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.