The p-Laplacian equation in thin domains: The unfolding approach

Arrieta, José M.; Nakasato, Jean Carlos; Pereira, Marcone C.

doi:10.1016/j.jde.2020.12.004

Cited by 16 publications

(11 citation statements)

References 18 publications

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“…Let us now deal with the convergence of the problem (2) at (1) and its dependence on α. Combining convergence results from Section 2 and [3,15], we obtain the following theorem. Theorem 3.2.…”

Section: The Linear Perturbation Casementioning

confidence: 71%

“…In order to accomplish our goal, we combine and adapt techniques developed to deal with asymptotic analysis for singular boundary value problems. We use the unfolding operator method for thin domains from [6,7], some monotone techniques used for the treatment of the p-Laplacian performed for instance in [3,12,15], tools to deal with concentrated phenomenas and singular integrals introduced in [1,2,4,5], and iterated homogenization technique given by [9,13].…”

Section: Introductionmentioning

confidence: 99%

“…) on open sets U ⊂ R 2 . Due to Minty-Browder's Theorem, one can show that problem (6) is well posed, and then, the constant q is well defined and positive for any α > 0 (see [3,15]). If α < 1, the homogenized coefficient q depends on p ∈ (1, ∞), which establishes the order of the p-Laplacian operator, and on function g, which sets the profile of R ε .…”

Section: Introductionmentioning

confidence: 99%

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The $$p\,$$-Laplacian equation in a rough thin domain with terms concentrating on the boundary

Nogueira

Nakasato

2020

Annali di Matematica

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In this work we use reiterated homogenization and unfolding operator approach to study the asymptotic behavior of the solutions of the p-Laplacian equation with Neumann boundary conditions set in a rough thin domain with concentrated terms on the boundary. We study weak, resonant and high roughness, respectively. In the three cases, we deduce the effective equation capturing the dependence on the geometry of the thin channel and the neighborhood where the concentrations take place.

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Section: The Linear Perturbation Casementioning

confidence: 71%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The $$p\,$$-Laplacian equation in a rough thin domain with terms concentrating on the boundary

Nogueira

Nakasato

2020

Annali di Matematica

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“…The analysis of eigenvalue problems for differential operators on thin domains has attracted noticeable interest in recent years, see e.g., [4,5,6,9,11,12,13,20,23,29,30,36,38,40] and references therein. A somehow complementary point of view is adopted in the asymptotic analysis of domains with small holes or perforations, see e.g., [1,16,19,33,39].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On the eigenvalues of the biharmonic operator with Neumann boundary conditions on a thin set

Ferraresso¹,

Provenzano²

2021

Preprint

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“…We now recollect some literature on the studying of p-Laplacian equation. This equation arises naturally in a boundary value problem of partial differential equations and has been widely used in various fields of science and technology; see [2,7,8] and the references therein. In the absence of noise, that is a deterministic p-Laplacian equations, many studies have been done on various aspects of attractors.…”

mentioning

confidence: 99%

Regularity of Wong-Zakai approximation for non-autonomous stochastic quasi-linear parabolic equation on <inline-formula><tex-math id="M1">$ {\mathbb{R}}^N $</tex-math></inline-formula>

Liu

Zhao

2021

era

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<p style='text-indent:20px;'>In this paper, we investigate a non-autonomous stochastic quasi-linear parabolic equation driven by multiplicative white noise by a Wong-Zakai approximation technique. The convergence of the solutions of quasi-linear parabolic equations driven by a family of processes with stationary increment to that of stochastic differential equation with white noise is obtained in the topology of <inline-formula><tex-math id="M2">\begin{document}$ L^2( {\mathbb{R}}^N) $\end{document}</tex-math></inline-formula> space. We establish the Wong-Zakai approximations of solutions in <inline-formula><tex-math id="M3">\begin{document}$ L^l( {\mathbb{R}}^N) $\end{document}</tex-math></inline-formula> for arbitrary <inline-formula><tex-math id="M4">\begin{document}$ l\geq q $\end{document}</tex-math></inline-formula> in the sense of upper semi-continuity of their random attractors, where <inline-formula><tex-math id="M5">\begin{document}$ q $\end{document}</tex-math></inline-formula> is the growth exponent of the nonlinearity. The <inline-formula><tex-math id="M6">\begin{document}$ L^l $\end{document}</tex-math></inline-formula>-pre-compactness of attractors is proved by using the truncation estimate in <inline-formula><tex-math id="M7">\begin{document}$ L^q $\end{document}</tex-math></inline-formula> and the higher-order bound of solutions.</p>

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The p-Laplacian equation in thin domains: The unfolding approach

Cited by 16 publications

References 18 publications

The $$p\,$$-Laplacian equation in a rough thin domain with terms concentrating on the boundary

The $$p\,$$-Laplacian equation in a rough thin domain with terms concentrating on the boundary

On the eigenvalues of the biharmonic operator with Neumann boundary conditions on a thin set

Regularity of Wong-Zakai approximation for non-autonomous stochastic quasi-linear parabolic equation on <inline-formula><tex-math id="M1">$ {\mathbb{R}}^N $</tex-math></inline-formula>

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