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

Hitoshi Ishii; Paola Loreti

doi:10.1137/S0036141004432827

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

Hitoshi Ishii^*, Paola Loreti

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

21 Citations (Scopus)

Abstract

We show that the convergence, as p → ∞, of the solution u _p of the Dirichlet problem for -Δ_pu(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) ∩ W^1,∞(ω) satisfying v| _∂ω =0 and ||Dv||_L∞(ω) ≤1.

Original language	English
Pages (from-to)	411-437
Number of pages	27
Journal	SIAM Journal on Mathematical Analysis
Volume	37
Issue number	2
DOIs	https://doi.org/10.1137/S0036141004432827
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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10.1137/S0036141004432827

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abstract = "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({\=ω}) ∩ W1,∞(ω) satisfying v| ∂ω =0 and ||Dv||L∞(ω) ≤1.",

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language = "English",

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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

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DO - 10.1137/S0036141004432827

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

Abstract

Keywords

ASJC Scopus subject areas

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