# Lipschitz regularity for viscous Hamilton-Jacobi equations with L terms

@article{Cirant2018LipschitzRF,
title={Lipschitz regularity for viscous Hamilton-Jacobi equations with L terms},
author={Marco Cirant and Alessandro Goffi},
journal={arXiv: Analysis of PDEs},
year={2018}
}
• Published 10 December 2018
• Mathematics
• arXiv: Analysis of PDEs
We provide Lipschitz regularity for solutions to viscous time-dependent Hamilton-Jacobi equations with right-hand side belonging to Lebesgue spaces. Our approach is based on a duality method, and relies on the analysis of the regularity of the gradient of solutions to a dual (Fokker-Planck) equation. Here, the regularizing effect is due to the non-degenerate diffusion and coercivity of the Hamiltonian in the gradient variable.
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#### References

SHOWING 1-10 OF 47 REFERENCES
Sobolev regularity for the first order Hamilton–Jacobi equation
• Mathematics
• 2014
We provide Sobolev estimates for solutions of first order Hamilton–Jacobi equations with Hamiltonians which are superlinear in the gradient variable. We also show that the solutions areExpand
Hölder Continuity to Hamilton-Jacobi Equations with Superquadratic Growth in the Gradient and Unbounded Right-hand Side
• Mathematics
• 2012
We show that solutions of time-dependent degenerate parabolic equations with super-quadratic growth in the gradient variable and possibly unbounded right-hand side are locally 𝒞0, α. Unlike theExpand
Adjoint and Compensated Compactness Methods for Hamilton–Jacobi PDE
We investigate the vanishing viscosity limit for Hamilton–Jacobi PDE with nonconvex Hamiltonians, and present a new method to augment the standard viscosity solution approach. The main idea is toExpand
Existence and uniqueness of solutions to parabolic equations with superlinear Hamiltonians
• A. Davini
• Mathematics
• Communications in Contemporary Mathematics
• 2019
We give a proof of existence and uniqueness of viscosity solutions to parabolic quasilinear equations for a fairly general class of nonconvex Hamiltonians with superlinear growth in the gradientExpand
Viscosity solutions of fully nonlinear second-order elliptic partial differential equations
• Mathematics
• 1990
We investigate comparison and existence results for viscosity solutions of fully nonlinear, second-order, elliptic, possibly degenerate equations. These results complement those recently obtained byExpand
Solvability and Maximal Regularity of Parabolic Evolution Equations with Coefficients Continuous in Time
• Mathematics
• 2001
Abstract We establish maximal regularity of type Lp for a parabolic evolution equation u′(t) = A(t)u(t) + f(t) with A( · ) ∈ C([0, T],  L (D(A(0)), X)) and construct the corresponding evolutionExpand
Viscosity solutions of general viscous Hamilton–Jacobi equations
• Mathematics
• 2013
We present comparison principles, Lipschitz estimates and study state constraints problems for degenerate, second-order Hamilton–Jacobi equations.
On the Large Time Behavior of Solutions of Hamilton-Jacobi Equations
• Mathematics, Computer Science
• SIAM J. Math. Anal.
• 2000
It is proved, under sharp conditions, that as time goes to infinity, solutions converge to solutions of the corresponding stationary equation. Expand
Lipschitz regularity results for nonlinear strictly elliptic equations and applications
• Mathematics
• 2016
Most of lipschitz regularity results for nonlinear strictly elliptic equations are obtained for a suitable growth power of the nonlinearity with respect to the gradient variable (subquadratic forExpand
On Some Existence Theorems for Semi-Linear Elliptic Equations.
• Mathematics
• 1977
Abstract : Boundary value problems for semilinear elliptic equations are considered. If the nonlinear terms do not grow too fast as a function of the gradient of the dependent variable and orderedExpand