An approximation result for solutions of Hessian equations

John Urbas

doi:10.1007/s00526-006-0064-7

An approximation result for solutions of Hessian equations

John Urbas^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

We show that W ^2,p weak solutions of the k-Hessian equation F _k (D ² u) = g(x) with k ≥ 2 can be approximated by smooth k-convex solutions v _j of similar equations with the right hands sides controlled uniformly in C ^0,1 norm, and so that the quantities ∫_Br(Δv_j)^{p-k+1}F _k-1(D²v_j) are bounded independently of j. This result simplifies the proof of previous interior regularity results for solutions of such equations. It also permits us to extend certain estimates for smooth solutions of degenerate two dimensional Monge-Ampère equations to W ^2,p solutions.

Original language	English
Pages (from-to)	219-230
Number of pages	12
Journal	Calculus of Variations and Partial Differential Equations
Volume	29
Issue number	2
DOIs	https://doi.org/10.1007/s00526-006-0064-7
Publication status	Published - Jun 2007

Access to Document

10.1007/s00526-006-0064-7

Cite this

@article{a80ac40d8a8841afb66772df25338697,

title = "An approximation result for solutions of Hessian equations",

abstract = "We show that W 2,p weak solutions of the k-Hessian equation F k (D 2 u) = g(x) with k ≥ 2 can be approximated by smooth k-convex solutions v j of similar equations with the right hands sides controlled uniformly in C 0,1 norm, and so that the quantities ∫Br(Δvj){p-k+1}F k-1(D2vj) are bounded independently of j. This result simplifies the proof of previous interior regularity results for solutions of such equations. It also permits us to extend certain estimates for smooth solutions of degenerate two dimensional Monge-Amp{\`e}re equations to W 2,p solutions.",

author = "John Urbas",

year = "2007",

month = jun,

doi = "10.1007/s00526-006-0064-7",

language = "English",

volume = "29",

pages = "219--230",

journal = "Calculus of Variations and Partial Differential Equations",

issn = "0944-2669",

publisher = "Springer New York",

number = "2",

}

TY - JOUR

T1 - An approximation result for solutions of Hessian equations

AU - Urbas, John

PY - 2007/6

Y1 - 2007/6

N2 - We show that W 2,p weak solutions of the k-Hessian equation F k (D 2 u) = g(x) with k ≥ 2 can be approximated by smooth k-convex solutions v j of similar equations with the right hands sides controlled uniformly in C 0,1 norm, and so that the quantities ∫Br(Δvj){p-k+1}F k-1(D2vj) are bounded independently of j. This result simplifies the proof of previous interior regularity results for solutions of such equations. It also permits us to extend certain estimates for smooth solutions of degenerate two dimensional Monge-Ampère equations to W 2,p solutions.

AB - We show that W 2,p weak solutions of the k-Hessian equation F k (D 2 u) = g(x) with k ≥ 2 can be approximated by smooth k-convex solutions v j of similar equations with the right hands sides controlled uniformly in C 0,1 norm, and so that the quantities ∫Br(Δvj){p-k+1}F k-1(D2vj) are bounded independently of j. This result simplifies the proof of previous interior regularity results for solutions of such equations. It also permits us to extend certain estimates for smooth solutions of degenerate two dimensional Monge-Ampère equations to W 2,p solutions.

UR - http://www.scopus.com/inward/record.url?scp=33947682120&partnerID=8YFLogxK

U2 - 10.1007/s00526-006-0064-7

DO - 10.1007/s00526-006-0064-7

M3 - Article

SN - 0944-2669

VL - 29

SP - 219

EP - 230

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

IS - 2

ER -

An approximation result for solutions of Hessian equations

Abstract

Access to Document

Other files and links

Fingerprint

Cite this