A regularity result for a class of non-uniformly elliptic operators

Ferrari, Fausto; Galise, Giulio

doi:10.1007/s00013-022-01715-3

Cited by 4 publications

(2 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For this equation with lower order terms, Ferrari and Vitolo [11] used methods from the viscosity theory to study ABP, Harnack and Hölder estimates, and later Ferrari and Galise [10] showed C 0,δ -regularity for δ = 1 − α 1 +α n ( √ α 1 + √ α n ) 2 ∈ (0, 1 2 ]. We note that δ = 1 2 holds if and only if α 1 = α n .…”

Section: Resultsmentioning

confidence: 99%

Game-Theoretic Approach to Hölder Regularity for PDEs Involving Eigenvalues of the Hessian

et al. 2022

View full text Add to dashboard Cite

We prove a local Hölder estimate for any exponent $0<\delta <\frac {1}{2}$ 0 < δ < 1 2 for solutions of the dynamic programming principle $ u^{\varepsilon } (x) = \sum \limits _{j=1}^{n} \alpha _{j} \underset {\dim (S)=j}{\inf } \underset {|v|=1}{\underset {v\in S}{\sup }} \frac {u^{\varepsilon } (x + \varepsilon v) + u^{\varepsilon } (x - \varepsilon v)}{2} $ u ε ( x ) = ∑ j = 1 n α j inf dim ( S ) = j sup v ∈ S | v | = 1 u ε ( x + ε v ) + u ε ( x − ε v ) 2 with α1,αn > 0 and α2,⋯ ,αn− 1 ≥ 0. The proof is based on a new coupling idea from game theory. As an application, we get the same regularity estimate for viscosity solutions of the PDE $ \sum \limits _{i=1}^{n} \alpha _{i} \lambda _{i}(D^{2}u)=0, $ ∑ i = 1 n α i λ i ( D 2 u ) = 0 , where λ1(D2u) ≤⋯ ≤ λn(D2u) are the eigenvalues of the Hessian.

show abstract

Section: Resultsmentioning

confidence: 99%

Game-Theoretic Approach to Hölder Regularity for PDEs Involving Eigenvalues of the Hessian

et al. 2022

View full text Add to dashboard Cite

show abstract

“…About such approach, there are many contributions in literature. Among them, we wish to recall the following works [BGI18], [BGL17], [FV20b], [FG21], where the regularity of viscosity solutions of truncated operators has been studied. Moreover, always in the frame of a degenerate situation but in a non-commutative structures, we point out the results contained in [Fer20], [FV20a] and [Gof20].…”

mentioning

confidence: 99%

Estimates for the $\infty$-Laplacian relative to vector fields

Ferrari¹,

Manfredi²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Lipschitz estimates for partial trace operators with extremal Hessian eigenvalues

Vitolo

2022

Advances in Nonlinear Analysis

View full text Add to dashboard Cite

We consider the Dirichlet problem for partial trace operators which include the smallest and the largest eigenvalue of the Hessian matrix. It is related to two-player zero-sum differential games. No Lipschitz regularity result is known for the solutions, to our knowledge. If some eigenvalue is missing, such operators are nonlinear, degenerate, non-uniformly elliptic, neither convex nor concave. Here we prove an interior Lipschitz estimate under a non-standard assumption: that the solution exists in a larger, unbounded domain, and vanishes at infinity. In other words, we need a condition coming from far away. We also provide existence results showing that this condition is satisfied for a large class of solutions. On the occasion, we also extend a few qualitative properties of solutions, known for uniformly elliptic operators, to partial trace operators.

show abstract

A regularity result for a class of non-uniformly elliptic operators

Abstract: We obtain an explicit Hölder regularity result for viscosity solutions of a class of second order fully nonlinear equations led by operators that are neither convex/concave nor uniformly elliptic.

Cited by 4 publications

References 23 publications

Game-Theoretic Approach to Hölder Regularity for PDEs Involving Eigenvalues of the Hessian

Game-Theoretic Approach to Hölder Regularity for PDEs Involving Eigenvalues of the Hessian

Estimates for the $\infty$-Laplacian relative to vector fields

Lipschitz estimates for partial trace operators with extremal Hessian eigenvalues

Contact Info

Product

Resources

About