So the result is sort of like the happy ending of one of those screwball romantic comedies that involve mistaken identity and the handsome vagabond turns out to be the prince in disguise; Alice can marry Ted who is really Bob and the bonds of matrimony extend smoothly across the edge of the black hole.
As ( v ˜ ϵ, β 0, w ˜ ϵ, β 0 ) remains in R p × E, one can extend smoothly F and Σ outside ℰ so that ( F, Σ ) satisfies Assumptions 2.1-2.2 2.1-2.2
WMLs tend to extend smoothly from the ventricular wall [ 42] and periventricular WMLs are often connected to deep WMLs on MR images.
Thus, ( v, w ) can be extended smoothly beyond t = T.
Thus, ( u, v, b ) can be extended smoothly beyond t = T.
The conductive feature extends smoothly to the west; however, there is a sharp resistivity contrast between C1 and resistor R2 in the east.
We can assume that g is extended smoothly on M 1 ⋑ M. Next, with g extended as above, we extend g ^ 1 so that g ^ 1 = g outside M.
If one chooses one of the two lobes of the light cone at a point O to be, say, future, that choice can be extended smoothly throughout the whole of the spacetime.
It is proved that if ∫ 0 T ∥ u x 3 ∥ L α β d t < ∞ with 3 α + 2 β ≤ 1 and α ≥ 3, then the solution ( u, v, b ) can be extended smoothly beyond t = T.
where 3 α + 2 β ≤ 1, α ≥ 3, then the solution ( u, v, b ) can be extended smoothly beyond t = T. Proof Multiplying the first equation of (1.1) by u and integrating with respect to x on R 3, using integration by parts, we obtain 1 2 d d t ∥ u ( t ) ∥ L 2 2 + ∥ ∇ u ( t ) ∥ L 2 2 = ∫ R 3 b ⋅ ∇ b ⋅ u d x + χ ∫ R 3 ( ∇ × v ) ⋅ u d x. (4.1).
Next, we can take a smooth function χ with support in U x that is identically equal 1 close to x. Extending χf by zero on the rest of M, we obtain a function, which we also call χf, defined everywhere and smooth away from x. Furthermore, ∂ ¯ (χ f ) = f = 0 near x, so ∂ ¯ χ f can be extended smoothly to the boundary.
