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<p>In this paper, we investigate the behavior of the solutions of a semilinear elliptic equation posed in a thin domain with an outward peak. Such peak is given by a nonnegative function which sets the profile of our region and makes the analysis challenging. In a first moment, we apply standard methods from asymptotic analysis and thin domains to show the strong convergence of the solutions of the linear version of the problem also determining the rate of convergence. In the sequel, we derive conditions under which the linear limit equation has a compact resolvent in order to analyze the semilinear equation. We obtain upper and lower semicontinuity of the solutions establishing rate of convergence under appropriated conditions on the geometry of the thin domain.</p>

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A semilinear elliptic equation with homogeneous Neumann boundary conditions posed in thin domains with outward peaks

  • 期刊论文

  • 发表日期：

    2025年10月3日

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  • Patrícia Neves de Araújo,
  • Jean Carlos Nakasato,
  • Marcone Corrêa Pereira

摘要

In this paper, we investigate the behavior of the solutions of a semilinear elliptic equation posed in a thin domain with an outward peak. Such peak is given by a nonnegative function which sets the profile of our region and makes the analysis challenging. In a first moment, we apply standard methods from asymptotic analysis and thin domains to show the strong convergence of the solutions of the linear version of the problem also determining the rate of convergence. In the sequel, we derive conditions under which the linear limit equation has a compact resolvent in order to analyze the semilinear equation. We obtain upper and lower semicontinuity of the solutions establishing rate of convergence under appropriated conditions on the geometry of the thin domain.