Asymptotic flatness of Morrey extremals
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Calculus of Variations
Asymptotic flatness of Morrey extremals Ryan Hynd1 · Francis Seuffert1 Received: 23 October 2019 / Accepted: 31 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We study the limiting behavior as |x| → ∞ of extremal functions u for Morrey’s inequality on Rn . In particular, we compute the limit of u(x) as |x| → ∞ and show |x||Du(x)| tends to 0. To this end, we exploit the fact that extremals are uniformly bounded and that they each satisfy a PDE of the form − p u = c(δx0 − δ y0 ) for some c ∈ R and distinct x0 , y0 ∈ Rn . More generally, we explain how to quantitatively deduce the asymptotic flatness of bounded p-harmonic functions on exterior domains of Rn for p > n.
1 Introduction For each n ∈ N and p > n, Morrey’s inequality asserts that there is a constant C > 0 such that 1/ p |u(x) − u(y)| p sup |Du| d x (1.1) ≤C 1−n/ p Rn x= y |x − y| for all continuously differentiable functions u : Rn → R. In particular, it provides control on the 1 − n/ p Hölder seminorm of any function whose first partial derivatives belong to L p (Rn ). In recent work [6], we showed that there is a smallest constant C∗ > 0 for which (1.1) holds and that there are nonconstant functions for which equality holds in (1.1) with C = C∗ . We call any such function an extremal. It turns out that for any nonconstant extremal function u, there is a unique pair of distinct points x0 , y0 ∈ Rn such that |u(x) − u(y)| |u(x0 ) − u(y0 )| sup = . (1.2) 1−n/ p |x − y| |x0 − y0 |1−n/ p x= y Moreover, u satisfies the PDE − p u = c(δx0 − δ y0 )
(1.3)
Communicated by O.Savin. R. Hynd: Partially supported by NSF Grant DMS-1554130.
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Ryan Hynd [email protected] Department of Mathematics, University of Pennsylvania, Philadelphia, USA 0123456789().: V,-vol
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R. Hynd, F. Seuffert
Fig. 1 The graph of a numerically approximated extremal u with n = 2, p = 4, x0 = (0, 1), y0 = (0, −1), u(x0 ) = 1 and u(y0 ) = −1. Note that u(x) ≈ 21 (u(x0 ) + u(y0 )) = 0 for larger values of |x|
in Rn for some nonzero constant c. Here p v := div(|Dv| p−2 Dv) is the p-Laplacian, and Eq. (1.3) is understood to mean |Du| p−2 Du · Dφd x = c(φ(x0 ) − φ(y0 )) Rn
Cc∞ (Rn ).
for each φ ∈ Equation (1.3) can be used to show that each extremal is bounded and has various symmetry properties. In this note, we will make use of these facts to prove the following theorem. We interpret the existence of limit (1.4) below as asserting that extremals are asymptotically flat. This result was also confirmed by numerical computations as observed in Fig. 1. Theorem 1.1 Suppose n ≥ 2 and that p > n. If u is an extremal which satisfies (1.2), then lim u(x) =
|x|→∞
1 (u(x0 ) + u(y0 )) 2
(1.4)
and lim |x||Du(x)| = 0.
|x|→∞
Furthermore,
|x|>r
x 2 |x| p−n |Du| p−2 Du · dx |x| |x|>r
r p−n
|Du| p d x = p
is nonincreasing in r ∈ (s, ∞) for some s > 0 and tends to 0 as r → ∞. In proving Theorem 1.1, we will first verify that any bounded p-harmonic function u on the exterior domain Rn \
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