A weak Kellogg property for quasiminimizers

Björn, Anders

doi:10.4171/cmh/75

Cited by 39 publications

(64 citation statements)

References 18 publications

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“…When X is complete this is equivalent to our definition, see A. Björn [2]. (Moreover, this definition was used by Ziemer [43] in R n , but he was no doubt aware of the equivalence in this case.)…”

Section: Quasi(super)minimizersmentioning

confidence: 91%

Moser iteration for (quasi)minimizers on metric spaces

Björn

Marola²

2006

manuscripta math.

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Abstract. We study regularity properties of quasiminimizers of the p-Dirichlet integral on metric measure spaces. We adapt the Moser iteration technique to this setting and show that it can be applied without an underlying differential equation. However, we have been able to run the Moser iteration fully only for minimizers. We prove Caccioppoli inequalities and local boundedness properties for quasisub-and quasisuperminimizers. This is done in metric spaces equipped with a doubling measure and supporting a weak (1, p)-Poincaré inequality. The metric space is not required to be complete.We also provide an example which shows that the dilation constant from the weak Poincaré inequality is essential in the condition on the balls in the Harnack inequality. This fact seems to have been overlooked in the earlier literature on nonlinear potential theory on metric spaces.

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Section: Quasi(super)minimizersmentioning

confidence: 91%

Moser iteration for (quasi)minimizers on metric spaces

Björn

Marola²

2006

manuscripta math.

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“…Furthermore, u is said to be a minimizer in V if (3.6) holds for all ϕ ∈ N 1,p 0 (V ). According to Proposition 3.2 in A. Björn [1], it is in fact only necessary to test (3.6) with (all nonnegative and all, respectively) ϕ ∈ Lip c (V ).…”

Section: Daniel Hansevimentioning

confidence: 99%

“…There are many equivalent definitions of (super)minimizers in the literature (see Proposition 3.2 in A. Björn [1]). The first definition for metric spaces was given by Kinnunen-Martio [25].…”

Section: Daniel Hansevimentioning

confidence: 99%

The obstacle and Dirichlet problems associated with p-harmonic functions in unbounded sets in R^n and metric spaces

Hansevi¹

2015

Ann. Acad. Sci. Fenn. Math.

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Abstract. The obstacle problem associated with p-harmonic functions is extended to unbounded open sets, whose complement has positive capacity, in the setting of a proper metric measure space supporting a (p, p)-Poincaré inequality, 1 < p < ∞, and the existence of a unique solution is proved. Furthermore, if the measure is doubling, then it is shown that a continuous obstacle implies that the solution is continuous, and moreover p-harmonic in the set where it does not touch the obstacle. This includes, as a special case, the solution of the Dirichlet problem for p-harmonic functions with Sobolev type boundary data.

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“…By Proposition 3.2 in A. Björn [3] it is enough to test (3.1) with (all and nonnegative, respectively) ϕ ∈ Lip c (Ω).…”

Section: Minimizers and Superharmonic Functionsmentioning

confidence: 99%