### Abstract

We show that the convergence, as p → ∞, of the solution u _{p} of the Dirichlet problem for -Δ_{p}u(x) = f(x) in a bounded domain ω ⊂ R^{n} with zero-Dirichlet boundary condition and with continuous f in the following cases: (i) one-dimensional case, radial cases; (ii) the case of no balanced family; and (iii) two cases with vanishing integral. We also give some properties of the maximizers for the functional ∫_{ω} f(x)v(x) dx in the space of functions v ∈ C(ω̄) ∩ W^{1,∞}(ω) satisfying v| _{∂ω} =0 and ||Dv||_{L∞}(ω) ≤1.

Original language | English |
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Pages (from-to) | 411-437 |

Number of pages | 27 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 37 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2006 |

### Keywords

- ∞-Laplace equation
- Asymptotic behavior
- Eikonal equation
- L variational problem
- P-Laplace equation
- Variational problem

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics
- Numerical Analysis

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## Cite this

Ishii, H., & Loreti, P. (2006). Limits of solutions of p-Laplace equations as p goes to infinity and related variational problems.

*SIAM Journal on Mathematical Analysis*,*37*(2), 411-437. https://doi.org/10.1137/S0036141004432827