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| We consider the celebrated $L^p$-Hardy inequality involving the distance $d_{\Omega}$ to the boundary of a domain $\Omega$ in ${\mathbb{R}}^n$
$$
\int_{\Omega }|\nabla u|^pdx \geq c \int_{\Omega }\frac{|u|^p}{d^p_{\Omega }}dx \, , \ {\rm for\ all}\ u\in C^{\infty }_c(\Omega).
$$
The $L^p$-Hardy constant is the best constant in the inequality above and is denoted by $H_p(\Omega )$.
We study the dependence of $H_p(\Omega)$ upon perturbation of $\Omega$ and we prove stability results.
Since for convex domains it is well-known that $H_p(\Omega)= ((p-1)/p)^p$, the focus is mainly on non-convex domains. |
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