On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations

Bardi, Martino; Mannucci, Paola

doi:10.3934/cpaa.2006.5.709

Cited by 34 publications

(43 citation statements)

References 23 publications

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“…To prove the property (3), it suffices to show that for each fixed x 0 ∈ Ω and each fixed v ∈ C 2 (Ω) with D 2 v(x 0 ) ∈ Θ(x 0 ) one has (3.16) w := u + v ∈ SA(x 0 ). 2 We recall that u * (x) := lim sup…”

Section: The Comparison Principle For Uniformly Continuous Elliptic Mapsmentioning

confidence: 99%

“…2) where Ω ⊂ R N is a bounded open domain with C 2 boundary and ϕ and F are given continuous functions. More precisely, we will examine the validity of the comparison principle for (1.1) and the well posedness of(1.1)-(1.2) by way of Perron's method for admissible viscosity solutions u of elliptic branches of the equation in the sense of Krylov [20].…”

Section: Introductionmentioning

confidence: 99%

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On viscosity solutions to the Dirichlet problem for elliptic branches of inhomogeneous fully nonlinear equations

Cirant¹,

Payne²

2017

Publicacions Matemàtiques

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For scalar fully nonlinear partial differential equations F (x, D 2 u(x)) = 0 with x ∈ Ω R N , we present a general theory for obtaining comparison principles and well posedness for the associated Dirichlet problem, where F (x, ·) need not be monotone on all of S(N ), the space of symmetric N ×N matrices. We treat admissible viscosity solutions u of elliptic branches of the equation in the sense of Krylov [20] and extend the program initiated by Harvey and Lawson [11] in the homogeneous case when F does not depend on x. In particular, for the set valued map Θ defining the elliptic branch by way of the differential inclusion D 2 u(x) ∈ ∂Θ(x), we identify a uniform continuity property which ensures the validity of the comparison principle and the applicability of Perron's method for the differential inclusion on suitably convex domains, where the needed boundary convexity is characterized by Θ. Structural conditions on F are then derived which ensure the existence of an elliptic map Θ with the needed regularity. Concrete applications are given in which standard structural conditions on F may fail and without the request of convexity conditions in the equation. Examples include perturbed Monge-Ampère equations and equations prescribing eigenvalues of the Hessian.

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Section: The Comparison Principle For Uniformly Continuous Elliptic Mapsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On viscosity solutions to the Dirichlet problem for elliptic branches of inhomogeneous fully nonlinear equations

Cirant¹,

Payne²

2017

Publicacions Matemàtiques

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“…Proof. The Comparison Principle in bounded sets under the first condition in (4.3) is standard [24], whereas under the second condition it is Corollary 4.1 in [12].…”

Section: Quasilinear Hypoelliptic Operatorsmentioning

confidence: 99%

Liouville properties and critical value of fully nonlinear elliptic operators

Bardi

Cesaroni

2016

Journal of Differential Equations

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Abstract. We prove some Liouville properties for sub-and supersolutions of fully nonlinear degenerate elliptic equations in the whole space. Our assumptions allow the coefficients of the first order terms to be large at infinity, provided they have an appropriate sign, as in OrnsteinUhlenbeck operators. We give two applications. The first is a stabilization property for large times of solutions to fully nonlinear parabolic equations. The second is the solvability of an ergodic Hamilton-Jacobi-Bellman equation that identifies a unique critical value of the operator.

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“…For a class of equations that can be written in Hamilton-Jacobi-Bellman form we can show that w := u − v is a subsolution of a homogeneous PDE F 0 (x, w, Dw, D 2 w) = 0 satisfying the SMP, and therefore we deduce immediately the SCP. A model problem is the equation (6) M + ((D 2 X u) * ) + H(x, Du) = 0, where M + denotes the Pucci's maximal operator (see Section 3.1 for the definition),X = (X 1 , ..., X m ) are Hörmander vector fields, and H(x, p) = sup α {p · b α (x) + f α (x)} with data b α , f α bounded and Lipschitz uniformly in α. Remarkably, this result implies the (weak) Comparison Principle also in some cases for which it was not yet known, see Section 4.…”

Section: Introductionmentioning

confidence: 99%

New strong maximum and comparison principles for fully nonlinear degenerate elliptic PDEs

Bardi

Goffi

2019

Calc. Var.

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We introduce a notion of subunit vector field for fully nonlinear degenerate elliptic equations. We prove that an interior maximum of a viscosity subsolution of such an equation propagates along the trajectories of subunit vector fields. This implies strong maximum and minimum principles when the operator has a family of subunit vector fields satisfying the Hörmander condition. In particular these results hold for a large class of nonlinear subelliptic PDEs in Carnot groups. We prove also a strong comparison principle for degenerate elliptic equations that can be written in Hamilton-Jacobi-Bellman form, such as those involving the Pucci's extremal operators over Hörmander vector fields.

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On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations

Cited by 34 publications

References 23 publications

On viscosity solutions to the Dirichlet problem for elliptic branches of inhomogeneous fully nonlinear equations

On viscosity solutions to the Dirichlet problem for elliptic branches of inhomogeneous fully nonlinear equations

Liouville properties and critical value of fully nonlinear elliptic operators

New strong maximum and comparison principles for fully nonlinear degenerate elliptic PDEs

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