TY - JOUR
T1 - On the mean value property of fractional harmonic functions
AU - Bucur, Claudia
AU - Dipierro, Serena
AU - Valdinoci, Enrico
PY - 2020/12
Y1 - 2020/12
N2 - As is well known, harmonic functions satisfy the mean value property, i.e. the average of such a function over a ball is equal to its value at the center. This fact naturally raises the question on whether this is a feature characterizing only balls, namely, is a set, for which all harmonic functions satisfy the mean value property, necessarily a ball? This question was investigated by several authors, including Bernard Epstein (1962), Bernard Epstein and Schiffer (1965), Myron Goldstein and Wellington (1971), who obtained a positive answer to this question under suitable additional assumptions. The problem was finally elegantly, completely and positively settled by Ülkü Kuran (1972), with an artful use of elementary techniques. This classical problem has been recently fleshed out by Giovanni Cupini, et al. (in press) who proved a quantitative stability result for the mean value formula, showing that a suitable “mean value gap” (measuring the normalized difference between the average of harmonic functions on a given set and their pointwise value) is bounded from below by the Lebesgue measure of the “gap” between the set and the ball (and, consequently, by the Fraenkel asymmetry of the set). That is, if a domain “almost” satisfies the mean value property for all harmonic functions, then that domain is “almost” a ball. The goal of this note is to investigate some nonlocal counterparts of these results. Some of our arguments rely on fractional potential theory, others on purely nonlocal properties, with no classical counterpart, such as the fact that “all functions are locally fractional harmonic up to a small error”.
AB - As is well known, harmonic functions satisfy the mean value property, i.e. the average of such a function over a ball is equal to its value at the center. This fact naturally raises the question on whether this is a feature characterizing only balls, namely, is a set, for which all harmonic functions satisfy the mean value property, necessarily a ball? This question was investigated by several authors, including Bernard Epstein (1962), Bernard Epstein and Schiffer (1965), Myron Goldstein and Wellington (1971), who obtained a positive answer to this question under suitable additional assumptions. The problem was finally elegantly, completely and positively settled by Ülkü Kuran (1972), with an artful use of elementary techniques. This classical problem has been recently fleshed out by Giovanni Cupini, et al. (in press) who proved a quantitative stability result for the mean value formula, showing that a suitable “mean value gap” (measuring the normalized difference between the average of harmonic functions on a given set and their pointwise value) is bounded from below by the Lebesgue measure of the “gap” between the set and the ball (and, consequently, by the Fraenkel asymmetry of the set). That is, if a domain “almost” satisfies the mean value property for all harmonic functions, then that domain is “almost” a ball. The goal of this note is to investigate some nonlocal counterparts of these results. Some of our arguments rely on fractional potential theory, others on purely nonlocal properties, with no classical counterpart, such as the fact that “all functions are locally fractional harmonic up to a small error”.
KW - Classification results
KW - Fractional harmonic functions
KW - Inverse problems
KW - Mean value formulas
UR - http://www.scopus.com/inward/record.url?scp=85090064486&partnerID=8YFLogxK
U2 - 10.1016/j.na.2020.112112
DO - 10.1016/j.na.2020.112112
M3 - Article
AN - SCOPUS:85090064486
VL - 201
JO - Nonlinear Analysis: Theory Methods & Applications
JF - Nonlinear Analysis: Theory Methods & Applications
SN - 0362-546X
M1 - 112112
ER -