The identification problem for complex-valued Ornstein-Uhlenbeck operators in L-p (R-d, C-N)

作者：Otten Denny^*

来源：Semigroup Forum, 2017, 95(1): 13-50.

DOI：10.1007/s00233-016-9804-y

• Complex-valued Ornstein-Uhlenbeck operator
• Identification problem in L-p
• L-p-dissipativity
• L-p-resolvent estimates
• Maximal domain
• Applications to rotating waves

In this paper we study perturbed Ornstein-Uhlenbeck operators
[L(infinity)v] (x) = A Delta v(x) + Sx, del v(x) - Bv(x), x is an element of R-d, d >= 2,
for simultaneously diagonalizable matrices A,B is an element of C-N,C-N . The unbounded drift term is defined by a skew-symmetric matrix S is an element of R-d,R-d . Differential operators of this form appear when investigating rotating waves in time-dependent reaction diffusion systems. We prove under certain conditions that the maximal domain D(A(p)) of the generator A(P) belonging to the Ornstein-Uhlenbeck semigroup coincides with the domain of L-infinity in L-P (R-d , C-N) given by
D-loc(p) (L-0) = {v is an element of W-loc(2,p) boolean AND L-p vertical bar A Delta v + S., del v is an element of L-p}, 1 < p < infinity.
One key assumption is a new L-p-dissipativity condition
vertical bar z vertical bar Re-2 omega, A omega + (P - 2)Re omega, z Re z, A omega >= gamma(A)vertical bar z vertical bar(2) vertical bar omega vertical bar(2) for all z, omega is an element of C-N
for some gamma(A) > 0 . The proof utilizes the following ingredients. First we show the closedness of L-infinity in L-p and derive L-p-resolvent estimates for L-infinity . Then we prove that the Schwartz space is a core of A(p) and apply an L-p-solvability result of the resolvent equation for A(p). In addition, we derive W-1,W-p -resolvent estimates. Our results may be considered as extensions of earlier works by Metafune, Pallara and Vespri to the vector-valued complex case.

• 出版日期2017-8