The present paper studies the bounded variation-type regularity for viscosity solutions of the Hamilton-Jacobi equation ut(t, x) + H ( Dxu(t, x) ) = 0, (t, x) (0,∞) × Rd, with a coercive and uniformly directionally convex Hamiltonian H. More precisely, we establish a BV bound on the slope of backward characteristics DH(Dxu(t, )) starting at a positive time t. Relying on the BV bound, we quantify the metric entropy in W1,1 loc ( Rd ) for the map St that associates, to every given initial data u0 Lip ( Rd ) , the corresponding solution Stu0. Finally, a counterexample is constructed to show that both Dxu(t, ) and DH(Dxu(t, )) fail to be in BVloc for a general strictly convex and coercive H ∈ C 2 ( Rd).

Metric Entropy for Hamilton-Jacobi Equations with Uniformly Directionally Convex Hamiltonian / Bianchini, Stefano; Dutta, Prerona; Nguyen, Khai T.. - In: SIAM JOURNAL ON MATHEMATICAL ANALYSIS. - ISSN 0036-1410. - 54:5(2022), pp. 5551-5575. [10.1137/22M1475430]

Metric Entropy for Hamilton-Jacobi Equations with Uniformly Directionally Convex Hamiltonian

Bianchini, Stefano;
2022-01-01

Abstract

The present paper studies the bounded variation-type regularity for viscosity solutions of the Hamilton-Jacobi equation ut(t, x) + H ( Dxu(t, x) ) = 0, (t, x) (0,∞) × Rd, with a coercive and uniformly directionally convex Hamiltonian H. More precisely, we establish a BV bound on the slope of backward characteristics DH(Dxu(t, )) starting at a positive time t. Relying on the BV bound, we quantify the metric entropy in W1,1 loc ( Rd ) for the map St that associates, to every given initial data u0 Lip ( Rd ) , the corresponding solution Stu0. Finally, a counterexample is constructed to show that both Dxu(t, ) and DH(Dxu(t, )) fail to be in BVloc for a general strictly convex and coercive H ∈ C 2 ( Rd).
2022
54
5
5551
5575
https://arxiv.org/abs/2012.10577
Bianchini, Stefano; Dutta, Prerona; Nguyen, Khai T.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/135417
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