Transfer from exterior internal control to outer boundary control

By the scale identification, the inclusion $H_D^s(\mathrm{\Omega })\to H^s(\mathrm{\Omega })$ is bounded. Restriction from $H^s(\mathrm{\Omega })$ to $H^s({\mathrm{\Omega }}_0)$ is bounded: every whole-space representative of the former restricts to a representative of the latter, so the quotient norm cannot increase. Consequently $u$ is norm-continuous in $H^s({\mathrm{\Omega }}_0)$. Applying the bounded restriction to the endpoint equalities and using $Ef{\mathrm{\mid }}_{{\mathrm{\Omega }}_0}=f$ gives $u(0)=u_0$ and $u(T)=u_1$.

The sphere ${\mathrm{\Gamma }}_1$ lies in the interior of $\mathrm{\Omega }$ and has a fixed bounded smooth collar disjoint from $\mathrm{\Theta }$. Restriction to that collar and its interior trace are bounded from $H^s(\mathrm{\Omega })$ to $H^{s-1\mathrm{/}2}({\mathrm{\Gamma }}_1)$. Hence $h$ is norm-continuous. The inner-side trace of $v{\mathrm{\mid }}_{{\mathrm{\Omega }}_0}$ agrees with this interior trace: both are the same classical restriction for smooth functions and the collar trace construction gives the common continuous extension. Thus ${\gamma }_{{\mathrm{\Gamma }}_1}u=h$.

For each $t$, the scale identification gives $v(t)\in H_0^1(\mathrm{\Omega })$. Choose a sequence in $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ converging to $v(t)$ in $H^1(\mathrm{\Omega })$. Its restrictions to ${\mathrm{\Omega }}_0$ converge to $u(t)$ in $H^1({\mathrm{\Omega }}_0)$ and vanish near ${\mathrm{\Gamma }}_2$. Boundedness of the first-order trace therefore makes the $H^{1\mathrm{/}2}$ trace of $u(t)$ vanish on ${\mathrm{\Gamma }}_2$. The order-$s$ trace agrees with the first-order trace as an $L^2$ boundary function, so the $H^{s-1\mathrm{/}2}$ trace also equals zero. This holds at every time.

Every test function compactly supported in $(0,T)\times {\mathrm{\Omega }}_0$ extends by zero to a test function on $(0,T)\times \mathrm{\Omega }$. Testing the exterior equation against that extension gives the restricted distributional equation: distributional derivatives commute with restriction, the locally integrable nonlinear expression restricts to $\lambda \mathrm{\mid }u{\mathrm{\mid }}^{m-1}u$, and $\phi k=0$ on ${\mathrm{\Omega }}_0$ because $R>R_0$. Together with the preceding trace and endpoint statements, this is the specified outer Dirichlet formulation for either $m=3$ or $m=5$.

The extension is nonzero: a nonzero smooth function compactly supported in ${\mathrm{\Omega }}_0$ belongs to $X^s({\mathrm{\Omega }}_0)$ and is recovered by restriction after applying $E$. Thus $\mathrm{\parallel }E\mathrm{\parallel }>0$. If $\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{X^s}\le {\delta }_1\mathrm{/}\mathrm{\parallel }E\mathrm{\parallel }$ for $j=0,1$, the operator-norm inequality gives $\mathrm{\parallel }E{u}_j{\mathrm{\parallel }}_{H_D^s}\le \mathrm{\parallel }E\mathrm{\parallel }\mathrm{\parallel }{u}_j{\mathrm{\parallel }}_{X^s}\le {\delta }_1$. The assumed exterior existence applies to these endpoints, and steps 1.1–2.1 then give the boundary solution. This is a conditional transfer and does not itself assert exterior controllability.