Page 546 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
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TOPICS IN SUB-ELLIPTIC AND ELLIPTIC PDES (MS-31)
Exponentially Subelliptic Harmonic Maps
Yuan Jen Chiang, ychiang@umw.edu
University of Mary Washington, Fredericksburg, VA, United States
Coauthor: Sorin Dragomir
Exponentially harmonic maps were first introduced by Eells and Lemaire [5] in 1990. Exponen-
tial wave maps are exponentially harmonic maps on Minkowski spaces, which were first studied
by Chiang and Yang [1, 4] since 2007. Firstly, we deal with the critical points of maps φ : Hn →
Sm from the Heiserberg group into a sphere with energy E1(φ) = Ω exp( 1 ||∇H φ||2θ )θ ∧ (dθ)n
2
for domains Ω ⊂⊂ Hn and a contact structure θ on Hn. They are solutions to the 2nd order
quasi-linear subelliptic PDE system
− bφj + 2eb(φ)φj + Gθ(∇H eb(φ), ∇H φj) = 0, 1 ≤ j ≤ m + 1,
and arise through Fefferman’s construction, i.e. as base maps φ : Hn → Sm associated to S1
invariant exponential wave maps Φ : C(Hn) → Sm from the total space of the canonical circle
bundle S1 → C(Hn) → Sm endowed with the Fefferman’s metric Fθ. We establish Caccioppoli
type estimates
exp Q ||∇H φ||2θ ||∇Hu||Qθ θ ∧ (dθ)n ≤ Crβ (0 < β < 1)
2
Br(x)
with Q = 2n + 2 (the homogeneous dimension of Hn), and show that any weak solution
φ ∈ p≥Q WH1,p(Ω, Sn) of finite p-energy Ep(φ) < ∞ for some p ≥ 2Q is locally Ho¨lder
continuous, i.e. φj ∈ Sl0o,cα(Ω) (Ho¨lder like spaces) for 0 < α ≤ 1, built in terms of the Carnot-
Carathe´odory metric ρθ. The main theorems and results are based on [2]. Secondly, we study
exponentially subelliptic harmonic (e.s.h.) maps from a compact pseudo hermitian manifold
(M, θ) into a Riemannian manifold (N, h), i.e. C2 solutions of φ : M → N to nonlinear PDE
system τb(φ) + φ∗∇Heb(φ) = 0 which are the Euler-Legrange equation of δEb(φ) = 0 with
Eb(φ) = M exp(eb(φ))θ ∧ dθn, where e.s.h. maps arise in a similar way as the first setting.
We study the second variation formula and stability of exponentially subelliptic harmonic maps
based on [3].
References
[1] Y. J. Chiang, Developments of harmonic maps, wave maps and Yang-Mills fields into bi-
harmonic maps, biwave maps and bi-Yang-Mills fields, Frontiers in Math, Birkha¨user,
Springer, Basel, xxi+399, 2013, Chapters 7 & 8 : Exponentially harmonic maps & Expo-
nential wave maps.
[2] Y. J. Chiang, S. Dragomir and F. Esposito, Exponentially subelliptic harmonic maps from
the Heisengberg group into a sphere, Cal. of Variations and PDEs, 1–45, online July 2019.
[3] Y. J. Chiang, S. Dragomir and F. Esposito, Second variation formula and stability of ex-
ponentially subelliptic harmonic maps, preprint.
[4] Y. J. Chiang and Y. H. Yang, Exponential wave maps, J. of Geom. Phys. 57 (12) (2007),
2521–2532.
[5] J. Eells and L. Lemaire, Some properties of exponentially harmonic maps, in: Partial
Differential Equations, Part 1, 2 (Warsaw, 1990), 129–136, Banach Center Publ. 27, Polish
Acad. Sci., Warsaw, 1992.
544
Exponentially Subelliptic Harmonic Maps
Yuan Jen Chiang, ychiang@umw.edu
University of Mary Washington, Fredericksburg, VA, United States
Coauthor: Sorin Dragomir
Exponentially harmonic maps were first introduced by Eells and Lemaire [5] in 1990. Exponen-
tial wave maps are exponentially harmonic maps on Minkowski spaces, which were first studied
by Chiang and Yang [1, 4] since 2007. Firstly, we deal with the critical points of maps φ : Hn →
Sm from the Heiserberg group into a sphere with energy E1(φ) = Ω exp( 1 ||∇H φ||2θ )θ ∧ (dθ)n
2
for domains Ω ⊂⊂ Hn and a contact structure θ on Hn. They are solutions to the 2nd order
quasi-linear subelliptic PDE system
− bφj + 2eb(φ)φj + Gθ(∇H eb(φ), ∇H φj) = 0, 1 ≤ j ≤ m + 1,
and arise through Fefferman’s construction, i.e. as base maps φ : Hn → Sm associated to S1
invariant exponential wave maps Φ : C(Hn) → Sm from the total space of the canonical circle
bundle S1 → C(Hn) → Sm endowed with the Fefferman’s metric Fθ. We establish Caccioppoli
type estimates
exp Q ||∇H φ||2θ ||∇Hu||Qθ θ ∧ (dθ)n ≤ Crβ (0 < β < 1)
2
Br(x)
with Q = 2n + 2 (the homogeneous dimension of Hn), and show that any weak solution
φ ∈ p≥Q WH1,p(Ω, Sn) of finite p-energy Ep(φ) < ∞ for some p ≥ 2Q is locally Ho¨lder
continuous, i.e. φj ∈ Sl0o,cα(Ω) (Ho¨lder like spaces) for 0 < α ≤ 1, built in terms of the Carnot-
Carathe´odory metric ρθ. The main theorems and results are based on [2]. Secondly, we study
exponentially subelliptic harmonic (e.s.h.) maps from a compact pseudo hermitian manifold
(M, θ) into a Riemannian manifold (N, h), i.e. C2 solutions of φ : M → N to nonlinear PDE
system τb(φ) + φ∗∇Heb(φ) = 0 which are the Euler-Legrange equation of δEb(φ) = 0 with
Eb(φ) = M exp(eb(φ))θ ∧ dθn, where e.s.h. maps arise in a similar way as the first setting.
We study the second variation formula and stability of exponentially subelliptic harmonic maps
based on [3].
References
[1] Y. J. Chiang, Developments of harmonic maps, wave maps and Yang-Mills fields into bi-
harmonic maps, biwave maps and bi-Yang-Mills fields, Frontiers in Math, Birkha¨user,
Springer, Basel, xxi+399, 2013, Chapters 7 & 8 : Exponentially harmonic maps & Expo-
nential wave maps.
[2] Y. J. Chiang, S. Dragomir and F. Esposito, Exponentially subelliptic harmonic maps from
the Heisengberg group into a sphere, Cal. of Variations and PDEs, 1–45, online July 2019.
[3] Y. J. Chiang, S. Dragomir and F. Esposito, Second variation formula and stability of ex-
ponentially subelliptic harmonic maps, preprint.
[4] Y. J. Chiang and Y. H. Yang, Exponential wave maps, J. of Geom. Phys. 57 (12) (2007),
2521–2532.
[5] J. Eells and L. Lemaire, Some properties of exponentially harmonic maps, in: Partial
Differential Equations, Part 1, 2 (Warsaw, 1990), 129–136, Banach Center Publ. 27, Polish
Acad. Sci., Warsaw, 1992.
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