Regularity and variationality of solutions to Hamilton-Jacobi equations. Part I : regularity
ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 3, pp. 426-451.

We formulate an Hamilton-Jacobi partial differential equation

H(x,Du(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 n-1 negligeable subset. These results can be applied to the cutlocus of a C 2 submanifold of a Finsler manifold.

DOI : 10.1051/cocv:2004014
Classification : 49L25, 53C22, 53C60
Mots clés : Hamilton-Jacobi equations, conjugate points
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     title = {Regularity and variationality of solutions to {Hamilton-Jacobi} equations. {Part} {I} : regularity},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {426--451},
     publisher = {EDP-Sciences},
     volume = {10},
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     mrnumber = {2084331},
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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, Tome 10 (2004) no. 3, pp. 426-451. doi : 10.1051/cocv:2004014. http://archive.numdam.org/articles/10.1051/cocv:2004014/

[1] G. Alberti, On the structure of singular sets of convex functions. Calc. Var. Partial Differ. Equ. 2 (1994) 17-27. | MR | Zbl

[2] A. Ambrosetti and G. Prodi, A primer of nonlinear analysis. Cambridge University Press, Cambridge (1993). | MR | Zbl

[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. | EuDML | Numdam | MR | Zbl

[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). | MR | Zbl

[5] R. Courant and D. Hilbert, Methods of Mathematical Physics, volume II. Interscience, New York (1963). | MR | Zbl

[6] T. Djaferis and I. Schick, Eds., Advances in System Theory. Kluwer Academic Publishers Boston, October (1999). | Zbl

[7] L.C. Evans, Partial Differential Equations. A.M.S. Grad. Stud. Math. 19 (2002).

[8] H. Federer, Geometric measure theory. Springer-Verlag, Berlin (1969). | MR | Zbl

[9] W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions. Springer-Verlag, Berlin (1993). | MR | Zbl

[10] P. Hartman, Ordinary Differential Equations. Wiley, New York (1964). | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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). | MR | Zbl

[14] P.L. Lions, Generalized Solutions of Hamilton-Jacobi Equations. Pitman, Boston (1982). | MR | Zbl

[15] C. Mantegazza and A.C. Mennucci, Hamilton-Jacobi equations and distance functions on Riemannian manifolds. Appl. Math. Optim. 47 (2003) 1-25. | MR | Zbl

[16] D. Mcduff and D. Salomon, Introduction to Symplectic Topology. Oxford Mathematical Monograph, Oxford University Press, Clarendon Press, Oxford (1995). | MR | Zbl

[17] A.C.G. Mennucci, Regularity and variationality of solutions to Hamilton-Jacobi equations. Part ii: variationality, existence, uniqueness (in preparation).

[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). | MR | Zbl

[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). | MR | Zbl

[20] C. Pignotti, Rectifiability results for singular and conjugate points of optimal exit time problems. J. Math. Anal. Appl. 270 (2002) 681-708. | MR | Zbl

[21] L. Simon, Lectures on Geometric Measure Theory, Vol. 3 of Proc. Center for Mathematical Analysis. Australian National University, Canberra (1983). | MR | Zbl

[22] Y. Yomdin, β-spreads of sets in metric spaces and critical values of smooth functions.

[23] Y. Yomdin, The geometry of critical and near-critical values of differential mappings. Math. Ann. 4 (1983) 495-515. | MR | Zbl

[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. | MR | Zbl

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