Let $(S(t))_{t\geq 0}$ be a semigroup generated by $\Delta$ operator in $L^{2}(\Omega)$ for a bounded smooth domain $\Omega \subset \mathbb{R}^{N}$. Here we assume \begin{align} \begin{cases} D(B) = \{u \in H_{0}^{1}(\Omega)\,|\, \Delta u \in L^{2}(\Omega)\}\\ \forall u \in D(B), \, Bu=\Delta u \end{cases} \end{align}

Recall that a one-parameter family $(S(t))_{t\geq 0}\subset \mathfrak{L}(L^{2}(\Omega))$ (the space of bounded linear operator from $L^{2}(\Omega)$ to $L^{2}(\Omega)$) is a contraction semigroup in $L^{2}(\Omega)$ provided that:
1 $\forall t \geq 0, ||S(t)||\leq 1$
2 $S(0) = I$
3 $\forall s,t\geq 0, \, S(t+s) = S(t)S(s)$
4 $\forall \phi \in L^{2}(\Omega)$, the function $t \mapsto S(t)\phi$ belongs to $C([0,\infty),L^{2}(\Omega))$.

Now, my question is for this setting, what are the conditions such that I can say that for $\phi \in L^{2}(\Omega)$, it holds that $\nabla S(t)\phi = S(t) \nabla \phi$? I actually want to show that $||(S(t)-I)\varphi||_{H_{0}^{1}(\Omega)}\to 0$ as $t\to0$ for a given $\varphi \in H_{0}^{1}(\Omega)$.

Any help is much appreciated! Thank you very much!


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