Mixed boundary condition for the Monge-Kantorovich equation
DOI:
https://doi.org/10.12775/TMNA.2015.088Keywords
Nonlinear PDE, tangential gradient, R^N-valued Radon measure flux, optimization problemAbstract
In this work we give some equivalent formulations for the optimization problem \begin{multline*} \max\bigg\{ \int_{\Omega} \xi \,d\mu + \int_{\Gamma_{N}}\xi \,d\nu;\ \xi \in W^{1,\infty}(\Omega) \text{ such that } \\ \xi_{/\Gamma_{D}}= 0, |\nabla\xi(x)|\leq 1 \text{ a.e. } x\in \Omega\bigg\}, \end{multline*} where the boundary of $\Omega$ is $\Gamma=\Gamma_{N}\cup\Gamma_{D}$.References
L. Ambrosio, Lecture notes on optimal transport, in mathematical aspect of evolving interfaces, in: Lecture Notes in Mathematics, LNM 1812 (2003), Springer, Berlin.
G. Aronson and L.G. Evans, An asymptotic model for compression molding, Indiana Univ. Math. J. 51 (2002), no. 1, 1-36.
J.W. Barrett and L. Prigozhin, A mixed formulation for the Monge-Kantorovich equations, M2AN Math. Model. Numer. Anal. 41 (6) (2007), 1041-1060.
G. Bouchitte, G. Buttazzo and P. Seppecher, Energies with respect to a measure and applications to low dimensional structures, Calc. Var. Partial Differential Equations 5, (1997), 37-54.
G. Bouchitte, G. Buttazzo and P. Seppecher, Shape Optimization Solutions via Monge-Kantorovich, C.R. Acad. Sci. Paris Ser. I 324 (1997), 1185-1991.
G. Bouchitte, Th. Champion and C. Jimenez, Completion of the space of measures in the Kantorovich norm, Riv. Math. Univ. Parma (N.S.) 7 (4), (2005), 127-139.
H. Brezis, Analyse Fonctionnelle Theorie et application, Masson, 1987.
G. Buttazzo and L. De Pascal, Optimal shapes and mass, and optimal transportation problems, Optimal Transportation and Applications, Martina Franca 2-8 September 2001, Lecture Notes in Mathematics, vol. 1813, Springer-Verlag, Berlin, 2003, 11-52.
S. Dumont, N. Igbida, On a dual formulation for growing sandpile problem, Euro. J. App. Math. 20 (2009), 169-185.
L.C. Evans, Partial differential equations and Monge-Kantorovich mass transfert, Current Developments in Mathematics, International Press, Boston, MA, (1997), 65-126.
L.C. Evans and W. Gangbo, Differential equations methods for the Monge-Kantorovich mass transfert problem, Mem. Amer. Math. Soc. 137 (1999).
N. Igdida, Equivalent formulations for Monge-Kantorovich equation, Nonlinear Anal. 71 (2009), 3805-3813.
N. Igdida, Evolution Monge-Kantorovich equation, J. Differential Equations 255, (2003), 1383-1407.
J.L. Lions, Quelques methodes de resolution des problemes aux limites non lineaires, Dunod. Gauthier-Villars, Paris, (1996).
G. Monge, Memoire sur la theorie des delais et des remblais, Histoire de L'Academie des Sciences de Paris (1781).
L. Prigozhin, Variational model of sandpile growth, Euro. J. App. Math. 7 (1996), 225-236.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 0
Number of citations: 0