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IIT Hyderabad

This paper deals with the homogenization of a mixed boundary value problem for the Laplace operator in a domain with locally periodic oscillating boundary. The Neumann condition is prescribed on the oscillating part of the boundary, and the Dirichlet condition on a separate part. It is shown that the homogenization result holds in the sense of weak L2 convergence of the solutions and their flows, under natural hypothesis on the regularity of the domain. The strong L2 convergence of average preserving extensions of the solutions and their flows is also considered. © 2022, The Author(s).

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Applied Mathematics and Optimization

Homogenization of a Locally Periodic Oscillating Boundary

In this paper, the asymptotic behavior of the solutions of a monotone problem posed in a locally periodic oscillating domain is studied. Nonlinear monotone boundary conditions are imposed on the oscillating part of the boundary whereas the Dirichlet condition is considered on the smooth separate part. Using the unfolding method, under natural hypothesis on the regularity of the domain, we prove the weak Lp-convergence of the zero-extended solutions of the nonlinear problem and their flows to the solutions of a limit distributional problem. © 2022 Elsevier Ltd

Nonlinear Analysis: Real World Applications

Homogenization of a nonlinear monotone problem in a locally periodic domain via unfolding method

The work is motivated by the Faraday cage effect. We consider the Helmholtz equation over a 3D domain containing a thin heterogeneous interface of thickness (Formula presented.). The layer has a δ-periodic structure in the in-plane directions and is cylindrical in the third direction. The periodic layer has one connected component and a collection of isolated regions. The isolated region in the thin layer represents air or liquid, and the connected component represents a solid metal grid with a δ thickness. The main issue is created by the contrast of the coefficients in the air and in the grid and that the zero-order term has a complex-valued coefficient in the connected faze while a real-valued in the complement. An asymptotic analysis with respect to (Formula presented.) is provided, and the limit Helmholtz problem is obtained with the Dirichlet condition on the interface. The periodic unfolding method is used to find the limit. © 2023 Informa UK Limited, trading as Taylor & Francis Group.

Complex Variables and Elliptic Equations

Homogenization of Helmholtz equation in a periodic layer to study Faraday cage-like shielding effects

We demonstrate the asymptotic analysis of a semi-linear optimal control problem posed on a smooth oscillating boundary domain in the present paper. We have considered a more general oscillating domain than the usual "pillar-type" domains. Consideration of such general domains will be useful in more realistic applications like circular domain with rugose boundary. We study the asymptotic behavior of the problem under consideration using a new generalized periodic unfolding operator. Further, we are studying the homogenization of a non-linear optimal control problem and such non-linear problems are limited in the literature despite the fact that they have enormous real-life applications. Among several other technical difficulties, the absence of a sufficient criteria for the optimal control is one of the most attention-grabbing issues in the current setting. We also obtain corrector results in this paper. © 2020 World Scientific Publishing Company.

Communications in Contemporary Mathematics

Semi-linear optimal control problem on a smooth oscillating domain

We consider a Dirichlet boundary control problem posed in an oscillating boundary domain governed by a biharmonic equation. Homogenization of a PDE with a non-homogeneous Dirichlet boundary condition on the oscillating boundary is one of the hardest problems. Here, we study the homogenization of the problem by converting it into an equivalent interior control problem. The convergence of the optimal solution is studied using periodic unfolding operator. © 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group.

Applicable Analysis

Biharmonic equation in a highly oscillating domain and homogenization of an associated control problem

This article aims to understand the locally periodic oscillating domain via unfolding operators. A three-dimensional rough domain Ω ε, ε&gt; 0 a small parameter, has been considered for the study where the boundary is rapidly oscillating with high amplitude. Though there are some articles with locally periodic boundary oscillations with small amplitude we do not see any literature with high-amplitude (O(1)) locally periodic oscillating domains. In this article, we attempt to study a problem in locally periodic rough domains with an eye towards the general oscillating domains without periodicity. With our experience of handling such domains and unfolding operators, we develop locally periodic unfolding operators to study our problems. We consider a nonlinear inhomogeneous Robin boundary value problem posed on this domain to demonstrate the utility of the newly defined operator. © 2019, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature.

Annali di Matematica Pura ed Applicata

Locally periodic unfolding operator for highly oscillating rough domains

We consider an optimal control problem posed on a domain with a highly oscillating smooth boundary where the controls are applied on the oscillating part of the boundary. There are many results on domains with oscillating boundaries where the oscillations are pillar-type (non-smooth) while the literature on smooth oscillating boundary is very few. In this article, we use appropriate scaling on the controls acting on the oscillating boundary leading to different limit control problems, namely, boundary optimal control and interior optimal control problem. In the last part of the article, we visualize the domains as a branched structure, and we introduce unfolding operators to get contributions from each level at every branch. © 2019 John Wiley & Sons, Ltd.

Mathematical Methods in the Applied Sciences

Asymptotic analysis of a boundary optimal control problem on a general branched structure

This paper is concerned with the asymptotic analysis of optimal control problems posed on a rough circular domain. The domain has two parts, namely a fixed outer part and an oscillating inner part. The period of the oscillation is of order ϵ > 0, a small parameter which approaches zero and the amplitude of the oscillation is fixed. We pose a periodic control on the oscillating part of the domain and study the homogenization of this problem using an unfolding operator suitably defined for this domain. One of the novelties of this paper is that we use the unfolding operator to characterize the optimal control in the non-homogenized level. © 2019 - IOS Press and the authors. All rights reserved.

Asymptotic Analysis

Control problem on a rough circular domain and homogenization

Unfolding operators have been introduced and used to study homogenization problems. Initially, they were introduced for problems with rapidly oscillating coefficients and porous domains. Later, this has been developed for domains with oscillating boundaries, typically with rectangular or pillar type boundaries which are classified as non-smooth. In this article, we develop new unfolding operators, where the oscillations can be smooth and hence they have wider applications. We have demonstrated by developing unfolding operators for circular domains with rapid oscillations with high amplitude of O(1) to study the homogenization of an elliptic problem. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

Calculus of Variations and Partial Differential Equations

Generalization of unfolding operator for highly oscillating smooth boundary domains and homogenization

We consider a linear parabolic problem in a thick junction domain which is the union of a fixed domain and a collection of periodic branched trees of height of order 1 and small width connected on a part of the boundary. We consider a three-branched structure, but the analysis can be extended to n-branched structures. We use unfolding operator to study the asymptotic behavior of the solution of the problem. In the limit problem, we get a multi-sheeted function in which each sheet is the limit of restriction of the solution to various branches of the domain. Homogenization of an optimal control problem posed on the above setting is also investigated. One of the novelty of the paper is the characterization of the optimal control via the appropriately defined unfolding operators. Finally, we obtain the limit of the optimal control problem. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.