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<p>In the present paper we consider a nonlinear transmission problem motivated by the question of homogenizing the brush problem. The equation studied is degenerated (only the vertical derivative is involved) in the upper part of the domain, while in the lower part the equation is a classical diffusion problem. A flux condition is added at the interface. We consider here a fairly general class of nonlinear operators with <i>p</i>-growth and a right-hand side in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>. We give an appropriate definition of renormalized solutions for which we prove existence, uniqueness and stability.</p>

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Existence and uniqueness results for a class of nonlinear transmission problem with \(L^1\) data

  • 期刊论文

  • 发表日期：

    2025年10月31日

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  • Silvio Bove,
  • Olivier Guibé

摘要

In the present paper we consider a nonlinear transmission problem motivated by the question of homogenizing the brush problem. The equation studied is degenerated (only the vertical derivative is involved) in the upper part of the domain, while in the lower part the equation is a classical diffusion problem. A flux condition is added at the interface. We consider here a fairly general class of nonlinear operators with p-growth and a right-hand side in \(L^1\) L 1 . We give an appropriate definition of renormalized solutions for which we prove existence, uniqueness and stability.