A priori estimates for harmonic mappings

Analysis ◽  
2007 ◽  
Vol 27 (4) ◽  
Author(s):  
Michael Pingen
Keyword(s):  
Riemannian Manifolds ◽  
A Priori Estimates ◽  
Hölder Continuity ◽  
A Priori ◽  
Regularity Result ◽  
Harmonic Mappings ◽  
Holder Continuity ◽  
Priori Estimates

SummaryWe give a new proof of a well known regularity result for harmonic mappings between Riemannian manifolds due to Giaquinta and Hildebrandt [3]. The proof uses a modification of a method due to L. Caffarelli [2] to show interior and boundary Hölder-continuity of harmonic mappings, whose images lie in a regular ball. In addition a priori estimates are established. We remark here that our proof completely avoids the use of Green´s functions.

Interior estimates of harmonic heat flow

2021 ◽  
pp. 2150039
Author(s):  
Xiangao Liu ◽  
Zixuan Liu ◽  
Kui Wang
Keyword(s):  
Riemannian Manifolds ◽  
A Priori Estimates ◽  
A Priori ◽  
Regularity Result ◽  
Harmonic Mappings ◽  
Geodesic Ball ◽  
Priori Estimates ◽  
Schauder Estimate ◽  
Harmonic Flow ◽  
Harmonic Heat Flow

Motivated by Giaquinta and Hildebrandt’s regularity result for harmonic mappings [M. Giaquinta and S. Hildebrandt, A priori estimates for harmonic mappings, J. Reine Angew. Math. 1982(336) (1982) 124–164, Theorems 3 and 4], we show a [Formula: see text]-regularity result of the harmonic flow between two Riemannian manifolds when the image is in a regular geodesic ball. The proof is based on De Giorgi–Moser’s iteration and Schauder estimate.

scholarly journals On Weyl’s Embedding Problem in Riemannian Manifolds

2018 ◽  
Vol 2020 (11) ◽  
pp. 3229-3259 ◽  
Author(s):  
Siyuan Lu
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Space Form ◽  
A Priori Estimates ◽  
A Priori ◽  
Three Dimensional ◽  
Embedding Theorem ◽  
Embedding Problem ◽  
Priori Estimates ◽  
Geometric Assumption

Abstract We consider a priori estimates of Weyl’s embedding problem of $(\mathbb{S}^2, g)$ in general three-dimensional Riemannian manifold $(N^3, \bar g)$. We establish interior $C^2$ estimate under natural geometric assumption. Together with a recent work by Li and Wang [18], we obtain an isometric embedding of $(\mathbb{S}^2,g)$ in Riemannian manifold. In addition, we reprove Weyl’s embedding theorem in space form under the condition that $g\in C^2$ with $D^2g$ Dini continuous.

EXISTENCE OF SOLUTIONS TO A THERMO-VISCOELASTIC CONTACT PROBLEM WITH COULOMB FRICTION

2002 ◽  
Vol 12 (10) ◽  
pp. 1491-1511 ◽  
Author(s):  
C. ECK
Keyword(s):  
Contact Problem ◽  
Coulomb Friction ◽  
A Priori Estimates ◽  
A Priori ◽  
Regularity Result ◽  
Existence Of A Solution ◽  
Moderate Growth ◽  
Priori Estimates ◽  
The Galerkin Method ◽  
Viscoelastic Contact

A thermo-viscoelastic contact problem with Coulomb friction is studied. The model involves a temperature-dependent heat conductivity with moderate growth in order to control the rapidly growing mixed terms as e.g. the heat generated by friction or by viscous deformation. The contact condition is formulated in velocities. The existence of a solution is proved by an approximation of the problem in several steps involving a penalty-approximation and a smoothing of the friction law. The solvability of the approximate problem is proved by the Galerkin method. A priori estimates uniform with respect to all approximation parameters make it possible to pass to the original problem. These estimates are based on a regularity result for the contact problem without heat transfer.

The Dirichlet problem for the uniformly elliptic equation in generalized weighted Morrey spaces

2020 ◽  
Vol 57 (1) ◽  
pp. 68-90 ◽  
Author(s):  
Tahir S. Gadjiev ◽  
Vagif S. Guliyev ◽  
Konul G. Suleymanova
Keyword(s):  
Elliptic Equations ◽  
A Priori Estimates ◽  
A Priori ◽  
Morrey Spaces ◽  
Smooth Bounded Domain ◽  
Weighted Morrey Spaces ◽  
Priori Estimates ◽  
Muckenhoupt Class ◽  
Morrey Estimates ◽  
Dirichlet Boundary Problem

Abstract In this paper, we obtain generalized weighted Sobolev-Morrey estimates with weights from the Muckenhoupt class Ap by establishing boundedness of several important operators in harmonic analysis such as Hardy-Littlewood operators and Calderon-Zygmund singular integral operators in generalized weighted Morrey spaces. As a consequence, a priori estimates for the weak solutions Dirichlet boundary problem uniformly elliptic equations of higher order in generalized weighted Sobolev-Morrey spaces in a smooth bounded domain Ω ⊂ ℝn are obtained.

Convergence of the Rosenau-Korteweg-de Vries Equation to the Korteweg-de Vries One

2020 ◽  
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
A Priori Estimates ◽  
Vries Equation ◽  
A Priori ◽  
Diffusion Parameter ◽  
Priori Estimates ◽  
De Vries ◽  
Wall Interactions

The Rosenau-Korteweg-de Vries equation describes the wave-wave and wave-wall interactions. In this paper, we prove that, as the diffusion parameter is near zero, it coincides with the Korteweg-de Vries equation. The proof relies on deriving suitable a priori estimates together with an application of the Aubin-Lions Lemma.

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