Maximal solutions for the ∞-eigenvalue problem.

In this article we prove that the first eigenvalue of the ∞ {\infty} -Laplacian { min ⁡ { - Δ ∞ ⁢ v , | ∇ ⁡ v | - λ 1 , ∞ ⁢ (Ω) ⁢ v } = 0 in ⁢ Ω , v = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle\min\{-\Delta_{\infty}v,|\nabla v|-\lambda% _{1,\infty}(\Omega)v\}&\displaystyle=0&&\displaystyle\text{in }\Omega,\\ \displaystyle v&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right. has a unique (up to scalar multiplication) maximal solution. This maximal solution can be obtained as the limit as ℓ ↗ 1 {\ell\nearrow 1} of concave problems of the form { min ⁡ { - Δ ∞ ⁢ v ℓ , | ∇ ⁡ v ℓ | - λ 1 , ∞ ⁢ (Ω) ⁢ v ℓ ℓ } = 0 in ⁢ Ω , v ℓ = 0 on ⁢ ∂ ⁡ Ω. \left\{\begin{aligned} \displaystyle\min\{-\Delta_{\infty}v_{\ell},|\nabla v_{% \ell}|-\lambda_{1,\infty}(\Omega)v_{\ell}^{\ell}\}&\displaystyle=0&&% \displaystyle\text{in }\Omega,\\ \displaystyle v_{\ell}&\displaystyle=0&&\displaystyle\text{on }\partial\Omega.% \end{aligned}\right. In this way we obtain that the maximal eigenfunction is the unique one that is the limit of the sub-homogeneous problems as happens for the usual eigenvalue problem for the p-Laplacian for a fixed 1 < p < ∞ {1<p<\infty}. [ABSTRACT FROM AUTHOR]