An Analogue of Garding's Inequality for Parabolic Operators and Its Applications

An energy inequality for higher order linear parabolic operators and its applications

“…We shall now prove the remainder of the proposition by induction on / For j= 1, we distinguish two cases: (a) For A e ( -k, 0] we have already shown the result, (b) For A e (0, k) we observe that r-2k e [0, k) which implies, by our previous results and Proposition 2(i) of [5] for all u e ¿f r'°(Q). Thus by Proposition 8 of [5] á Cr0H|r>o.n r-ik,0,Si for all u e 3iï*-\iï). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use…”

“…With only slight modifications the proof is exactly like that of Theorem 5 of [5] with 7? replacing P, our form of the energy inequality replacing Theorem 3 of [5], and our Theorem 6 replacing Theorem 4 of [5]. Proof.…”

An Energy Inequality for Higher Order Linear Parabolic Operators and Its Applications

“…To prove this result we employ our estimates from [6] and a commutator estimate from [5]. By employing the energy inequality once again we deduce that Theorem 7.…”

“…We present a Hubert space treatment for proving existence and uniqueness in the Cauchy problem for a general linear 2A>parabolic differential operator. We shall follow in rough outline the Hubert space approach to the Cauchy problem for parabolic operators of the form P = |-£(0 = |-2 "Áx*t)D", where L(t) is uniformly strongly elliptic on 7?n (here x = (xx,..., xn}, «=<«!,...,«">, |«| =<*!+• ••+«", and Da = ((lli) 8l8Xl)"i-■ -((l/i) 8l8xny»), as given in [5]. As in [5] we shall make use of the Hubert spaces Jfr'$ (=äS2k in the notation of [4,Chapter II], where k(£, r)=A:r,s(í, t) is the temperate weight function defined for <£ r>=<fi,..., £n, r> e R»'+1 by *,..(£ r)=ßU *M£)-Here #(i) = {l + |f|2}1/2, with |f|2 = 2?=i if, is the usual elliptic weight function in Rn and Q(£, T) = {r2+q4k($)}llik).…”

Pseudo-Differential Estimates for Linear Parabolic Operators

“…We also employed the maps M a, p, a real, which are the natural isometric isomorphisms of KTfS onto ¡V'p,s~cr. In extending our "energy inequality" for R to the Kr,,s(fi) spaces we employed a proposition of S. Kaplan (Proposition 5 of [3] …”