The Identification Problem for complex-valued Ornstein-Uhlenbeck Operators in L^p(ℝ^d,ℂ^N)

2015/10/03 by Denny Otten, Otten, Denny
Mathematics · #35A02 #35J47 (35K57 #47A05 #47A10 #47B44) #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35A02 #msc:35J47 #msc:47A05 #msc:47A10

paper · pdf · doi:10.48550/arxiv.1510.00827

32 pages

arxiv created 2015/10/03 · arxiv updated 2015/10/06

Abstract

In this paper we study perturbed Ornstein-Uhlenbeck operators [L_∞ v](x)=A\triangle v(x)+⟨ Sx,∇ v(x)⟩-B v(x), x∈ℝ^d, d\geqslant 2, for simultaneously diagonalizable matrices A,B∈ℂ^N,N. The unbounded drift term is defined by a skew-symmetric matrix S∈ℝ^d,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_∞ in L^p(ℝ^d,ℂ^N) given by D^p_loc(L₀)=\v∈ W^2,p_loc∩ L^p| A\triangle v+⟨ S⋅,∇ v⟩∈ L^p\, 1<p<∞. One key assumption is a new L^p-antieigenvalue condition μ₁(A) > (|p-2|)/(p), 1<p<∞, μ₁(A) first antieigenvalue of A. The proof utilizes the following ingredients. First we show the closedness of L_∞ in L^p and derive L^p-resolvent estimates for L_∞. 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. A second characterization shows that the maximal domain even coincides with D^p_max(L₀)=\v∈ W^2,p| ⟨ S⋅,∇ v⟩∈ L^p\, 1<p<∞. This second characterization is based on the first one, and its proof requires L^p-regularity for the Cauchy problem associated with A_p. Finally, we show a W^2,p-resolvent estimate for L_∞ and an L^p-estimate for the drift term ⟨ S⋅,∇ v⟩.

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