C. Peskine and L. Szpiro in "Dimension projective finie et cohomologie locale",(Proposition 4.1) proved the following vanishing theorem for local cohomology: 
Let$R$ be a regular local ring of characteristic $p\gneqq 0$. Let $I$ be an ideal of $R$ such that $R/I$ is a CM(i.e. Cohen-Macaulay) ring; then:
 $$H_I^i(M) = 0~~ \text{for each}~ i> \text{dim} R-\text{dim} R/I.$$
It is to be noted that this theorem does not extend to characteristic $0$.
 
But I think  to a restricted version, when the characteristic is zero.
My question is that if $R= K [[ x_1, x_2, ...x_n]]$, where $K$ is a field with $\text{char}~K=0$ and $R/I$ is a CM ring, is it true that:
$$\text{inj}~\text{dim}~H_I^i(R) =\text{dim}~ H_I^i(R)?  $$
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Michael Hellus in http://arxiv.org/pdf/math/0703126v1.pdf  has attacked the question and in this case has established the following facts:

Let $R$ be a noetherian local regular ring
containing a field, $I$ an ideal of $R$ and $i$ a natural number.
Here is a summary of results:

(i) If $H^i_I(R)$ is $I$-cofinite then $\text{inj dim} _R(H^i_I(R))=\dim
_R(H^i_I(R))$ holds.

(ii) There are examples where $H^i_I(R)$ is not $I$-cofinite, but
$\text{inj dim} _R(H^i_I(R))=\dim _R(H^i_I(R))$ holds.

(iii) There are examples where $\text{inj dim} _R(H^i_I(R))=\dim
_R(H^i_I(R))$ does not hold.