# analysis – Adjoint operator of a partial differential operator used to do the carleman estimate

I am reading Carleman estimate for parabolic equations and applications written by Masahiro Yamamoto.
Let $$P$$ be an operator defined by
$$Pw=partial_tw-Delta w+2snabla varphicdotnabla w+(-spartial_t varphi-s^2|nablavarphi|^2+sDelta varphi)w.$$
The article on page 4 says that the way is the decomposition of $$P$$ into the symmetric part $$P_+$$ and antisymmetric part $$P_-$$. To do this, he consider the formal ajoint operator $$P^*$$ to $$P$$:
$$(Pw,v)_{L^2(D)}=(w,P^*v)_{L^2(D)},quad v,win C_0^infty(D).$$
By integration by parts, the Green’s theorem and $$v,w in C_0^infty(D)$$, he obtains
$$P^*=-partial_t w-Delta w -2snablavarphicdotnabla w-(sDelta varphi+s^2|nablavarphi|^2+s(partial_tvarphi))w.$$
It can be seen that the term $$sDeltavarphi w$$ change to $$-sDelta varphi w$$. I cannot figure out why this change happens. According to my understanding (may not be true),
$$(sDelta varphi w,v)=(w,sDelta varphi v),$$
so there must be something wrong with my thinking.