Remarks on hyperbolic polynomials

Wakabayashi, Seiichiro

doi:10.21099/tkbjm/1496160384

Cited by 43 publications

(37 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Then (4.4), (4.5) and (4.7) are also valid. Therefore, applying the same arguments as in the proof of Proposition 4.1 we have (4.10), (4.12) From Theorem 3.4 of [7] and Theorem 3 of [9] it follows that K þ z is outer semicontinuous with respect to z 1 ðx; xÞ for x 0 b 0. Therefore, there is j 0 A N satisfying …”

mentioning

confidence: 79%

Singularities of Solutions to the Cauchy Problem for a Class of Second-Order Hyperbolic Operators

Wakabayashi

2014

Self Cite

View full text Add to dashboard Cite

Abstract. In this paper we deal with the Cauchy problem for second-order hyperbolic operators with the coe‰cients of their principal parts depending only on the time variable. We shall give outer estimates of the wave front sets of solutions to the Cauchy problem when the Cauchy problem is C y well-posed. More precisely, we shall show that the wave front sets of solutions are included in the union of broken null bicharacteristics emanating from the singularities of data in non-trivial cases.

show abstract

mentioning

confidence: 79%

Singularities of Solutions to the Cauchy Problem for a Class of Second-Order Hyperbolic Operators

Wakabayashi

2014

Self Cite

View full text Add to dashboard Cite

show abstract

“…For hyperbolic systems, τ max (y, ξ), Γ + (y), and T (y) depend on y. The smoothness in (3), implies by a Theorem of Bronstein (see [1], [15]) that τ max is uniformly lipschitzean on compacts. The propagation and time like cones inherit this regularity.…”

Section: Hyperbolic Systemsmentioning

confidence: 99%

Precise Finite Speed with Bare Hands

Rauch¹

2005

Methods and Applications of Analysis

View full text Add to dashboard Cite

Dedication.I am pleased to dedicate this article to my friend and colleague Joel Smoller. Joel advised and recruited me at the beginning of my career and is largely responsible for the warm welcome I received at the University of Michigan. His seminar, teaching, and research have been an inspiration to me for more than three decades. AbstractA precise finite speed result is proved for general hyperbolic systems using neither Hamilton-Jacobi Theory as in [9] nor the theorem of Marchaud [12] as in Leray [11]. The proof is shorter than both [11] and [9], and more general than the first. It does not give the additional information about spacelike deformations proved in [9].

show abstract

“…We note that the ƒÉj(x, ƒÌ') are locally Lipschitz continuous (see [2], [21]). In addition to (P-1) we impose the following conditions on p for every z0=(x0, ƒÌ0) •¸ T*Rn with |ƒÌ0'|=1 and dp(z0)=0.…”

Section: Remark (I)mentioning

confidence: 99%

“…We may assume that ai(y, r1') < A2(y, r1') < ... < ar(y, r1'). It follows from Lemma 2.2 in [21] that there are positive constants c1 and c2 such that for (y,ƒÅ') •¸ C' with |ƒÅ|_? ?_1, where pk0 (y, rl) = {I=(i ~1rl A° (y, r1')) and A° (y, ri') ~ <)4(y, r1').…”

Section: Proof It Is Obvious That (12)m Is Valid If (13) Is Validmentioning

confidence: 99%