Fritz John pp 460-461 | Cite as

# Commentary on [39] and [47]

## Abstract

On page 553 in [47] John remarks that “there appears to be a connection between *well-behaved* character of a problem and *regularity* of the solution for regular data”. An example of this is given for the wave equation. *u* _{ tt } - *u* _{ xx } - *u* _{ yy } = 0 in **R** ^{3}. On one hand, the continuation of solutions from a cylinder *Φ* = {(*x, y, t*); *x* ^{2} + *y* ^{2}< *r* ^{2}} to the complement is shown to be only very weakly continuous. On the other hand, a solution is constructed which is analytic in *Φ*, of class *C* ^{ m } exactly on ∂*Φ* and of class *C* ^{ m } ^{+1} exactly in the complement of *Φ*. John comments on page 574: “What is remarkable is that this cylinder is not a characteristic surface for the differential equation. Apparently not *all* types of singularities propagate along characteristic surfaces.”

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