Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I : regularity

Andrea C. G. Mennucci

ESAIM: Control, Optimisation and Calculus of Variations (2004)

  • Volume: 10, Issue: 3, page 426-451
  • ISSN: 1292-8119

Abstract

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We formulate an Hamilton-Jacobi partial differential equation H ( x , D u ( x ) ) = 0 on a n dimensional manifold M , with assumptions of convexity of H ( x , · ) and regularity of H (locally in a neighborhood of { H = 0 } in T * M ); we define the “min solution” u , a generalized solution; to this end, we view T * M as a symplectic manifold. The definition of “min solution” is suited to proving regularity results about u ; in particular, we prove in the first part that the closure of the set where u is not regular may be covered by a countable number of n - 1 dimensional manifolds, but for a ℋ n - 1 negligeable subset. These results can be applied to the cutlocus of a C 2 submanifold of a Finsler manifold.

How to cite

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Mennucci, Andrea C. G.. "Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I : regularity." ESAIM: Control, Optimisation and Calculus of Variations 10.3 (2004): 426-451. <http://eudml.org/doc/244619>.

@article{Mennucci2004,
abstract = {We formulate an Hamilton-Jacobi partial differential equation\[ H( x, D u(x))=0\]on a $n$ dimensional manifold $M$, with assumptions of convexity of $H(x,\cdot )$ and regularity of $H$ (locally in a neighborhood of $\lbrace H=0\rbrace $ in $T^*M$); we define the “min solution” $u$, a generalized solution; to this end, we view $T^*M$ as a symplectic manifold. The definition of “min solution” is suited to proving regularity results about $u$; in particular, we prove in the first part that the closure of the set where $u$ is not regular may be covered by a countable number of $n-1$ dimensional manifolds, but for a $\{\mathcal \{H\}\}^\{n-1\}$ negligeable subset. These results can be applied to the cutlocus of a $C^2$ submanifold of a Finsler manifold.},
author = {Mennucci, Andrea C. G.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Hamilton-Jacobi equations; conjugate points; Hamilton-Jacobi equation; manifold; min solution; viscosity solution; geodesics; regularity; Finsler manifold},
language = {eng},
number = {3},
pages = {426-451},
publisher = {EDP-Sciences},
title = {Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I : regularity},
url = {http://eudml.org/doc/244619},
volume = {10},
year = {2004},
}

TY - JOUR
AU - Mennucci, Andrea C. G.
TI - Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I : regularity
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2004
PB - EDP-Sciences
VL - 10
IS - 3
SP - 426
EP - 451
AB - We formulate an Hamilton-Jacobi partial differential equation\[ H( x, D u(x))=0\]on a $n$ dimensional manifold $M$, with assumptions of convexity of $H(x,\cdot )$ and regularity of $H$ (locally in a neighborhood of $\lbrace H=0\rbrace $ in $T^*M$); we define the “min solution” $u$, a generalized solution; to this end, we view $T^*M$ as a symplectic manifold. The definition of “min solution” is suited to proving regularity results about $u$; in particular, we prove in the first part that the closure of the set where $u$ is not regular may be covered by a countable number of $n-1$ dimensional manifolds, but for a ${\mathcal {H}}^{n-1}$ negligeable subset. These results can be applied to the cutlocus of a $C^2$ submanifold of a Finsler manifold.
LA - eng
KW - Hamilton-Jacobi equations; conjugate points; Hamilton-Jacobi equation; manifold; min solution; viscosity solution; geodesics; regularity; Finsler manifold
UR - http://eudml.org/doc/244619
ER -

References

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  1. [1] G. Alberti, On the structure of singular sets of convex functions. Calc. Var. Partial Differ. Equ. 2 (1994) 17-27. Zbl0790.26010MR1384392
  2. [2] A. Ambrosetti and G. Prodi, A primer of nonlinear analysis. Cambridge University Press, Cambridge (1993). Zbl0781.47046MR1225101
  3. [3] L. Ambrosio, P. Cannarsa and H.M. Soner, On the propagation of singularities of semi-convex functions. Ann. Scuola. Norm. Sup. Pisa XX (1993) 597-616. Zbl0874.49041MR1267601
  4. [4] P. Cannarsa, A. Mennucci and C. Sinestrari, Regularity results for solutions of a class of Hamilton-Jacobi equations. Arch. Ration. Mech. Anal. 140 (1997) 197-223 (or, preprint 13-95 Dip. Mat. Univ Tor Vergata, Roma). Zbl0901.70013MR1486892
  5. [5] R. Courant and D. Hilbert, Methods of Mathematical Physics, volume II. Interscience, New York (1963). Zbl0099.29504MR65391
  6. [6] T. Djaferis and I. Schick, Eds., Advances in System Theory. Kluwer Academic Publishers Boston, October (1999). 
  7. [7] L.C. Evans, Partial Differential Equations. A.M.S. Grad. Stud. Math. 19 (2002). 
  8. [8] H. Federer, Geometric measure theory. Springer-Verlag, Berlin (1969). Zbl0176.00801MR257325
  9. [9] W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions. Springer-Verlag, Berlin (1993). Zbl0773.60070MR1199811
  10. [10] P. Hartman, Ordinary Differential Equations. Wiley, New York (1964). Zbl0125.32102MR171038
  11. [11] J. Itoh and M. Tanaka, The Lipschitz continuity of the distance function to the cut locus. Trans. Amer. Math. Soc. 353 (2000) 21-40. Zbl0971.53031MR1695025
  12. [12] S.N. Kružhkov, The cauchy problem in the large for certain non-linear first order differential equations. Soviet Math. Dockl. 1 (1960) 474-475. Zbl0128.32303MR121575
  13. [13] Yan yan Li and L. Nirenberg, The distance function to the boundary, finsler geometry and the singular set of viscosity solutions of some Hamilton-Jacobi equations (2003) (preprint). Zbl1062.49021MR2094267
  14. [14] P.L. Lions, Generalized Solutions of Hamilton-Jacobi Equations. Pitman, Boston (1982). Zbl0497.35001MR667669
  15. [15] C. Mantegazza and A.C. Mennucci, Hamilton-Jacobi equations and distance functions on Riemannian manifolds. Appl. Math. Optim. 47 (2003) 1-25. Zbl1048.49021MR1941909
  16. [16] D. McDuff and D. Salomon, Introduction to Symplectic Topology. Oxford Mathematical Monograph, Oxford University Press, Clarendon Press, Oxford (1995). Zbl0844.58029MR1702941
  17. [17] A.C.G. Mennucci, Regularity and variationality of solutions to Hamilton-Jacobi equations. Part ii: variationality, existence, uniqueness (in preparation). Zbl1216.49025
  18. [18] C. Sinestrari and P. Cannarsa, Semiconcave functions, Hamilton-Jacobi equations and optimal control problems, in Progress in Nonlinear Differential Equations and Their Applications, Vol. 58, Birkhauser Boston (2004). Zbl1095.49003MR2041617
  19. [19] G.J. Galloway, P.T. Chruściel, J.H.G. Fu and R. Howard, On fine differentiability properties of horizons and applications to Riemannian geometry (to appear). Zbl1023.53054MR1872378
  20. [20] C. Pignotti, Rectifiability results for singular and conjugate points of optimal exit time problems. J. Math. Anal. Appl. 270 (2002) 681-708. Zbl1008.49020MR1916603
  21. [21] L. Simon, Lectures on Geometric Measure Theory, Vol. 3 of Proc. Center for Mathematical Analysis. Australian National University, Canberra (1983). Zbl0546.49019MR756417
  22. [22] Y. Yomdin, β -spreads of sets in metric spaces and critical values of smooth functions. 
  23. [23] Y. Yomdin, The geometry of critical and near-critical values of differential mappings. Math. Ann. 4 (1983) 495-515. Zbl0507.57019MR716263
  24. [24] Y. Yomdin, Metric properties of semialgebraic sets and mappings and their applications in smooth analysis, in Géométrie algébrique et applications, III (la Rábida, 1984), Herman, Paris (1987) 165-183. Zbl0632.58009MR907941

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