×

On gradient flows. (English) Zbl 0927.37007

The purpose of this paper is to contribute to the theory of existence of solutions to ordinary differential equations in \(\mathbb{R}^n\), when the right hand side of the equation is not necessarily continuous. More precisely the authors study existence and uniqueness of solutions for the equation \(x'= \nabla u(x)\) when \(u\) is not necessarily differentiable everywhere. This problem is closely related with (multidimensional) calculus of variations.

MSC:

37B35 Gradient-like behavior; isolated (locally maximal) invariant sets; attractors, repellers for topological dynamical systems
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34A36 Discontinuous ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aubin, J. P.; Cellina, A., Differential Inclusions (1984), Springer: Springer New York
[2] M. Bardi, I. Capuzzo Dolcetta, Deterministic optimal and viscosity solutions of Hamilton-Jacobi-Bellman equations; M. Bardi, I. Capuzzo Dolcetta, Deterministic optimal and viscosity solutions of Hamilton-Jacobi-Bellman equations · Zbl 0890.49011
[3] Brézis, H., Operateurs maximaux monotones et semigroupes de contractions dans les espaces de Hilbert (1973), North Holland: North Holland Amsterdam · Zbl 0252.47055
[4] Caldiroli, P.; Treu, G., Measure properties of the set of initial data yielding non uniqueness for a class of differential inclusions, NODEA, 3, 499-507 (1996) · Zbl 0859.34010
[5] P. Celada, S. Perrotta, G. Treu, Existence of solutions for a class of non convex minimum problems, Math. Zeitschrift; P. Celada, S. Perrotta, G. Treu, Existence of solutions for a class of non convex minimum problems, Math. Zeitschrift · Zbl 0936.49010
[6] Cellina, A., On uniqueness almost everywhere for monotonic differential inclusions, Nonlinear Anal., 25, 899-903 (1995) · Zbl 0837.34023
[7] Cellina, A., Minimizing a functional depending on ∇\(uu\), Ann. Inst. H. Poincaré, Anal. Non Linéaire, 14, 339-352 (1997) · Zbl 0876.49001
[8] Cellina, A.; Perrotta, S., On the validity of the Euler Lagrange equations, preprint S.I.S.S.A. (1996)
[9] Cohn, D. L., Measure Theory (1980), Birkhäuser: Birkhäuser Stuttgart · Zbl 0436.28001
[10] Elliot, R. J., Viscosity solutions and optimal control, Pitman Research Notes in Mathematics (1987)
[11] Evans, L. C.; Gariepy, R. F., Measure Theory and Fine Properties of Functions (1992), CRC Press: CRC Press Boca Raton · Zbl 0626.49007
[12] Kawohl, B., On a family of torsional creep problems, J. Reine Angew. Math., 410, 1-22 (1990) · Zbl 0701.35015
[13] Rockafellar, T., Convex Analysis (1970), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0193.18401
[14] Sakaguchi, S., Concavity properties of solutions to some degenerate quasi-linear elliptic Dirichlet problems, Ann. Scuola Norm. Sup. Pisa (IV), 14, 404-421 (1987)
[15] M. Vornicescu, A variational problem on subsets of \(R^n\); M. Vornicescu, A variational problem on subsets of \(R^n\) · Zbl 0920.49002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.