Stability of the mean value formula for harmonic functions in Lebesgue spaces
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Stability of the mean value formula for harmonic functions in Lebesgue spaces Giovanni Cupini1 · Ermanno Lanconelli1 Received: 30 April 2020 / Accepted: 17 August 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Let D be an open subset of ℝn with finite measure, and let x0 ∈ D . We introduce the p-Gauss gap of D w.r.t. x0 to measure how far are the averages over D of the harmonic functions u ∈ Lp (D) from u(x0 ) . We estimate from below this gap in terms of the ball gap of D w.r.t. x0 , i.e., the normalized Lebesgue measure of D ⧵ B , being B the biggest ball centered at x0 contained in D. From these stability estimates of the mean value formula for harmonic functions in Lp-spaces, we straightforwardly obtain rigidity properties of the Euclidean balls. We also prove a continuity result of the p-Gauss gap in the Sobolev space ′ W 1,p , where p′ is the conjugate exponent of p. Keywords Gauss mean value formula · Stability · Harmonic functions · Rigidity · Lebesgue spaces Mathematics Subject Classification Primary: 35B05 · Secondary: 31B05
1 Introduction Let D be an open subset of ℝn , n ≥ 2 , and let B(x0 , r) be the open Euclidean ball with center x0 and radius r > 0 . If B(x0 , r) ⊆ D , by the Gauss mean value Theorem,
u(x0 ) =
1 u(y) dy, |B(x0 , r)| ∫B(x0 ,r)
∀u ∈ H(D),
where H(D) denotes the linear space of the harmonic functions in D and |B(x0 , r)| stands for the Lebesgue measure of B(x0 , r) . By the dominated convergence theorem, it follows that
* Giovanni Cupini [email protected] Ermanno Lanconelli [email protected] 1
Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy
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G. Cupini, E. Lanconelli
u(x0 ) =
1 u(y) dy, |B(x0 , r)| ∫B(x0 ,r)
∀u ∈ H(B(x0 , r)) ∩ L1 (B(x0 , r)).
(1.1)
In the literature, the stability and the rigidity properties of (1.1) have been studied. The question of rigidity, a sort of an inverse problem for (1.1), is the oldest one treated in the literature and is related to the following question: if D is an open set with finite Lebesgue measure containing x0 , such that the mean integral of harmonic functions in L1 on D equals the value of these functions at x0 , then is D a ball centered at x0? The historical development of this problem, together with a comprehensive collection of results, is contained in the excellent survey [21] by Netuka and Veselý. Here, we only quote a theorem by Kuran, who definitely gave a positive answer to the previous question, providing a short and elegant proof, see [18]. In his paper, Kuran introduced a harmonic test function (see (1.5)) which will play a crucial role in the present paper. We also remark that a domain satisfying the mean value property for any harmonic functions is the simplest instance of a so-called quadrature domain. Quadrature domains have been extensively studied ever since the 1960’s. A good source of reference is the survey article [17]. The oth
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