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arXiv · 2111.03875

A stability result for elliptic equations with singular nonlinearity and its applications to homogenization problems

Takanobu Hara
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Abstract

We consider model semilinear elliptic equations of the type \[ \begin{cases} - \mathrm{div} (A(x) \nabla u) = f u^{- λ}, \quad u > 0 \quad \text{in} \ Ω, \\ u \in H_{0}^{1}(Ω), \end{cases} \] where $Ω$ is a bounded domain in $\mathbf{R}^{N}$, $N \ge 1$, $A \in L^{\infty}(Ω)^{N \times N}$ is a coercive matrix, $0 < λ\le 1$ and $f$ is a nonnegative function in $L^{1}_{loc}(Ω)$, or more generally, nonnegative Radon measure on $Ω$. We discuss $H^{1}$-stability of $u$ under a minimal assumption on $f$. Additionally, we apply the result to homogenization problems.

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Takanobu Hara. 2021-11-06. A stability result for elliptic equations with singular nonlinearity and its applications to homogenization problems. https://doi.org/10.1016/j.jmaa.2023.127509

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