Regularity and global structure of solutions to hamiltonjacobi equations II: Convex initial data

Yinchuan Zhao*, Tao TANG, Jinghua Wang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

The paper is concerned with the HamiltonJacobi (HJ) equations of multidimensional space variables with convex initial data and general Hamiltonians. Using Hopf's formula (II), we will study the differentiability of the HJ solutions. For any given point, we give a sufficient and necessary condition such that the solutions are Ck smooth in some neighborhood of this point. We also study the characteristics of the equations which play important roles in our analysis. It is shown that there are only two kinds of characteristics, one never touches the singularity point, but the other one touches the singularity point in a finite time. Based on these results, we study the global structure of the set of singularity points for the solutions. It is shown that there exists a one-to-one correspondence between the path connected components of the set of singularity points and path connected component of the set {(Dg(y),H(Dg(y)))| y ∈ ℝn}\ {(Dg(y), conv H(Dg(y)))| y ∈ ℝn}, where conv, H is the convex hull of H. A path connected component of the set of singularity points never terminates as t increases. Moreover, our results depend only on H and its domain of definition.

Original languageEnglish
Pages (from-to)709-723
Number of pages15
JournalJournal of Hyperbolic Differential Equations
Volume6
Issue number4
DOIs
Publication statusPublished - Dec 2009

Scopus Subject Areas

  • Analysis
  • Mathematics(all)

User-Defined Keywords

  • Global structure
  • Hamilton-Jacobi equations
  • Hopf's formula (II)
  • Singularity point

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