A singular perturbation problem for A quasi-linear operator satisfying the natural growth condition of lieberman
In this paper we study the following problem. For ε > 0, take uε as a solution of Luε := div(g(ι∇u ει/ι∇uει)∇u ε) = βε(uε), u ε ≥ 0. A solution to (Pε) is a function uε Ε W1,G(Ω) ∩ L&infin(Ω) such that ∫ωg(ι∇uει) ∇u ε/ι∇uει ∇ℓ dx = ̄ ∫ω ℓβε(uε) dx for every ℓ Ε C&infin 0(Ω). H...
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todo:paper_00361410_v41_n1_p318_Sandra2023-10-03T14:47:36Z A singular perturbation problem for A quasi-linear operator satisfying the natural growth condition of lieberman Sandra, M. Noemi, W. Free boundaries Orlicz spaces Singular perturbation Following problem Free boundary Free-boundary problems Growth conditions Limiting problem Orlicz spaces Quasi-linear Singular perturbation problems Singular perturbations Weak solution Differential equations Mathematical operators In this paper we study the following problem. For ε > 0, take uε as a solution of Luε := div(g(ι∇u ει/ι∇uει)∇u ε) = βε(uε), u ε ≥ 0. A solution to (Pε) is a function uε Ε W1,G(Ω) ∩ L&infin(Ω) such that ∫ωg(ι∇uει) ∇u ε/ι∇uει ∇ℓ dx = ̄ ∫ω ℓβε(uε) dx for every ℓ Ε C&infin 0(Ω). Here βε (s) = 1/εβ(s/ε), with β Ε Lip(rdbl), β > 0in(0, 1) and β = 0 otherwise. We are interested in the limiting problem, when ε → 0. As in previous work with L = Δ or L = Δp we prove, under appropriate assumptions, that any limiting function is a weak solution to a free boundary problem. Moreover, for nondegenerate limits we prove that the reduced free boundary is a C 1, α surface. This result is new even for Δp. Throughout the paper, we assume that g satisfies the conditions introduced by Lieberman in [ Comm. Partial Differential Equations, 16 (1991), pp. 311-361]. © 2009 Society for Industrial and Applied Mathematics. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00361410_v41_n1_p318_Sandra |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Free boundaries Orlicz spaces Singular perturbation Following problem Free boundary Free-boundary problems Growth conditions Limiting problem Orlicz spaces Quasi-linear Singular perturbation problems Singular perturbations Weak solution Differential equations Mathematical operators |
| spellingShingle |
Free boundaries Orlicz spaces Singular perturbation Following problem Free boundary Free-boundary problems Growth conditions Limiting problem Orlicz spaces Quasi-linear Singular perturbation problems Singular perturbations Weak solution Differential equations Mathematical operators Sandra, M. Noemi, W. A singular perturbation problem for A quasi-linear operator satisfying the natural growth condition of lieberman |
| topic_facet |
Free boundaries Orlicz spaces Singular perturbation Following problem Free boundary Free-boundary problems Growth conditions Limiting problem Orlicz spaces Quasi-linear Singular perturbation problems Singular perturbations Weak solution Differential equations Mathematical operators |
| description |
In this paper we study the following problem. For ε > 0, take uε as a solution of Luε := div(g(ι∇u ει/ι∇uει)∇u ε) = βε(uε), u ε ≥ 0. A solution to (Pε) is a function uε Ε W1,G(Ω) ∩ L&infin(Ω) such that ∫ωg(ι∇uει) ∇u ε/ι∇uει ∇ℓ dx = ̄ ∫ω ℓβε(uε) dx for every ℓ Ε C&infin 0(Ω). Here βε (s) = 1/εβ(s/ε), with β Ε Lip(rdbl), β > 0in(0, 1) and β = 0 otherwise. We are interested in the limiting problem, when ε → 0. As in previous work with L = Δ or L = Δp we prove, under appropriate assumptions, that any limiting function is a weak solution to a free boundary problem. Moreover, for nondegenerate limits we prove that the reduced free boundary is a C 1, α surface. This result is new even for Δp. Throughout the paper, we assume that g satisfies the conditions introduced by Lieberman in [ Comm. Partial Differential Equations, 16 (1991), pp. 311-361]. © 2009 Society for Industrial and Applied Mathematics. |
| format |
JOUR |
| author |
Sandra, M. Noemi, W. |
| author_facet |
Sandra, M. Noemi, W. |
| author_sort |
Sandra, M. |
| title |
A singular perturbation problem for A quasi-linear operator satisfying the natural growth condition of lieberman |
| title_short |
A singular perturbation problem for A quasi-linear operator satisfying the natural growth condition of lieberman |
| title_full |
A singular perturbation problem for A quasi-linear operator satisfying the natural growth condition of lieberman |
| title_fullStr |
A singular perturbation problem for A quasi-linear operator satisfying the natural growth condition of lieberman |
| title_full_unstemmed |
A singular perturbation problem for A quasi-linear operator satisfying the natural growth condition of lieberman |
| title_sort |
singular perturbation problem for a quasi-linear operator satisfying the natural growth condition of lieberman |
| url |
http://hdl.handle.net/20.500.12110/paper_00361410_v41_n1_p318_Sandra |
| work_keys_str_mv |
AT sandram asingularperturbationproblemforaquasilinearoperatorsatisfyingthenaturalgrowthconditionoflieberman AT noemiw asingularperturbationproblemforaquasilinearoperatorsatisfyingthenaturalgrowthconditionoflieberman AT sandram singularperturbationproblemforaquasilinearoperatorsatisfyingthenaturalgrowthconditionoflieberman AT noemiw singularperturbationproblemforaquasilinearoperatorsatisfyingthenaturalgrowthconditionoflieberman |
| _version_ |
1807316001424932864 |