A regularity result for critical points of conformally invariant functionals

Philippe Choné

doi:10.1007/BF01071697

A regularity result for critical points of conformally invariant functionals

Philippe Choné

Université Paris-Saclay

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

Abstract

Let Ω be an open set in R² and I be a conformally invariant functional defined on H¹(Ω, R^d). Let u be a critical point of I. We show that, if u is a priori assumed to be bounded, then u is smooth in Ω, up to ∂Ω (if u_|δΩ is smooth). This is a partial (positive) answer to a conjecture of S. Hildebrandt [13]. As an application, we establish a regularity result for weak solutions to the equation of surfaces of prescribed mean curvature in a three-dimensional compact riemannian manifold.

Original language	English
Pages (from-to)	269-296
Number of pages	28
Journal	Potential Analysis
Volume	4
Issue number	3
DOIs	https://doi.org/10.1007/BF01071697
Publication status	Published - 1 Jun 1995
Externally published	Yes

Keywords

Conformally invariant functional
Lorentz spaces
Mathematics Subject Classifications (1991): 35B65, 53A10, 58E12
compensation phenomena
jacobians
mobile frame
prescribed mean curvature

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10.1007/BF01071697

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title = "A regularity result for critical points of conformally invariant functionals",

abstract = "Let Ω be an open set in R2 and I be a conformally invariant functional defined on H1(Ω, Rd). Let u be a critical point of I. We show that, if u is a priori assumed to be bounded, then u is smooth in Ω, up to ∂Ω (if u|δΩ is smooth). This is a partial (positive) answer to a conjecture of S. Hildebrandt [13]. As an application, we establish a regularity result for weak solutions to the equation of surfaces of prescribed mean curvature in a three-dimensional compact riemannian manifold.",

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N2 - Let Ω be an open set in R2 and I be a conformally invariant functional defined on H1(Ω, Rd). Let u be a critical point of I. We show that, if u is a priori assumed to be bounded, then u is smooth in Ω, up to ∂Ω (if u|δΩ is smooth). This is a partial (positive) answer to a conjecture of S. Hildebrandt [13]. As an application, we establish a regularity result for weak solutions to the equation of surfaces of prescribed mean curvature in a three-dimensional compact riemannian manifold.

AB - Let Ω be an open set in R2 and I be a conformally invariant functional defined on H1(Ω, Rd). Let u be a critical point of I. We show that, if u is a priori assumed to be bounded, then u is smooth in Ω, up to ∂Ω (if u|δΩ is smooth). This is a partial (positive) answer to a conjecture of S. Hildebrandt [13]. As an application, we establish a regularity result for weak solutions to the equation of surfaces of prescribed mean curvature in a three-dimensional compact riemannian manifold.

KW - Conformally invariant functional

KW - Lorentz spaces

KW - Mathematics Subject Classifications (1991): 35B65, 53A10, 58E12

KW - compensation phenomena

KW - jacobians

KW - mobile frame

KW - prescribed mean curvature

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SN - 0926-2601

VL - 4

SP - 269

EP - 296

JO - Potential Analysis

JF - Potential Analysis

IS - 3

ER -

A regularity result for critical points of conformally invariant functionals

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Keywords

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