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Djitte","submitted_at":"2025-05-30T22:06:21Z","abstract_excerpt":"We use shape derivative approach to prove that balls are the only convex and $C^{1,1}$ regular domains in which the fractional overdetermined problem \\begin{equation*} \\left\\{\\begin{aligned} \\Ds u&= \\lambda_{s, p} u^{p-1}\\quad\\text{in}\\quad\\Om \\\\ u &= 0\\quad \\text{in}\\quad\\R^N\\setminus \\Om\\\\ u/d^s&=C_0\\quad\\text{on\\;\\; $\\partial\\O$} \\end{aligned} \\right. \\end{equation*} admits a nontrivial solution for $p\\in [1, 2]$ and where $\\lambda_{s, p}= \\lambda_{s, p}(\\O)$ is the best constant in the family of Subcritical Sobolev inequalities. 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