Boundary regularity for solutions to the linearized Monge-Amp\`ere equations

Nam Le; Ovidiu Savin

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Boundary regularity for solutions to the linearized Monge-Amp\`ere equations

Nam Le, Ovidiu Savin

Analysis of PDEs · Submitted 26 Sep 2011

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The authors' main claim We obtain boundary Holder gradient estimates and regularity for solutions to the linearized Monge-Ampere equations under natural assumptions on the domain, Monge-Ampere measures and boundary data.

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We obtain boundary Holder gradient estimates and regularity for solutions to the linearized Monge-Ampere equations under natural assumptions on the domain, Monge-Ampere measures and boundary data. Our results are affine invariant analogues of the boundary Holder gradient estimates of Krylov.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We obtain boundary Holder gradient estimates and regularity for solutions to the linearized Monge-Ampere equations under natural assumptions on the domain, Monge-Ampere measures and boundary data.	result 0.69	0.96	0.93	asserted 1.00
s1	Our results are affine invariant analogues of the boundary Holder gradient estimates of Krylov.	interpretation 0.93	0.69	0.09	asserted 1.00

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Original abstract

We obtain boundary Holder gradient estimates and regularity for solutions to the linearized Monge-Ampere equations under natural assumptions on the domain, Monge-Ampere measures and boundary data. Our results are affine invariant analogues of the boundary Holder gradient estimates of Krylov.

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2 arXiv papers cite this in 12 sentences

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We handle the degeneracy of $ℒ_ϕ$ by working as in [ref] with sections of solutions to the Monge-Ampère equations.

Global $W^{2,p}$ estimates for solutions to the linearized Monge--Amp\`ere equations · Introduction and Statement of the Main Results · 1.00

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Uses the method or tool of 2 papers

Our main tools are a localization theorem at the boundary for solutions to the Monge-Ampère equation which was obtained in [ref] , and the interior Harnack estimates for solutions to the linearized Monge-Ampère equations which were established in [ref] (see Section [ref]).

A localization property at the boundary for the Monge-Ampere equation · Statement of the main results · 1.00
To overcome the singularity of $L_u$ near the boundary, we use a Localization Theorem at the boundary for solutions to the Monge-Ampère equations which was obtained in [ref] .

Pointwise $C^{2,\alpha}$ estimates at the boundary for the Monge-Ampere equation · Introduction · 1.00

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