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A non-local regularization of first order Hamilton-Jacobi equations. (English) Zbl 1073.35059
This paper is concerned with the non-local first-order Hamilton-Jacobi equation $\partial_t u+ H(t,x,u,\nabla u)+ g[u]= 0\quad\text{in }[0,\infty)\times \mathbb{R}^N,\tag{1}$
$u(0,x)= u_0(x)\quad\text{for all }x\in\mathbb{R}^N,\tag{2}$ with $$u_0\in W^{1,\infty}(\mathbb{R}^N)$$, where $$\nabla u$$ denotes the gradient w.r.t. $$x$$, and $$g[u]$$ denotes the pseudodifferential operator defined by the symbol $$|\xi|^\lambda$$, $$1< \lambda< 2$$.
The main result asserts that there exists a solution of (1) with bounded Lipschitz continuous initial condition that is twice continuously differentiable in $$x$$ and continuously differentiable in $$t$$, i.e. is regular. Firstly, the viscosity solution theory is used to give a notion of merely continuous solution of (1) and to construct a bounded Lipschitz continuous one. Secondly, using Duhamel’s integral representation of (1), an “integral” solution that is $$C^2$$ in $$x$$ is constructed by a fixed point method and is proved that the “integral” solution is $$C^1$$ in $$t$$. It finally turns out to be a viscosity solution of (1) (with classical derivatives); the comparison result (which implies uniqueness) permits to conclude.
The author also proves that higher regularity (in fact $$C^\infty$$ regularity in $$(t,x)$$) can be obtained if the assumptions on $$H$$ are strengthened. For $$\lambda= 2$$, this method for proving regularity results is new. In the last section, thinking of the vanishing viscosity method, a vanishing Lévy operator is considered and an error estimate is given.

##### MSC:
 35F25 Initial value problems for nonlinear first-order PDEs 35B65 Smoothness and regularity of solutions to PDEs 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35G25 Initial value problems for nonlinear higher-order PDEs 35K55 Nonlinear parabolic equations
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##### References:
 [1] Alvarez, O.; Tourin, A., Viscosity solutions of nonlinear integro-differential equations, Ann. inst. H. Poincaré anal. non linéaire, 13, 3, 293-317, (1996) · Zbl 0870.45002 [2] Amadori, A.L., Nonlinear integro-differential evolution problems arising in option pricinga viscosity solutions approach, Differential integral equations, 16, 7, 787-811, (2003) · Zbl 1052.35083 [3] Barles, G., Uniqueness and regularity results for first-order hamilton – jacobi equations, Indiana univ. math. J., 39, 2, 443-466, (1990) · Zbl 0709.35024 [4] Barles, G.; Perthame, B., Discontinuous solutions of deterministic optimal stopping time problems, RAIRO modél. math. anal. numér., 21, 4, 557-579, (1987) · Zbl 0629.49017 [5] Barles, G.; Perthame, B., Comparison principle for Dirichlet-type hamilton – jacobi equations and singular perturbations of degenerated elliptic equations, Appl. math. optim., 21, 1, 21-44, (1990) · Zbl 0691.49028 [6] Barles, G.; Buckdahn, R.; Pardoux, E., Backward stochastic differential equations and integral-partial differential equations, Stochastics stochastics rep., 60, 1-2, 57-83, (1997) · Zbl 0878.60036 [7] Benth, F.E.; Karlsen, K.H.; Reikvam, K., Optimal portfolio management rules in a non-Gaussian market with durability and intertemporal substitution, Finance stochastics, 5, 4, 447-467, (2001) · Zbl 1049.91059 [8] Benth, F.E.; Karlsen, K.H.; Reikvam, K., Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constrainta viscosity solution approach, Finance stochastics, 5, 3, 275-303, (2001) · Zbl 0978.91039 [9] Benth, F.E.; Karlsen, K.H.; Reikvam, K., Portfolio optimization in a Lévy market with intertemporal substitution and transaction costs, Stochastics stochastics rep., 74, 3-4, 517-569, (2002) · Zbl 1035.91027 [10] Clarke, F.H.; Ledyaev, Yu.S.; Stern, R.J.; Wolenski, P.R., Nonsmooth analysis and control theory, graduate texts in mathematics, vol. 178, (1998), Springer New York · Zbl 1047.49500 [11] Clavin, P., Diamond patterns in the cellular front of an overdriven detonation, Phys. rev. lett., 88, (2002) [12] Crandall, M.G.; Ishii, H.; Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations, Bull. amer. math. soc. (N.S.), 27, 1, 1-67, (1992) · Zbl 0755.35015 [13] Crandall, M.G.; Lions, P.-L., Viscosity solutions of hamilton – jacobi equations, Trans. amer. math. soc., 277, 1, 1-42, (1983) · Zbl 0599.35024 [14] Droniou, J.; Gallouet, T.; Vovelle, J., Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. evol. equation, 3, 3, 499-521, (2003) · Zbl 1036.35123 [15] J. Droniou, Vanishing non-local regularization of a scalar conservation law, Electron. J. Differential Equations (2003) 117, 20pp. (electronic). · Zbl 1039.35061 [16] Fleming, W.H., Nonlinear partial differential equations—probabilistic and game theoretic methods, (), 95-128 · Zbl 0225.35020 [17] Friedman, A., Uniqueness for the Cauchy problem for degenerate parabolic equations, Pacific J. math., 46, 131-147, (1973) · Zbl 0256.35040 [18] Ishii, H., Uniqueness of unbounded viscosity solution of hamilton – jacobi equations, Indiana univ. math. J., 33, 5, 721-748, (1984) · Zbl 0551.49016 [19] E.R. Jakobsen, K.H. Karlsen, Continuous dependence estimates for viscosity solutions of integro-pdes, preprint. · Zbl 1082.45008 [20] E.R. Jakobsen, K.H. Karlsen, A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations, preprint. · Zbl 1105.45006 [21] Lions, P.-L., Generalized solutions of hamilton – jacobi equations, research notes in mathematics, vol. 69, (1982), Pitman (Advanced Publishing Program) Boston, MA [22] Sayah, A., Équations d’hamilton – jacobi du premier ordre avec termes intégro-différentiels. I. unicité des solutions de viscosité, II. existence de solutions de viscosité, Comm. partial differential equations, 16, 6-7, 1057-1093, (1991) · Zbl 0742.45004 [23] Souganidis, P.E., Existence of viscosity solutions of hamilton – jacobi equations, J. differential equations, 56, 3, 345-390, (1985) · Zbl 0506.35020 [24] Woyczyński, W.A., Lévy processes in the physical sciences, (), 241-266 · Zbl 0982.60043
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