<p>In this paper, we consider the Lavrent’ev–Bitsadze equation in a partially perforated model domain with a characteristic size of micro-inhomogeneities <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varepsilon .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We study the boundary-value problem with a boundary condition of the third kind on boundaries of cavities (the Fourier condition), which has a small multiplicative parameter <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varepsilon ^\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>ε</mi> <mi>α</mi> </msup> </math></EquationSource> </InlineEquation> in coefficients, and the Dirichlet condition on the outer part of boundary. We construct a homogenized problem and prove the convergence of the solutions of the original problem to the solution of the homogenized problem in three cases. The subcritical case: <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha >1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and dissipation at the boundary of the cavities is negligibly small; the critical case: <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha =1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and a potential appears in the equation due to dissipation; the supercritical case: <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\alpha <1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and the dissipation plays the major role, it leads to degeneracy of the solution of the entire problem.</p>

ABOUT HOMOGENIZATION OF THE LAVRENT’EV–BITSADZE EQUATION IN A PARTIALLY PERFORATED DOMAIN WITH THE THIRD BOUNDARY CONDITION ON BOUNDARIES OF CAVITIES: SUBCRITICAL, CRITICAL, AND SUPERCRITICAL CASES

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发表日期：

2025年9月1日

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G. A. Chechkin

摘要

In this paper, we consider the Lavrent’ev–Bitsadze equation in a partially perforated model domain with a characteristic size of micro-inhomogeneities $\varepsilon .$ $ε .$ We study the boundary-value problem with a boundary condition of the third kind on boundaries of cavities (the Fourier condition), which has a small multiplicative parameter $\varepsilon ^\alpha $ $ε^{α}$ in coefficients, and the Dirichlet condition on the outer part of boundary. We construct a homogenized problem and prove the convergence of the solutions of the original problem to the solution of the homogenized problem in three cases. The subcritical case: $\alpha >1$ $α > 1$ and dissipation at the boundary of the cavities is negligibly small; the critical case: $\alpha =1$ $α = 1$ and a potential appears in the equation due to dissipation; the supercritical case: $\alpha <1$ $α < 1$ and the dissipation plays the major role, it leads to degeneracy of the solution of the entire problem.