RT Journal Article T1 Weak geodesics on prox-regular subsets of Riemannian manifolds A1 Ferrera Cuesta, Juan A1 Pouryayevali, Mohamad R. A1 Radmanesh, Hajar AB We give a definition of weak geodesics on prox-regular subsets of Riemannian manifolds as continuous curves with some weak regularities. Then obtaining a suitable Lipschitz constant of the projection map, we characterize weak geodesics on a prox-regular set with assigned end points as viscosity critical points of the energy functional. YR 2022 FD 2022 LK https://hdl.handle.net/20.500.14352/71711 UL https://hdl.handle.net/20.500.14352/71711 LA eng NO [1] Adams, R.A., Fournier J.J.: Sobolev Spaces. Elsevier (2003)[2] Alexander, S.B., Berg, I.D., Bishop, R.L.: The Riemannian obstacle problem. Illinois J. Math. 31, 167-184 (1987)[3] Azagra, D., Ferrera, J.: Proximal calculus on Riemannian manifolds. Mediterr. J. Math. 2, 437-450 (2005)[4] Azagra, D., Ferrera, J., López-Mesas, F.: Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds. J. Funct. Anal. 220, 304-361 (2005)[5] Barani, A., Hosseini, S., Pouryayevali, M.R.: On the metric projection onto ϕ-convex subsets of Hadamard manifolds. Rev. Mat. Complut. 26, 815-826 (2013)[6] Canino, A.: Existence of a closed geodesic on p-convex sets. Ann. Inst. H. Poincaré C Anal. Non Linéaire. 5, 501-518 (1988)[7] Canino, A.: Local properties of geodesics on p-convex sets. Ann. Mat. Pura Appl. 159(1), 17-44 (1991)[8] Canino, A.: On p-convex sets and geodesics. J. Differential Equations. 75, 118-157 (1988)[9] Clarke, F.H., Ledyaev, Yu.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Graduate Texts in Mathematics 178, Springer, New York (1998)[10] Colombo, G., Thibault, L.: Prox-regular sets and applications. Handbook of Nonconvex Analysis and Applications, D.Y. Gao and D. Motreanu Eds., International Press, Boston, 99-182 (2010)[11] Convent, A., Van Schaftingen, J.: Higher order intrinsic weak differentiability and Sobolev spaces between manifolds. Adv. Calc. Var. 12(3), 303-332 (2019)[12] Convent, A., Van Schaftingen, J.: Intrinsic colocal weak derivatives and Sobolev spaces between manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16, 97-128 (2016)[13] Degiovanni, M., Marzocchi, M.: A critical point theory for nonsmooth functionals. Ann. Mat. Pura Appl.(4) 167, 73-100 (1994)[14] Degiovanni, M., Morbini, L.: Closed geodesics with Lipschitz obstacle. J. Math. Anal. Appl. 233, 767-789 (1999)[15] do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Boston (1992)[16] Ghimenti, M.: Geodesics in conical manifolds. Topol. Methods Nonlinear Anal. 25(2),235-261 (2005)[17] Hardering, H., Intrinsic discretization error bounds for geodesic finite elements. Doctoral dissertation, Freie Universität Berlin (2015)[18] Hosseini, S., Pouryayevali, M.R.: On the metric projection onto prox-regular subsetsof Riemannian manifolds. Proc. Amer. Math. Soc. 141, 233-244 (2013)[19] Klingenberg, W.P.: Riemannian Geometry. Walter de Gruyter (2011)[20] Lancelotti, S., Marzocchi, M.: Lagrangian systems with Lipschitz obstacle on manifolds. Topol. Methods Nonlinear Anal. 27(2), 229-253 (2006)[21] Lee, J.M.: Introduction to Riemannian Manifolds. Graduate Texts in Mathematics 176, Springer, New York (2018)[22] Marino, A., Scolozzi, D.: Geodetiche con ostacolo. Boll. Un. Mat. Ital. B(6) 2, 1-31(1983)[23] Maury, B., Venel, J.: A mathematical framework for a crowd motion model. C. R. Math. Acad. Sci. Paris. 346, 1245–1250 (2008)[24] Poliquin, R.A., Rockafellar, R.T.: Prox-regular functions in variational analysis. Trans. Amer. Math. Soc.348, 1805-1838 (1996)[25] Pouryayevali, M.R., Radmanesh, H.: Minimizing curves in proxregular subsets of Riemannian manifolds. Set-Valued and Var. Anal. (2021). https://doi.org/10.1007/s11228-021-00614-z[26] Pouryayevali, M.R., Radmanesh, H.: Sets with the unique footpoint property and ϕ-convex subsets of Riemannian manifolds. J. Convex Anal. 26, 617-633 (2019)[27] Sakai, T.: Riemannian Geometry. Translations of Mathematical Monographs 149. American Mathematical Society (1996)[28] Schwartz, J.T.: Generalizing the Lusternik-Schnirelman theory of critical points. Comm. Pure Appl. Math. 17, 307-315 (1964)[29] Scolozzi, D.: Un risultato di locale unicità per le geodetiche su varietà con bordo. Boll. Un. Mat. Ital. B (6) 5, 309-327 (1986)[30] Serre, J.P.: Homologie singulière des espaces fibrés. Ann. of Math. 54, 425-505 (1951)[31] Tanwani, A., Brogliato, B., Prieur, C.: Stability and observer design for Lur’e systems with multivalued, nonmonotone, time-varying nonlinearities and state jumps. SIAM J. Control Optim. 52, 3639–3672 (2014)[32] Wehrheim, K.: Uhlenbeck compactness. European Mathematical Society (2004)[33] Wolter, F.E.: Interior metric shortest paths and loops in Riemannian manifolds with not necessarily smooth boundary. preprint (1979) NO Iran National Science Foundation (INSF) DS Docta Complutense RD 3 may 2024