Corollary 1.6: quintic control near zero

PDF p. 5: Corollary 1.6; PDF p. 17: Proof of Corollary 1.6

Assume the Axiom of Choice. Let $n\in \{2,3\}$ and let $\mathrm{\Theta }\subset {\mathbb{R}}^n$ be a nonempty compact obstacle with smooth boundary, connected exterior $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$, and the non-trapping property that every generalized reflected or glancing ray leaves every compact set in finite forward time. Let $T>0$, $s\in \{1,2\}$ and $\lambda \in \mathbb{R}\setminus \{0\}$. Let $H_D^s(\mathrm{\Omega })=D((I-{\mathrm{\Delta }}_D{)}^{s\mathrm{/}2})$ be the Dirichlet spectral space; in particular $H_D^1=H_0^1$ and $H_D^2=H^2\cap H_0^1$ with equivalent norms. Here $H_0^1$ is the $H^1$ closure of compactly supported smooth functions.

Choose $R>0$ containing $\mathrm{\Theta }$ and a smooth real cutoff $\phi \in C^{\mathrm{\infty }}({\mathbb{R}}^n;[0,1])$ which is zero on $\mathrm{\mid }x\mathrm{\mid }\le R$, positive on $\mathrm{\mid }x\mathrm{\mid }>R$, and one on $\mathrm{\mid }x\mathrm{\mid }\ge R+1$. There is ${\delta }_{\mathrm{i}\mathrm{n}\mathrm{t}}>0$ such that, for every $v_0,v_T\in H_D^s(\mathrm{\Omega })$ of spectral norms at most ${\delta }_{\mathrm{i}\mathrm{n}\mathrm{t}}$, a control $h\in C([0,T];H_D^s(\mathrm{\Omega }))$ and an integral solution $v\in C([0,T];H_D^s(\mathrm{\Omega }))$ satisfy $v(0)=v_0$, $v(T)=v_T$, the homogeneous Dirichlet condition, and $i{\mathrm{\partial }}_{t}v+\mathrm{\Delta }v+\lambda \mathrm{\mid }v{\mathrm{\mid }}^{4}v=\phi h$ in distributions on $(0,T)\times \mathrm{\Omega }$.

For the boundary conclusion, choose $R_0>0$ with $\mathrm{\Theta }\subset B_{R_0}(0)$, set ${\mathrm{\Omega }}_0=B_{R_0}(0)\setminus \mathrm{\Theta }$, ${\mathrm{\Gamma }}_1=\mathrm{\partial }{B}_{R_0}(0)$ and ${\mathrm{\Gamma }}_2=\mathrm{\partial }\mathrm{\Theta }$, and let $X^s({\mathrm{\Omega }}_0)=\{f\in H^s({\mathrm{\Omega }}_0):{\gamma }_{{\mathrm{\Gamma }}_2}f=0\}$ with its inherited $H^s$ norm. The trace ${\gamma }_{{\mathrm{\Gamma }}_2}$ is the Sobolev value trace. There is ${\delta }_{\mathrm{b}\mathrm{d}\mathrm{r}\mathrm{y}}>0$ such that any $u_0,u_1\in X^s({\mathrm{\Omega }}_0)$ of norms at most ${\delta }_{\mathrm{b}\mathrm{d}\mathrm{r}\mathrm{y}}$ can be joined by $u\in C([0,T];H^s({\mathrm{\Omega }}_0))$ with $u(0)=u_0$, $u(T)=u_1$, and a boundary control $k\in C([0,T];H^{s-1\mathrm{/}2}({\mathrm{\Gamma }}_1))$. They satisfy $i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u+\lambda \mathrm{\mid }u{\mathrm{\mid }}^{4}u=0$ in distributions, ${\gamma }_{{\mathrm{\Gamma }}_1}u=k$, and ${\gamma }_{{\mathrm{\Gamma }}_2}u=0$. Thus the local internal and outer Dirichlet boundary conclusions remain valid for the quintic nonlinearity.

Exterior and truncated obstacle domains

Control region and nested real cutoffs

Linear and nonlinear internally controlled exterior systems

Dirichlet spectral Sobolev scale and duality

States compatible with the uncontrolled obstacle boundary

Distributional equation convention for complex Schrödinger solutions

Continuous and mixed time-space function spaces

The Axiom of Choice

The frozen article is cited for attribution of Corollary 1.6. The argument above uses the separately stated quintic control, spectral, extension and transfer suppliers; it does not use the corollary as its own proof.