American Institute of Mathematical Sciences

Advanced
July  2010, 9(4): 987-1010. doi: 10.3934/cpaa.2010.9.987

Coercive energy estimates for differential forms in semi-convex domains

Dorina Mitrea 1, , Irina Mitrea 2, , Marius Mitrea 1, and  Lixin Yan 3,

1. 

Department of Mathematics, University of Missouri, Columbia, MO 65211, United States, United States
2. 

Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester MA 01609-2280, United States
3. 

Department of Mathematics, Zhongshan University, Guangzhou, 510275, China

Received  July 2009 Revised  February 2010 Published  April 2010

In this paper, we prove a $H^1$-coercive estimate for differential forms of arbitrary degrees in semi-convex domains of the Euclidean space. The key result is an integral identity involving a boundary term in which the Weingarten matrix of the boundary intervenes, established for any Lipschitz domain $\Omega\subseteq \mathcal{R}^n$ whose outward unit normal $\nu$ belongs to $L^{n-1}_1(\partial\Omega)$, the $L^{n-1}$-based Sobolev space of order one on $\partial\Omega$.
Keywords: Semi-convex domains, differential forms, Weingarten map, coercive estimates..
Mathematics Subject Classification: Primary: 35B65, 35J46, 46E35; Secondary: 35B45, 35J50, 35J57, 53C4.
Citation: Dorina Mitrea, Irina Mitrea, Marius Mitrea, Lixin Yan. Coercive energy estimates for differential forms in semi-convex domains. Communications on Pure & Applied Analysis, 2010, 9 (4) : 987-1010. doi: 10.3934/cpaa.2010.9.987
[1]

Wan-Tong Li, Bin-Guo Wang. Attractor minimal sets for nonautonomous type-K competitive and semi-convex delay differential equations with applications. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 589-611. doi: 10.3934/dcds.2009.24.589
[2]

Holger Heumann, Ralf Hiptmair. Eulerian and semi-Lagrangian methods for convection-diffusion for differential forms. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1471-1495. doi: 10.3934/dcds.2011.29.1471
[3]

Dorina Mitrea and Marius Mitrea. Boundary integral methods for harmonic differential forms in Lipschitz domains. Electronic Research Announcements, 1996, 2: 92-97.

[4]

Giuseppe Cordaro. Existence and location of periodic solutions to convex and non coercive Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 983-996. doi: 10.3934/dcds.2005.12.983
[5]

Murat Adivar, Shu-Cherng Fang. Convex optimization on mixed domains. Journal of Industrial & Management Optimization, 2012, 8 (1) : 189-227. doi: 10.3934/jimo.2012.8.189
[6]

R.D.S. Oliveira, F. Tari. On pairs of differential $1$-forms in the plane. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 519-536. doi: 10.3934/dcds.2000.6.519
[7]

Chiara Caracciolo, Ugo Locatelli. Computer-assisted estimates for Birkhoff normal forms. Journal of Computational Dynamics, 2020, 7 (2) : 425-460. doi: 10.3934/jcd.2020017
[8]

Carlos Gutierrez, Víctor Guíñez, Alvaro Castañeda. Quartic differential forms and transversal nets with singularities. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 225-249. doi: 10.3934/dcds.2010.26.225
[9]

Holger Heumann, Ralf Hiptmair, Cecilia Pagliantini. Stabilized Galerkin for transient advection of differential forms. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 185-214. doi: 10.3934/dcdss.2016.9.185
[10]

Olivier Hénot. On polynomial forms of nonlinear functional differential equations. Journal of Computational Dynamics, 2021, 8 (3) : 307-323. doi: 10.3934/jcd.2021013
[11]

Frank Natterer. Photo-acoustic inversion in convex domains. Inverse Problems & Imaging, 2012, 6 (2) : 315-320. doi: 10.3934/ipi.2012.6.315
[12]

Dyi-Shing Ou, Kenneth James Palmer. A constructive proof of the existence of a semi-conjugacy for a one dimensional map. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 977-992. doi: 10.3934/dcdsb.2012.17.977
[13]

Stephen Baigent. Convex geometry of the carrying simplex for the May-Leonard map. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1697-1723. doi: 10.3934/dcdsb.2018288
[14]

Laura Caravenna. Regularity estimates for continuous solutions of α-convex balance laws. Communications on Pure & Applied Analysis, 2017, 16 (2) : 629-644. doi: 10.3934/cpaa.2017031
[15]

Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745
[16]

Raffaela Capitanelli, Maria Agostina Vivaldi. Uniform weighted estimates on pre-fractal domains. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1969-1985. doi: 10.3934/dcdsb.2014.19.1969
[17]

Lipu Zhang, Yinghong Xu, Zhengjing Jin. An efficient algorithm for convex quadratic semi-definite optimization. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 129-144. doi: 10.3934/naco.2012.2.129
[18]

Xiaojin Zheng, Zhongyi Jiang. Tighter quadratically constrained convex reformulations for semi-continuous quadratic programming. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2331-2343. doi: 10.3934/jimo.2020071
[19]

Weigu Li, Jaume Llibre, Hao Wu. Polynomial and linearized normal forms for almost periodic differential systems. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 345-360. doi: 10.3934/dcds.2016.36.345
[20]

Ovidiu Carja, Victor Postolache. A Priori estimates for solutions of differential inclusions. Conference Publications, 2011, 2011 (Special) : 258-264. doi: 10.3934/proc.2011.2011.258

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (117)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences