Existence and Uniqueness Results for a Class of Singular Elliptic Problems in Two-Component Domains

Donato, Patrizia; Raimondi, Federica

doi:10.1007/978-3-319-59384-5_8

Cited by 4 publications

(4 citation statements)

References 9 publications

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“…Proposition 4.3. [26] Under assumptions H 1 )-H 3 ), let u ε ∈ H ε γ be the solution of problem (2.7) and δ a xed positive real number. Then…”

Section: The Main Resultsmentioning

confidence: 99%

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Homogenization of a class of singular elliptic problems in two-component domains

Raimondi

2023

ASY

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This paper deals with the homogenization of a quasilinear elliptic problem having a singular lower order term and posed in a two-component domain with an ε-periodic imperfect interface. We prescribe a Dirichlet condition on the exterior boundary, while we assume that the continuous heat flux is proportional to the jump of the solution on the interface via a function of order ε γ . We prove an homogenization result for − 1 < γ < 1 by means of the periodic unfolding method (see SIAM J. Math. Anal. 40 (2008) 1585–1620 and The Periodic Unfolding Method (2018) Springer), adapted to two-component domains in (J. Math. Sci. 176 (2011) 891–927). One of the main tools in the homogenization process is a convergence result for a suitable auxiliary linear problem, associated with the weak limit of the sequence { u ε } of the solutions, as ε → 0. More precisely, our result shows that the gradient of u ε behaves like that of the solution of the auxiliary problem, which allows us to pass to the limit in the quasilinear term, and to study the singular term near its singularity, via an accurate a priori estimate.

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“…Proposition 4.3. [26] Under assumptions H 1 )-H 3 ), let u ε ∈ H ε γ be the solution of problem (2.7) and δ a xed positive real number. Then…”

Section: The Main Resultsmentioning

confidence: 99%

“…For this case we refer to [23] where dierent a priori estimates needed. As for our problem, existence and uniqueness results has been proved in [26]- [27] for…”

Section: Introductionmentioning

confidence: 99%

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Homogenization of a class of singular elliptic problems in two-component domains

Raimondi

2023

ASY

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“…The exact controllability and exact boundary controllability for semilinear wave equations can be found in [41] and [61], respectively. Finally, for what concerns transmission problems in the nonlinear case we quote [28] and [29] (see also [25] and [26]).…”

Section: γ Interface Boundary ∂ωmentioning

confidence: 99%

Homogenization and exact controllability for problems with imperfect interface

Monsurrò¹,

Perugia²

2019

Networks &Amp; Heterogeneous Media

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In this paper we study the internal exact controllability for a second order linear evolution equation defined in a two-component domain. On the interface we prescribe a jump of the solution proportional to the conormal derivatives, meanwhile a homogeneous Dirichlet condition is imposed on the exterior boundary. Due to the geometry of the domain, we apply controls through two regions which are neighborhoods of a part of the external boundary and of the whole interface, respectively. Our approach to internal exact controllability consists in proving an observability inequality by using the Lagrange multipliers method. Eventually we apply the Hilbert Uniqueness Method, introduced by J. -L. Lions, which leads to the construction of the exact control through the solution of an adjoint problem. Finally we find a lower bound for the control time depending not only on the geometry of our domain and on the matrix of coefficients of our problem but also on the coefficient of proportionality of the jump with respect to the conormal derivatives.

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Exact Internal Controllability for a Problem with Imperfect Interface

2022

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In this paper, we study the internal exact controllability for a second order linear evolution equation defined in a two-component domain. On the interface, we prescribe a jump of the solution proportional to the conormal derivatives, meanwhile a homogeneous Dirichlet condition is imposed on the exterior boundary. Due to the geometry of the domain, we apply controls through two regions which are neighborhoods of a part of the external boundary and of the whole interface, respectively. Our approach to internal exact controllability consists in proving an observability inequality by using the Lagrange multipliers method. Eventually, we apply the Hilbert Uniqueness Method, introduced by Lions, which leads to the construction of the exact control through the solution of an adjoint problem. Finally, we find a lower bound for the control time depending not only on the geometry of our domain and on the matrix of coefficients of our problem but also on the coefficient of proportionality of the jump with respect to the conormal derivatives.

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Existence and Uniqueness Results for a Class of Singular Elliptic Problems in Two-Component Domains

Cited by 4 publications

References 9 publications

Homogenization of a class of singular elliptic problems in two-component domains

Homogenization of a class of singular elliptic problems in two-component domains

Homogenization and exact controllability for problems with imperfect interface

Exact Internal Controllability for a Problem with Imperfect Interface

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