Liouville properties for 𝑝-harmonic maps with finite 𝑞-energy

Chang, Shu-Cheng; Chen, Jui Tang; Wei, Shihshu Walter

doi:10.1090/tran/6351

Cited by 24 publications

(15 citation statements)

References 41 publications

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“…maps to R) that can give is some orientation. It has been proven recently by Chang, Chen, and Wei [CCW,Lemma 5.4] and reads |∇Du| 2 ≥ (1 +κ)|D|Du|| 2 , where herẽ κ := min{ (p−1) 2 m−1 , 1}, and m is the domain dimension. We do not see how the proof could carry over to the S 3 -valued case, but we do get some improvement of Kato's inequality, which is probably not optimal, but for p ց 2 reproduces Okayasu's result.…”

Section: Dimension Reduction For the Singular Setmentioning

confidence: 99%

Regularity Issues for Cosserat Continua and $p$-Harmonic Maps

Gastel¹

2019

SIAM J. Math. Anal.

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For minimizers in a geometrically nonlinear Cosserat model for micropolar elasticity of continua, we prove interior Hölder regularity, up to isolated singular points that may be possible if the exponent p from the model is 2 or in ( 32 15 , 3). The obstacle to full continuity turns out to be the existence of certain minimizing homogeneous p-harmonic maps to S 3 . For those, we slightly improve existing regularity theorems in order to achieve our result on the Cosserat model. MSC 2020. 58E20; 74G40; 74B20.

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Section: Dimension Reduction For the Singular Setmentioning

confidence: 99%

Regularity Issues for Cosserat Continua and $p$-Harmonic Maps

Gastel¹

2019

SIAM J. Math. Anal.

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“…, p > 1, they become p-harmonic maps, or harmonic maps if p = 2) (see Theorem 10.1). In contrast to the work of Chang et al [8] on Liouville properties for a p-harmonic morphism or a p-harmonic function on a manifold that supports a weighted Poincaré inequality, we have Liouville theorems for p-harmonic maps (see Theorem 10.2).…”

Section: )mentioning

confidence: 61%

Dualities in comparison theorems and bundle-valued generalized harmonic forms on noncompact manifolds

Wei

2021

Sci. China Math.

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We observe, utilize dualities in differential equations and differential inequalities (see Theorem 2.1), dualities between comparison theorems in differential equations (see Theorems E and 2.2), and obtain dualities in "swapping" comparison theorems in differential equations. These dualities generate comparison theorems on differential equations of mixed types I and II (see Theorems 2.3 and 2.4) and lead to comparison theorems in Riemannian geometry (see Theorems 2.5 and 2.8) with analytic, geometric, PDE's and physical applications. In particular, we prove Hessian comparison theorems (see Theorems 3.1-3.5) and Laplacian comparison theorems (see Theorems 2.6, 2.7 and 3.1-3.5) under varied radial Ricci curvature, radial curvature, Ricci curvature and sectional curvature assumptions, generalizing and extending the work of Han-Li-Ren-Wei (2014) and Wei (2016).We also extend the notion of function or differential form growth to bundle-valued differential form growth of various types and discuss their interrelationship (see Theorem 5.4). These provide tools in extending the notion, integrability and decomposition of generalized harmonic forms to those of bundle-valued generalized harmonic forms, introducing Condition W for bundle-valued differential forms, and proving the duality theorem and the unity theorem, generalizing the work of Andreotti and Vesentini (1965) and Wei (2020). We then apply Hessian and Laplacian comparison theorems to obtain comparison theorems in mean curvature, generalized sharp Caffarelli-Kohn-Nirenberg type inequalities on Riemannian manifolds, the embedding theorem for weighted Sobolev spaces of functions on manifolds, geometric differential-integral inequalities, generalized sharp Hardy type inequalities on Riemannian manifolds, monotonicity formulas and vanishing theorems for differential forms of degree k with values in vector bundles, such as F -Yang-Mills fields (when F is the identity map, they are Yang-Mills fields), generalized Yang-Mills-Born-Infeld fields on manifolds, Liouville type theorems for Fharmonic maps (when F (t) = 1 p (2t) p 2 , p > 1, they become p-harmonic maps or harmonic maps if p = 2), and Dirichlet problems on starlike domains for vector bundle valued differential 1-forms and F -harmonic maps (see

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“…For the case p = 2 our characterization of the equality cases seems to be new. However, refined Kato's inequalities for p-harmonic functions are provided also in [11].…”

Section: Proof Of Theorem 34mentioning

confidence: 99%

Geometric aspects of p-capacitary potentials

Fogagnolo

Mazzieri

Pinamonti

2019

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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We provide monotonicity formulas for solutions to the p-Laplace equation defined in the exterior of a convex domain. A number of analytic and geometric consequences are derived, including the classical Minkowski inequality as well as new characterizations of rotationally symmetric solutions and domains. The proofs rely on the conformal splitting technique introduced by the second author in collaboration with V. Agostiniani.

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Liouville properties for 𝑝-harmonic maps with finite 𝑞-energy

Cited by 24 publications

References 41 publications

Regularity Issues for Cosserat Continua and $p$-Harmonic Maps

Regularity Issues for Cosserat Continua and $p$-Harmonic Maps

Dualities in comparison theorems and bundle-valued generalized harmonic forms on noncompact manifolds

Geometric aspects of p-capacitary potentials

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