On the generalized Dirichlet problem for viscous Hamilton–Jacobi equations

Barles, Guy; Lio, Francesca Da

doi:10.1016/s0021-7824(03)00070-9

Cited by 67 publications

(110 citation statements)

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“…This means that in the study of the Dirichlet problem the Hölder regularity of the boundary datum ϕ is necessary in order to find a strong solution, satisfying pointwise the condition u = ϕ. This allows us to complement previous results obtained in [4] concerning the relaxed formulation of the Dirichlet problem associated to (1.1), in case A(x) does not degenerate in the normal direction at ∂Ω. More precisely, under some compatibility conditions between f and ϕ and assuming an upper bound on the Hölder constant of ϕ, we prove (Theorem 2.12) the existence of a C 0, p−2 p−1 (Ω) viscosity solution of (1.1) satisfying pointwise the Dirichlet condition u = ϕ.…”

Section: J(x A)mentioning

confidence: 81%

“…By contrast, when p > 2, the results of [4,10] imply that problem (2.22), even in the uniformly elliptic case and posed in a smooth domain, cannot be solved in general. The best one can obtain is the existence of a viscosity solution satisfying the boundary condition in the relaxed viscosity formulation (see [5,8]).…”

Section: Solvability Of the Dirichlet Problemmentioning

confidence: 99%

“…This is equivalent to proving the existence of a viscosity subsolution v ∈ C(Ω) satisfying v = ϕ on ∂Ω. Indeed, if such a subsolution v exists, then the generalized solution u ∈ C(Ω) proved in [4] to exist satisfies u ≥ v in Ω and u ≤ ϕ on ∂Ω; hence ϕ = v ≤ u ≤ ϕ on ∂Ω.…”

Section: Solvability Of the Dirichlet Problemmentioning

confidence: 99%

“…The best one can obtain is the existence of a viscosity solution satisfying the boundary condition in the relaxed viscosity formulation (see [5,8]). Precisely, in [4], the authors prove that (2.22) has a unique generalized viscosity solution, that is, a function u ∈ C(Ω) which is a viscosity solution in Ω, u ≤ ϕ on ∂Ω and it satisfies, in the viscosity sense,…”

Section: Solvability Of the Dirichlet Problemmentioning

confidence: 99%

See 3 more Smart Citations

Hölder estimates for degenerate elliptic equations with coercive Hamiltonians

Dolcetta

Leoni

Porretta

2010

Trans. Amer. Math. Soc.

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Abstract. We prove a priori estimates and regularity results for some quasilinear degenerate elliptic equations arising in optimal stochastic control problems. Our main results show that strong coerciveness of gradient terms forces bounded viscosity subsolutions to be globally Hölder continuous, and solutions to be locally Lipschitz continuous. We also give an existence result for the associated Dirichlet problem.

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Section: J(x A)mentioning

confidence: 81%

Section: Solvability Of the Dirichlet Problemmentioning

confidence: 99%

Section: Solvability Of the Dirichlet Problemmentioning

confidence: 99%

Section: Solvability Of the Dirichlet Problemmentioning

confidence: 99%

See 2 more Smart Citations

Hölder estimates for degenerate elliptic equations with coercive Hamiltonians

Dolcetta

Leoni

Porretta

2010

Trans. Amer. Math. Soc.

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“…Indeed, the classical Dirichlet problem (with boundary data being really satisfied by the solution) cannot be solved in general and one has to use the generalized Dirichlet problem in the sense of viscosity solutions. We refer to the Users' guide [18] and references therein for an introduction of this concept and to [6,25] for the applications to (1.6)- (1.5) where it is shown that the generalized Dirichlet problem is well-posed in C(Ω) or C(Ω × [0, T ]), and with examples where the solution is different from ϕ at points on ∂Ω.…”

Section: Introductionmentioning

confidence: 99%

Lipschitz regularity for censored subdiffusive integro-differential equations with superfractional gradient terms

Barles

Topp

2016

Nonlinear Analysis

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Abstract. In this paper we are interested in integro-differential elliptic and parabolic equations involving nonlocal operators with order less than one, and a gradient term whose coercivity growth makes it the leading term in the equation. We obtain Lipschitz regularity results for the associated stationary Dirichlet problem in the case when the nonlocality of the operator is confined to the domain, feature which is known in the literature as censored nonlocality. As an application of this result, we obtain strong comparison principles which allow us to prove the well-posedness of both the stationary and evolution problems, and steady/ergodic large time behavior for the associated evolution problem.

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Hölder estimates for viscous Hamilton–Jacobi equations

Dolcetta

Leoni

Porretta

2007

Proc Appl Math and Mech

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Presentation and statement of the results.We present here new results concerning regularity and a priori estimates for solutions (or subsolutions) of second order degenerate elliptic equations, whose simplest model is the followingHere Ω is an open bounded subset of R N , A : Ω → S + N is a continuous map into the space of symmetric non-negative matrices of order N , λ > 0 and p > 1 are given numbers and f : Ω → R is a continuous function. It is well-known that equations of the form (1) arise in connection to optimal control problems for degenerate diffusion processes governed by the Ito's equationwhere a(X t ) is interpreted as a feedback control and W t is a standard Brownian motion. Indeed, the classical Dynamic Programming argument (see [7]) indicates that given a cost functional of the formwhere τ x is the first exit time, then the value function u(x) =: inf a J(x, a) is a viscosity solution of equation (1), supplemented either by the Dirichlet boundary condition u = ϕ (exit-time problem) or by the state constraint condition τ x = +∞ (the process stays in Ω for all time, with probability one). Of course, both these conditions have to be formulated in a suitable generalized sense, which is possible in the framework of viscosity solutions (see [6]); indeed, one has to take into account the possibility that the boundary condition is not verified at every point (as for first order equations) but only in a relaxed sense. Similar problems have been investigated under various sets of assumptions (see e.g. [2], [3], [4], [8], [9], [12]). In particular, the state constraint problem for equation (1) was deeply studied in [10] in case of a purely Brownian motion (i.e. A(x) ≡ Id); it turns out that there is a significant difference between the cases p ≤ 2 and p > 2, since in this latter case the maximal solution of (1) is bounded and loss of boundary condition can occur in the study of the Dirichlet problem (see also [2]).Our main contribution concerns the case of superquadratic growth p > 2, where we prove that not only solutions, but even subsolutions of (1) are Hölder continuous. The precise result reads as follows:Theorem 1.

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On the generalized Dirichlet problem for viscous Hamilton–Jacobi equations

Cited by 67 publications

References 16 publications

Hölder estimates for degenerate elliptic equations with coercive Hamiltonians

Hölder estimates for degenerate elliptic equations with coercive Hamiltonians

Lipschitz regularity for censored subdiffusive integro-differential equations with superfractional gradient terms

Hölder estimates for viscous Hamilton–Jacobi equations

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