Suppose $\lambda$ is a strong limit cardinal of cofinality $\omega$ and for $A$ a transitive set, define $L(A)$ in the usual fashion by setting 
$$L\_0(A)=A;$$
$$L\_{\alpha+1}(A) = L\_\alpha (A)\cup \mathcal P\_{Def}(L\_\alpha(A));$$
$$L(A)=\bigcup\_{\alpha\in Ord} L\_\alpha (A).$$


In Woodin's longer article "The Continuum Hypothesis" (in LNL 19, Logic Colloquium 2000), the following facts are stated regarding $L(V\_{\lambda+1})$: 


(1) If $c$ is Cohen generic over $V$ then very likely $$(L(V\_{\lambda+1}))^{V[c]}\neq L(V\_{\lambda+1})[c].$$


(2) On the other hand, if $G\subset Coll(\omega\_1,\mathbb{R})$ is $V$-generic then $$(L(V\_{\lambda+1}))^{V[G]}= L(V\_{\lambda+1})[G].$$ 


Can anyone give a (sketch of) proof of either (1) or (2)? Are these results given only in the context of a non-trivial elementary embedding $j:L(V\_{\lambda+1})\prec L(V\_{\lambda+1})$ with $crit(j)<\lambda$?


More generally, for a partial order $\mathbb{P}$ and a $G\subset \mathbb{P}$ which is $V$-generic, which properties of $\mathbb{P}$ are sufficient to ensure the equality $$(L(V\_{\lambda+1}))^{V[G]}= L(V\_{\lambda+1})[G]$$ holds? Fails? Is this even known? 



Generic Extensions and $L(V_{\lambda+1})$
They are immediate consequences of Theorem 175 of W. Hugh Woodin, "Suitable extender models II: beyond $\omega$-huge", J. Math. Log., vol. 11 (2011), no. 2, pp. 115–436, which says 



> 
> If there is a (proper) elementary embedding $j: L(V\_{\lambda+1}) \to L(V\_{\lambda+1})$ with $\text{crit}(j)<\lambda$ and if $G\subset \mathbb{P}$ is $V$-generic for some poset $\mathbb{P}\in V\_\lambda$, then $V\_{\lambda+1} \in L\_\lambda(V\_{\lambda+1})^{V[G]}$ if and only if $(\lambda^\omega)^V = (\lambda^\omega)^{V[G]}$.
> 
> 
> 


This theorem can be proved using the Large Perfect Set Theorem for subsets of $V\_{\lambda+1}$ in $L\_\lambda(V\_{\lambda+1})$, which says assuming there is a (proper) elementary embedding $j: L(V\_{\lambda+1}) \to L(V\_{\lambda+1})$ with $\text{crit}(j)<\lambda$, those subsets of $V\_{\lambda+1}$ either has size $\leq \lambda$ or contains a large perfect subset, i.e. a homeomorphic copy of $\lambda^\omega$.
Cramer recently improves the Large Perfect Set Theorem to all subsets of $V\_{\lambda+1}$ in $L(V\_{\lambda+1})$, so $L\_\lambda(V\_{\lambda+1})$ can be replaced by $L(V\_{\lambda+1})$.


