Linearized equation and its real-duality adjoint

PDF p. 4: (1.10); PDF p. 5: (1.11)

Assume the Axiom of Choice. Let $T>0$, let $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$ be the smooth connected exterior of the stated compact obstacle, let $\lambda \in \mathbb{R}\setminus \{0\}$, and let $w$ satisfy hypothesis (W): $w\in C([0,T];H_D^2(\mathrm{\Omega })\cap W^{2,\mathrm{\infty }}(\mathrm{\Omega }))$ and $i{\mathrm{\partial }}_{t}w+\mathrm{\Delta }w+\lambda \mathrm{\mid }w{\mathrm{\mid }}^{2}w=0$ in distributions. The condition $w(t)\in H_D^2=H^2\cap H_0^1$ supplies its homogeneous Dirichlet condition. Let $\phi$ be the real control cutoff, zero near the obstacle and supported as a control multiplier in the exterior region. Define $V_1=2\lambda \mathrm{\mid }w{\mathrm{\mid }}^2$ and $V_2=\lambda w^2$.

For a control $h$ and initial state $z_0$, the linearized controlled system is

$i{\mathrm{\partial }}_{t}z+\mathrm{\Delta }z+V_{1}z+V_2\bar{z}=\phi h\text{in }\mathrm{\Omega }\times (0,T),z{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=0,z(0)=z_0\mathrm{.}$

The real-duality adjoint system is

$i{\mathrm{\partial }}_{t}v+\mathrm{\Delta }v+V_{1}v-V_2\bar{v}=0\text{in }\mathrm{\Omega }\times (0,T),v{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=0.$

The real pairing on complex $L^2(\mathrm{\Omega })$ is $(z,v{)}_{\mathbb{R}}=\mathrm{Re}{\int }_{\mathrm{\Omega }}z\bar{v}dx$. Here $\bar{z}$ denotes complex conjugation and $\mathrm{Re}$ the real part. An operator is real-linear if it preserves addition and multiplication by real scalars. The conjugate terms are real-linear; when $V_2$ is not identically zero they are not complex-linear. The term adjoint refers to the cancellation identity for this real pairing explained below, with the sign of the $V_2$ term as displayed.

These formulas specify systems rather than asserting existence or uniqueness. Interior equations may be interpreted distributionally when every displayed product is defined as a distribution; initial and boundary conditions require a solution class in which the corresponding values or traces have a meaning. The $L^2$ integral above is not asserted to define a pairing between arbitrary negative-order distributions.

Reference trajectory hypothesis W

Control region and nested real cutoffs

Linear and nonlinear internally controlled exterior systems

Dirichlet spectral Sobolev scale and duality

Distributional equation convention for complex Schrödinger solutions

Realification of Hilbert pairings and conjugate-linear coefficients

Real and imaginary parts, complex conjugation, and modulus

Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive

The Axiom of Choice

The exterior, real coupling and trajectory are those of Reference trajectory hypothesis W; the cutoff and control region are those of Control region and nested real cutoffs. The cubic controlled equation being linearized is the $m=3$ case in Linear and nonlinear internally controlled exterior systems. Under the Choice assumption recorded in The Axiom of Choice, Dirichlet spectral Sobolev scale and duality supplies the Dirichlet realization and its self-adjointness, the order-two domain, and the $L^2$ pairing convention. The distributional convention and its limitation on traces are those of Distributional equation convention for complex Schrödinger solutions.

Using the conjugation and modulus conventions of Real and imaginary parts, complex conjugation, and modulus and the conjugation laws of Conjugation is an involutive real-field automorphism, zz̄=|z|², and modulus is definite, multiplicative, and subadditive, direct expansion gives $\mathrm{\mid }w+z{\mathrm{\mid }}^2(w+z)=\mathrm{\mid }w{\mathrm{\mid }}^{2}w+2\mathrm{\mid }w{\mathrm{\mid }}^{2}z+w^2\bar{z}+2w\mathrm{\mid }z{\mathrm{\mid }}^2+\bar{w}{z}^2+\mathrm{\mid }z{\mathrm{\mid }}^{2}z$. Thus the displayed coefficients are exactly the part linear over the reals in $z$; the remaining terms have degree at least two. Conjugation is additive and commutes with real scalars, whereas $\overline{iz}=-i\bar{z}$, which establishes the stated linearity distinction.

For the adjoint terminology, suppose in addition that $z,v$ are continuously differentiable as $L^2$-valued functions, continuously take values in $D({\mathrm{\Delta }}_D)$, and satisfy the displayed equations in $L^2$, with $h(t)\in L^2$. The trajectory hypothesis makes $V_1,V_2$ essentially bounded at each time, and $V_1$ is real. By Realification of Hilbert pairings and conjugate-linear coefficients, the real adjoint of $z\mapsto i(V_{1}z+V_2\bar{z})$ is $v\mapsto -i{V}_{1}v+i{V}_2\bar{v}$. Self-adjointness of ${\mathrm{\Delta }}_D$ gives $(i{\mathrm{\Delta }}_{D}z,v{)}_{\mathbb{R}}+(z,i{\mathrm{\Delta }}_{D}v{)}_{\mathbb{R}}=0$. For the product rule, put $H=L^2(\mathrm{\Omega }{)}_{\mathbb{R}}$, the real Hilbert space supplied by Realification of Hilbert pairings and conjugate-linear coefficients, and $B(u,q)=(u,q{)}_{\mathbb{R}}$. This is a real-bilinear map $H\times H\to \mathbb{R}$; by Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors, $\mathrm{\mid }B(u,q)\mathrm{\mid }\le \mathrm{\parallel }u{\mathrm{\parallel }}_2\mathrm{\parallel }q{\mathrm{\parallel }}_2$, so it is bounded. On the open interval $U=(0,T)$, the assumed $L^2$ differentiability of the two curves is real Fréchet differentiability, with $Dz(t)s=s{\mathrm{\partial }}_{t}z(t)$ and $Dv(t)s=s{\mathrm{\partial }}_{t}v(t)$ for $s\in \mathbb{R}$. Applying the bounded-bilinear product rule in Chain sum product and composition rules for Banach derivatives with $X=W=\mathbb{R}$ and $Y=Z=H$ gives $\frac{d}{dt}(z,v{)}_{\mathbb{R}}=({\mathrm{\partial }}_{t}z,v{)}_{\mathbb{R}}+(z,{\mathrm{\partial }}_{t}v{)}_{\mathbb{R}}$ for $0<t<T$. Substituting the two equations and the preceding cancellation identities yields $\frac{d}{dt}(z,v{)}_{\mathbb{R}}=(-i\phi h,v{)}_{\mathbb{R}}$ on this interval. This is a calculation for the stated regularity, not an asserted extension to every solution space.

The original article supplies equations (1.10) and (1.11) and their attribution as a linearized pair. The expansion and the stated real-pairing cancellation above justify the terminology using the named mathematical suppliers; the article is not used as proof of its own claim.