A Counterexample to Kenig's Interpolation Problem for Sobolev Spaces with Zero Boundary Conditions

Abstract

Let $n\in \mathbb N\cap[2,\infty)$. 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. Ser. in Math. 83, 1994]. As applications, we prove that for such a domain $\Omega$ \begin{align*} H^2(\Omega)\cap H_0^1(\Omega)\subsetneqq D(-\Delta_D) \end{align*} (the domain of the Dirichlet Laplacian operator $-\Delta_D$ on $\Omega$) and construct a solution of the homogeneous heat equation with zero Dirichlet boundary condition, which does not belong to $L^2((0,T);H^2(\Omega)\cap H_0^1(\Omega))$ for any given $T\in(0,\infty)$.