Regularity of minimizers of the area functional in metric spaces

Hakkarainen, Heikki; Kinnunen, Juha; Lahti, Panu

doi:10.1515/acv-2013-0022

Cited by 10 publications

(29 citation statements)

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“…[13,42,47], and in the metric setting in [28]. In this paper we first prove standard De Giorgi-type and weak Harnack inequalities for 1subminimizers and certain solutions of obstacle problems, following especially [2,19,25], and then use these to show the aforementioned semicontinuity property of 1-superminimizers as well as a similar property for solutions of obstacle problems at points where the obstacle is continuous, see Theorem 3.18.…”

Section: Introductionmentioning

confidence: 99%

Superminimizers and a weak Cartan property for p = 1 in metric spaces

Lahti

2020

JAMA

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We study functions of least gradient as well as related superminimizers and solutions of obstacle problems in metric spaces that are equipped with a doubling measure and support a Poincaré inequality. We show a standard weak Harnack inequality and use it to prove semicontinuity properties of such functions. We also study some properties of the fine topology in the case p = 1. Then we combine these theories to prove a weak Cartan property of superminimizers in the case p = 1, as well as a strong version at points of nonzero capacity. Finally we employ the weak Cartan property to show that any topology that makes the upper representative u ∨ of every 1-superminimizer u upper semicontinuous in open sets is stronger (in some cases, strictly) than the 1-fine topology. * 2010 Mathematics Subject Classification: 30L99, 31E05, 26B30.

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Section: Introductionmentioning

confidence: 99%

Superminimizers and a weak Cartan property for p = 1 in metric spaces

Lahti

2020

JAMA

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“…For f (t) = t, this gives the de nition of functions of bounded variation, or BV functions, on metric measure spaces, see [1], [3] and [24]. For f (t) = √ + t , we get the generalized surface area functional, which has been considered previously in [17] and [18]. Our rst result shows that if F(u, Ω) < ∞, then F(u, ·) is a Borel regular outer measure on Ω.…”

Section: Introductionmentioning

confidence: 99%

Relaxation and Integral Representation for Functionals of Linear Growth on Metric Measure spaces

Hakkarainen

Kinnunen²,

Lahti³

et al. 2016

Analysis and Geometry in Metric Spaces

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Abstract:This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincaré inequality. Such a functional is de ned via relaxation, and it de nes a Radon measure on the space. For the singular part of the functional, we get the expected integral representation with respect to the variation measure. A new feature is that in the representation for the absolutely continuous part, a constant appears already in the weighted Euclidean case. As an application we show that in a variational minimization problem involving the functional, boundary values can be presented as a penalty term.

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“…Let us also note that the proofs for fine continuity given in [8] and [23] involve the theory of p-harmonic functions, for p > 1. While some results on 1-harmonic functions, known as functions of least gradient, have been derived in [14,15,19], we do not use this theory, relying on a geometric tool known as the boxing inequality instead.…”

Section: Introductionmentioning

confidence: 99%