W2,-estimates for fully nonlinear elliptic equations with oblique boundary conditions

Byun, Sun‐Sig; Han, Jeongmin

doi:10.1016/j.jde.2019.09.018

Cited by 10 publications

(5 citation statements)

References 63 publications

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“…Proof of Theorem 2.3. Before proving our main result, we introduce the global 2, -estimates for fully nonlinear elliptic equations with oblique boundary condition when the nonlinearity ( , , , ) satisfies the regular assumptions A1-A3, see [2,Theorem 4.6].…”

Section: -Regularity For Fully Nonlinear Elliptic and Parabolic Equations 3307mentioning

confidence: 99%

“…Next, we introduce a recent result [3] regarding the Calderón-Zygmund estimate for the Hessian of viscosity solutions to fully nonlinear parabolic oblique boundary value problem with nonlinearity ( , , , , ) satisfying the following regular structure conditions:…”

Section: -Regularity For Fully Nonlinear Elliptic and Parabolic Equations 3313mentioning

confidence: 99%

“…As a natural extension of the result in [2,3], we will deal with in this paper the fully nonlinear elliptic and parabolic oblique boundary value problem while the nonlinearity is merely supposed to be asymptotically -regular. In fact, the concept of asymptotically regular was introduced by Chipot and Evans [8].…”

mentioning

confidence: 98%

“…As for the oblique derivative problems, Milakis-Silvestre [20] and Li-Zhang [17] demonstrated that the 1, and 2, regularity were valid for fully nonlinear equations with Neumann boundary condition or the general oblique boundary condition, respectively. Recently, Byun and Han [2,3] further gave a global 2, -regularity for the oblique boundary value problem (1). For more researches on this topic of fully nonlinear problems, see [1,7,9,10,11,14,15,18,19,24,25] and the references therein.…”

mentioning

confidence: 99%

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$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values

Zhang¹,

Zheng²,

Zuo³

2021

DCDS-S

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<p style='text-indent:20px;'>We prove a global <inline-formula><tex-math id="M1">\begin{document}$ W^{2, p} $\end{document}</tex-math></inline-formula>-estimate for the viscosity solution to fully nonlinear elliptic equations <inline-formula><tex-math id="M2">\begin{document}$ F(x, u, Du, D^{2}u) = f(x) $\end{document}</tex-math></inline-formula> with oblique boundary condition in a bounded <inline-formula><tex-math id="M3">\begin{document}$ C^{2, \alpha} $\end{document}</tex-math></inline-formula>-domain for every <inline-formula><tex-math id="M4">\begin{document}$ \alpha\in (0, 1) $\end{document}</tex-math></inline-formula>. Here, the nonlinearities <inline-formula><tex-math id="M5">\begin{document}$ F $\end{document}</tex-math></inline-formula> is assumed to be asymptotically <inline-formula><tex-math id="M6">\begin{document}$ \delta $\end{document}</tex-math></inline-formula>-regular to an operator <inline-formula><tex-math id="M7">\begin{document}$ G $\end{document}</tex-math></inline-formula> that is <inline-formula><tex-math id="M8">\begin{document}$ (\delta, R) $\end{document}</tex-math></inline-formula>-vanishing with respect to <inline-formula><tex-math id="M9">\begin{document}$ x $\end{document}</tex-math></inline-formula>. We employ the approach of constructing a regular problem by an appropriate transformation. With a similar argument, we also obtain a global <inline-formula><tex-math id="M10">\begin{document}$ W^{2, p} $\end{document}</tex-math></inline-formula>-estimate for the viscosity solution to fully nonlinear parabolic equations <inline-formula><tex-math id="M11">\begin{document}$ F(x, t, u, Du, D^{2}u)-u_{t} = f(x, t) $\end{document}</tex-math></inline-formula> with oblique boundary condition in a bounded <inline-formula><tex-math id="M12">\begin{document}$ C^{3} $\end{document}</tex-math></inline-formula>-domain.</p>

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Section: -Regularity For Fully Nonlinear Elliptic and Parabolic Equations 3307mentioning

confidence: 99%

Section: -Regularity For Fully Nonlinear Elliptic and Parabolic Equations 3313mentioning

confidence: 99%

mentioning

confidence: 98%

mentioning

confidence: 99%

See 2 more Smart Citations

$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values

Zhang¹,

Zheng²,

Zuo³

2021

DCDS-S

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“…For oblique boundary value problem for F (D 2 u, x) = f (x), Li and Zhang [9] obtained pointwise C α , C 1,α and C 2,α regularity. Byun and Han [1] derived the boundary W 2,p estimates on flat domains based on [9] and then on C 2,α domains by straightening out boundary. However, there is little literature concerning boundary W 2,p regularity for u ∈ S(λ, Λ, f ).…”

Section: Introductionmentioning

confidence: 99%

${{W}^{2,δ}}$ Estimates for Solution Sets of Fully Nonlinear Elliptic inequalities on ${C^{1,α}}$ Domains

Li¹,

Li²

2022

Preprint

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In this paper, we establish boundary W 2,δ estimates for u ∈ S(λ, Λ, f ) with f ∈ L p for n < p < ∞ and C 1,α boundary values on C 1,α domains, where δ depends on n, λ, Λ, α and p. Caffarelli [2] established interior W 2,δ 0 estimates for u ∈ S(λ, Λ, f ) with f ∈ L n , where δ 0 depends on n, λ and Λ. Instead of straightening out boundary, our main idea is to obtain local boundary W 2,δ estimates from interior W 2,δ 0 estimates by Whitney decomposition.

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On $$W^{2,p}$$-estimates for solutions of obstacle problems for fully nonlinear elliptic equations with oblique boundary conditions

Byun

Han

2022

Calc. Var.

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This paper concerns fully nonlinear elliptic obstacle problems with oblique boundary conditions. We investigate the existence, uniqueness and $$W^{2,p}$$ W 2 , p -regularity results by finding approximate non-obstacle problems with the same oblique boundary condition and then making a suitable limiting process.

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W2,-estimates for fully nonlinear elliptic equations with oblique boundary conditions

Cited by 10 publications

References 63 publications

$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values

$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values

${{W}^{2,δ}}$ Estimates for Solution Sets of Fully Nonlinear Elliptic inequalities on ${C^{1,α}}$ Domains

On $$W^{2,p}$$-estimates for solutions of obstacle problems for fully nonlinear elliptic equations with oblique boundary conditions

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