Au as A 0 for all u E D(A). Proof. Let u E D(A). 10, one has JAAu-Au—*0 asAJ0. 11 that Aau = J)Au. Thus, hence the result. 3. Extrapolation In this section, we show that, given an m-dissipative operator A on X with dense domain, one can extend it to an m-dissipative operator A on a larger space X. This result will be very useful for characterizing the weak solutions in Chapters 3 and 4. 1. Let A be an m-dissipative operator in X with dense domain.
Similarly, we have ((u,v),B(w,z)) t,2 XH -1 = J zudx - J wvdx. Therefore, (B(u,v),(w,x))L xH-1= - ((u,v),B(w,z))L2xx-1. 7) with (u, v) = (w, z), it follows that (B(u, u), (u, v)) = 0. 2). Now let (f, g) E Y. 3. 4). 5) and we obtain v E L 2 (fl). Therefore (u, v) E D(B) and (u, v) — B(u, v) = (f, g); hence B is m-dissipative. Similarly, we show that —B is m-dissipative. 7), we have G(B) C G(—B`). 11 proves that B is skew-adjoint. 11. 4. 1. Proof. Properties (i), (iii), and (iv) are clearly satisfied.