A nonlinear fabes—stroock result

Fok, Ken

doi:10.1080/03605309808821375

Cited by 29 publications

(19 citation statements)

References 10 publications

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“…It is known (see [3,10,11,13]) that there exists p 0 = p 0 (n, /λ) satisfying n/2 ≤ p 0 < n such that for p > p 0 there is a constant C = C(n, p, λ, ) such that if f ∈ L p ( ) and u ∈ C( ) ∩ W 2, p loc ( ) is an L p -strong subsolution of…”

Section: Remark 24mentioning

confidence: 99%

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Maximum principle for fully nonlinear equations via the iterated comparison function method

Koike

Święch

2007

Math. Ann.

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We present various versions of generalized Aleksandrov-Bakelman-Pucci (ABP) maximum principle for L p -viscosity solutions of fully nonlinear second-order elliptic and parabolic equations with possibly superlinear-growth gradient terms and unbounded coefficients. We derive the results via the "iterated" comparison function method, which was introduced in our previous paper (Koike andŚwie ch in Nonlin.

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Section: Remark 24mentioning

confidence: 99%

“…We will show several results. For the elliptic PDE (1.1) we will first consider the case when m = 1 and we will extend the result of [12] (see also [13]). When m > 1, we will show that the maximum principle holds provided that µ or f is small enough in a certain norm.…”

Section: Introductionmentioning

confidence: 99%

Maximum principle for fully nonlinear equations via the iterated comparison function method

Koike

Święch

2007

Math. Ann.

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“…In fact, the proof in [11] is a modification of that in [4]. We also refer to Remark 1.4 in [12] (see also [13,18] for maximum principle estimates for linear equations with unbounded gradient coefficients).…”

Section: Shigeaki Koike and Andrzejświȩch Nodeamentioning

confidence: 99%

“…where p 0 = p 0 (n, Λ λ ) is a constant for which the generalized maximum principle holds [10,9,12,8]. More precisely there exists a constant p 0 (n,…”

Section: Introductionmentioning

confidence: 99%

Maximum principle and existence of Lp-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms

Koike

Święch

2004

Nonlinear differ. equ. appl.

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We study L p -viscosity solutions of fully nonlinear, second-order, uniformly elliptic partial differential equations (PDE) with measurable terms and quadratic nonlinearity. We present a sufficient condition under which the maximum principle holds for L p -viscosity solution. We also prove stability and existence results for the equations under consideration.2000 Mathematics Subject Classification: 49L25, 35J60, 35D10.

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“…We will always assume that p > p 0 , where p 0 = p 0 (n, Λ λ ) ∈ [n/2, n) is the constant giving the range where the maximum principle holds (see e.g. [10,11,12,14]). The constants 0 < λ ≤ Λ are fixed ellipticity constants of F .…”

mentioning

confidence: 99%

Pointwise properties of <inline-formula><tex-math id="M1">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms

Święch¹

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We prove various pointwise properties of L p-viscosity solutions of fully nonlinear uniformly elliptic equations with unbounded measurable terms and possibly quadratically growing gradient terms. These include differentiability properties and "pointwise maximum principle". In particular we discuss an equivalent pointwise definition of L p-viscosity solution which, together with pointwise properties, allows to operate on L p-viscosity solutions almost as if they were strong solutions and move them freely from one equation to another. Such results were proved before in [6, 8, 30] for equations with linearly growing gradient terms, however it appears that they have not been widely used. Here we generalize pointwise properties of L p-viscosity solutions to a larger class of equations and show that they create a very powerful set of tools which can be used for instance to prove regularity results for equations and differential inequalities.

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A nonlinear fabes—stroock result

Cited by 29 publications

References 10 publications

Maximum principle for fully nonlinear equations via the iterated comparison function method

Maximum principle for fully nonlinear equations via the iterated comparison function method

Maximum principle and existence of Lp-viscosity solutions for fully nonlinear uniformly elliptic equations with measurable and quadratic terms

Pointwise properties of <inline-formula><tex-math id="M1">\begin{document}$ L^p $\end{document}</tex-math></inline-formula>-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms

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