Mean value properties and unique continuation

Llorente, José G.

doi:10.3934/cpaa.2015.14.185

Cited by 9 publications

(9 citation statements)

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“…The opposite question, whether one need to require mean value property to hold for all radii of balls centered at the given point has also been investigated by several mathematicians, to mention results due to Koebe, Volterra and Kellogg, Hansen and Nadirashvili, and Blaschke, Privaloff and Zaremba. We refer to Section 2 in Llorente [22] for an interesting historical account on the mean value property and harmonicity; also to Heath [14] for further studies on to what extent the restricted mean value property is sufficient for harmonicity in the Euclidean setting. In order to motivate Definition 3.2 below more thoroughly, let us just mention that Koebe, for instance, showed that in order for a continuous function in a domain Ω ⊂ R n to be harmonic it is enough to satisfy the mean value property at every x ∈ Ω with respect to some family of radii r x with inf r x = 0.…”

Section: Harmonic Functionsmentioning

confidence: 99%

Harmonic functions on metric measure spaces

2018

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We introduce and study strongly and weakly harmonic functions on metric measure spaces defined via the mean value property holding for all and, respectively, for some radii of balls at every point of the underlying domain. Among properties of such functions we investigate various types of Harnack estimates on balls and compact sets, weak and strong maximum principles, comparison principles, the Hölder and the Lipshitz estimates and some differentiability properties. The latter one is based on the notion of a weak upper gradient. The Dirichlet problem for functions satisfying the mean value property is studied via the dynamical programming method related to stochastic games. We employ the Perron method to construct a harmonic function with continuous boundary data. Finally, we discuss and prove the Liouville type theorems.Our results are obtained for various types of measures: continuous with respect to a metric, doubling, uniform, measures satisfying the annular decay condition. Relations between such measures are presented as well. The presentation is illustrated by examples.

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Section: Harmonic Functionsmentioning

confidence: 99%

Harmonic functions on metric measure spaces

2018

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“…Moreover, the Harnack inequality and the strong maximum principle hold for strongly harmonic functions as well as the local Hölder continuity and even local Lipschitz continuity under more involved assumptions, see [1]. It is important to mention here that similar type of problems were studied for a more general, nonlinear mean value property by Manfredi-Parvainen-Rossi and Arroyo-Llorente, see [3,4,19,20].…”

Section: Strongly Harmonic Functions On Open Subsets Of R Nmentioning

confidence: 96%

Characterization of mean value harmonic functions on norm induced metric measure spaces with weighted Lebesgue measure

Kijowski

2018

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We study the mean-value harmonic functions on open subsets of R n equipped with weighted Lebesgue measures and norm induced metrics. Our main result is a necessary condition saying that all such functions solve a certain homogeneous system of elliptic PDEs. Moreover, a converse result is established in case of analytic weights. Assuming Sobolev regularity of weight w ∈ W l,∞ we show that strongly harmonic functions are as well in W l,∞ and that they are analytic, whenever the weight is analytic.The analysis is illustrated by finding all mean-value harmonic functions in R 2 for the l p -distance 1 ≤ p ≤ ∞. The essential outcome is a certain discontinuity with respect to p, i.e. that for all p = 2 there are only finitely many linearly independent mean-value harmonic functions, while for p = 2 there are infinitely many of them. We conclude with a remarkable observation that strongly harmonic functions in R n possess the mean value property with respect to infinitely many weight functions obtained from a given weight.

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“…If S is the operator given by (1.2), associated to some admissible radius function in Ω, then functions satisfying Su = u are called harmonious functions. The functional equation Su = u appears in different contexts, related to the problem of extending a continous function on a closed subset to the whole space respecting its modulus of continuity( [9]), as a Dynamic Programming Principle in tug-of-war games ( [15], [16]), as a mean value property related to the infinity laplacian ( [11], [8]) and also in connection with problems of image processing ( [3]).…”

Section: Introductionmentioning

confidence: 99%

“…is the so called infinity laplacian of u (see [2], [8]). Another important differential operator is the p-laplacian:…”

Section: Introductionmentioning

confidence: 99%