \] with $f$ nonnegative, continuous and nondecreasing. A striking consequence is the emergence of a \emph{stabilization phenomenon}: for every such nonlinearity there exists a threshold $p_0 \colon = p_0(f,λ,Ω)$ such that for all $p \geq p_0$ solutions exist. To our knowledge, this stabilization effect with respect to $p$, that apparently has not been observed before, suggests a connection to the $\infty$-Laplacian. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_16307 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A maximum principle for the $p$-Laplacian, an eigenvalue estimate and a stabilization phenomenon for the large-$p$ regime Carrillo-Reina, Kevin Cortissoz, Jean C. Analysis of PDEs 35J92, 35B50, 35J70, 35J60 We establish an explicit maximum principle for the Dirichlet problem associated with the $p$-Laplacian ($p>1$), where the constant depends on both $p$ and the geometry of the domain. From this result we derive two main applications. First, we obtain a new lower bound for the first nontrivial eigenvalue of the $p$-Laplacian, which improves upon existing estimates in certain parameter regimes and for thin domains. Second, we prove an existence theorem for nonlinear boundary value problems of the form \[ -Δ_p u = λf(u) \quad \text{in } Ω, \qquad u=0 \quad \text{on } \partial Ω, \] with $f$ nonnegative, continuous and nondecreasing. A striking consequence is the emergence of a \emph{stabilization phenomenon}: for every such nonlinearity there exists a threshold $p_0 \colon = p_0(f,λ,Ω)$ such that for all $p \geq p_0$ solutions exist. To our knowledge, this stabilization effect with respect to $p$, that apparently has not been observed before, suggests a connection to the $\infty$-Laplacian. |
| title | A maximum principle for the $p$-Laplacian, an eigenvalue estimate and a stabilization phenomenon for the large-$p$ regime |
| topic | Analysis of PDEs 35J92, 35B50, 35J70, 35J60 |
| url | https://arxiv.org/abs/2605.16307 |