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In this article, we show that there exists a bounded $C^1$ domain $\\Omega\\subset \\mathbb R^n$ such that, for any given $s\\in(1,2)\\setminus\\{\\frac32\\}$, \\begin{align*} \\left[H_0^1(\\Omega),H^2(\\Omega)\\cap H_0^1(\\Omega)\\right]_{s-1} =H^s(\\Omega)\\cap H_0^1(\\Omega)=H_0^s(\\Omega) \\end{align*} with equivalent norms, but \\begin{align*} \\left[H_0^1(\\Omega),H^2(\\Omega)\\cap H_0^1(\\Omega)\\right]_{\\frac12} \\subsetneqq H^{\\frac32}(\\Omega)\\cap H_0^1(\\Omega), \\end{align*} which provides a counterexample to Problem 3.3.19 of Kenig in [CBMS Regional Conf. 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In this article, we show that there exists a bounded $C^1$ domain $\\Omega\\subset \\mathbb R^n$ such that, for any given $s\\in(1,2)\\setminus\\{\\frac32\\}$, \\begin{align*} \\left[H_0^1(\\Omega),H^2(\\Omega)\\cap H_0^1(\\Omega)\\right]_{s-1} =H^s(\\Omega)\\cap H_0^1(\\Omega)=H_0^s(\\Omega) \\end{align*} with equivalent norms, but \\begin{align*} \\left[H_0^1(\\Omega),H^2(\\Omega)\\cap H_0^1(\\Omega)\\right]_{\\frac12} \\subsetneqq H^{\\frac32}(\\Omega)\\cap H_0^1(\\Omega), \\end{align*} which provides a counterexample to Problem 3.3.19 of Kenig in [CBMS Regional Conf. 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