Abstract
We obtain the homogenized problem associated with the Poisson equation in a domain perforated by "tiny" balls (or in a domain defined as the exterior to a periodic set of very small particles) of radius \(a_{\varepsilon }=C_{0}\varepsilon ^{\gamma }\) with \(\gamma =\frac{ n}{n-2},\quad C_{0}>0\) (the so-called, "critical case"). On the boundary of these balls, we assume a dynamic unilateral boundary condition (the so-called "Signorini boundary condition"). We prove that the homogenized problem consists of an elliptic equation coupled with an ordinary differential unilateral problem: in contrast with the case of "big perforations" (or "big particles") for which the equation is a unilateral parabolic problem. In particular, we prove that the solution to the homogenized problem may become regionally negative (at least in the interior of some subset of \(Q^{T}=\Omega \times (0,T)\) on which f(x, t) is negative). Nothing similar may happen in the case of big particles since the corresponding homogenized problem imply that the the solution is always non-negative, even for very negative data f(x, t)
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Acknowledgements
The research of J.I. Díaz was partially supported by the project ref. PID2020-112517GB-I00 of the DGISPI (Spain) and the Research Group MOMAT (Ref. 910480) of the UCM.
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Díaz, J.I., Podolskiy, A.V. & Shaposhnikova, T.A. Unexpected regionally negative solutions of the homogenization of Poisson equation with dynamic unilateral boundary conditions: critical symmetric particles. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 9 (2024). https://doi.org/10.1007/s13398-023-01503-w
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DOI: https://doi.org/10.1007/s13398-023-01503-w
Keywords
- Critically scaled homogenization
- Perforated domain
- The exterior of periodic particles
- Unilateral dynamic boundary conditions
- Strange terms
- Regionally negative solutions