### 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 |

### Fingerprint

### 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

### Cite this

*SIAM Journal on Mathematical Analysis*,

*37*(2), 411-437. https://doi.org/10.1137/S0036141004432827

**Limits of solutions of p-Laplace equations as p goes to infinity and related variational problems.** / Ishii, Hitoshi; Loreti, Paola.

Research output: Contribution to journal › Article

*SIAM Journal on Mathematical Analysis*, vol. 37, no. 2, pp. 411-437. https://doi.org/10.1137/S0036141004432827

}

TY - JOUR

T1 - Limits of solutions of p-Laplace equations as p goes to infinity and related variational problems

AU - Ishii, Hitoshi

AU - Loreti, Paola

PY - 2006

Y1 - 2006

N2 - We show that the convergence, as p → ∞, of the solution u p of the Dirichlet problem for -Δpu(x) = f(x) in a bounded domain ω ⊂ Rn 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) ∩ W1,∞(ω) satisfying v| ∂ω =0 and ||Dv||L∞(ω) ≤1.

AB - We show that the convergence, as p → ∞, of the solution u p of the Dirichlet problem for -Δpu(x) = f(x) in a bounded domain ω ⊂ Rn 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) ∩ W1,∞(ω) satisfying v| ∂ω =0 and ||Dv||L∞(ω) ≤1.

KW - ∞-Laplace equation

KW - Asymptotic behavior

KW - Eikonal equation

KW - L variational problem

KW - P-Laplace equation

KW - Variational problem

UR - http://www.scopus.com/inward/record.url?scp=31644441878&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=31644441878&partnerID=8YFLogxK

U2 - 10.1137/S0036141004432827

DO - 10.1137/S0036141004432827

M3 - Article

AN - SCOPUS:31644441878

VL - 37

SP - 411

EP - 437

JO - SIAM Journal on Mathematical Analysis

JF - SIAM Journal on Mathematical Analysis

SN - 0036-1410

IS - 2

ER -