TY - JOUR

T1 - Regularity and global structure of solutions to hamiltonjacobi equations II

T2 - Convex initial data

AU - Zhao, Yinchuan

AU - TANG, Tao

AU - Wang, Jinghua

N1 - Funding Information:
The first author was supported by China Postdoctoral Science Foundation under contract 148028 and National Natural Foundation of China under contract 70901025. He also thanks Professor Pingwen Zhang for support and encouragements. The second author was partially supported by CERG Grants of Hong Kong Research Grant Council, FRG grants of Hong Kong Baptist University, and International Research Team of Complex Systems of Chinese Academy of Sciences. The third author was supported by National Natural Foundation of China under contract 10671116 and 10871133.

PY - 2009/12

Y1 - 2009/12

N2 - 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.

AB - 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.

KW - Global structure

KW - Hamilton-Jacobi equations

KW - Hopf's formula (II)

KW - Singularity point

UR - http://www.scopus.com/inward/record.url?scp=76449108458&partnerID=8YFLogxK

U2 - 10.1142/S0219891609001976

DO - 10.1142/S0219891609001976

M3 - Article

AN - SCOPUS:76449108458

VL - 6

SP - 709

EP - 723

JO - Journal of Hyperbolic Differential Equations

JF - Journal of Hyperbolic Differential Equations

SN - 0219-8916

IS - 4

ER -