Pointwise Regularity for Fully Nonlinear Elliptic Equations in General Forms

Lian, Yuanyuan; Wang, Lihe; Zhang, Kai

doi:10.48550/arxiv.2012.00324

Cited by 6 publications

(18 citation statements)

References 23 publications

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“…This allows us to apply Theorem 2.1 in Lian et al (2020) (see also Theorem 2.1 in Świech (1997)) to conclude that sup x∈B r 1+γ |∇u(x)| ≤ C for some C independent of r, completing the proof.…”

Section: Exploratory Temperature Control Problemmentioning

confidence: 75%

“…The following result concerns higher regularity of bounded solutions to concave operators; see e.g. Caffarelli and Cabré (1995) and Lian et al (2020). As a consequence, viscosity solutions to concave operators are classical solutions.…”

Section: Analysis Of the Exploratory Hjb Equationmentioning

confidence: 98%

“…Lemma 5 (Theorems 2.1 and 2.6, Lian et al (2020)). Assume that F = F (X, p, s, x) satisfies (a)-(c), and let…”

Section: Analysis Of the Exploratory Hjb Equationmentioning

confidence: 99%

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Exploratory HJB equations and their convergence

Tang¹,

Zhang²

2021

Preprint

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We study the exploratory Hamilton-Jacobi-Bellman (HJB) equation arising from the entropy-regularized exploratory control problem, which was formulated by Wang, Zariphopoulou and Zhou (J. Mach. Learn. Res., 21, 2020) in the context of reinforcement learning in continuous time and space. We establish the well-posedness and regularity of the viscosity solution to the equation, as well as the convergence of the exploratory control problem to the classical stochastic control problem when the level of exploration decays to zero. We then apply the general results to the exploratory temperature control problem, which was introduced by Gao, Zhou (arXiv:2005.04057, 2020) to design an endogenous temperature schedule for simulated annealing (SA) in the context of non-convex optimization. We derive an explicit rate of convergence for this problem as exploration diminishes to zero, and find that the steady state of the optimally controlled process exists, which is however neither a Dirac mass on the global optimum nor a Gibbs measure.

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Section: Exploratory Temperature Control Problemmentioning

confidence: 75%

Section: Analysis Of the Exploratory Hjb Equationmentioning

confidence: 98%

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Exploratory HJB equations and their convergence

Tang¹,

Zhang²

2021

Preprint

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“…This can be proved by Caffarelli's perturbation method [13], [28], more precisely, by seeing the problem as "tangential" to [26,Lemma 4.1]. Results of similar vein, even in larger generality (for unbounded coefficients and L p -viscosity solutions), have recently appeared in [12] for the so-called uniformly elliptic S * -class where the optimal boundary exponent depends also on α F , as well as in the recent preprints [22], [18]. We thus assume Theorem 1.2 for γ = 0 might be known to the experts; however, since this particular case is interesting in itself and deserves a quotable source, and since we use it in the proof of the full Theorem 1.2 and strive to make this work self-contained, we give a complete proof in Section 2.…”

Section: Introductionmentioning

confidence: 90%

Sharp boundary and global regularity for degenerate fully nonlinear elliptic equations

Araújo¹,

Sirakov²

2021

Preprint

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We obtain optimal boundary and global regularity estimates for viscosity solutions of fully nonlinear elliptic equations whose ellipticity degenerates at the critical points of a given solution. We show that any solution is C 1,α on the boundary of the domain, for an optimal and explicit α given only in terms of the regularity of the boundary datum and the elliptic degeneracy degree, no matter how possibly low is the interior regularity for that class of equations. We also obtain sharp global estimates. Our findings are new even for model equations, involving only a degenerate Laplacian; all previous results of global nature give C 1,α regularity only for some small α > 0.

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“…Pointwise regularity has been the subject of various papers, see for example [8] and [14]. These are useful when a certain property is not verified locally but instead only at a point.…”

Section: Introductionmentioning

confidence: 99%