American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Local and global existence results for the Navier-Stokes equations in the rotational framework
  • CPAA Home
  • This Issue
  • Next Article
    Note on global regularity of 3D generalized magnetohydrodynamic-$\alpha$ model with zero diffusivity
March  2015, 14(2): 597-607. doi: 10.3934/cpaa.2015.14.597

$W$-Sobolev spaces: Higher order and regularity

Alexandre B. Simas 1, and  Fábio J. Valentim 2,

1. 

Departamento de Matemática, Universidade Federal da Paraíba, Cidade Universitária - Campus I, 58051-970, João Pessoa - PB, Brazil
2. 

Departamento de Matemática, Universidade Federal do Espírito Santo, Av. Fernando Ferrari, 514, Goiabeiras, 29075-910, Vitória - ES, Brazil

Received  June 2014 Revised  September 2014 Published  December 2014

Fix a function $W(x_1,\ldots,x_d) = \sum_{k=1}^d W_k(x_k)$ where each $W_k: R \to R$ is a right continuous with left limits and strictly increasing function, and consider the $W$-laplacian given by $\Delta_W = \sum_{i=1}^d \partial_{x_i}\partial_{W_i}$, which is a generalization of the laplacian operator. In this work we introduce the $W$-Sobolev spaces of higher order, thus extending the notion of $W$-Sobolev spaces introduced in Simas and Valentim (2011) [7]. We then provide a characterization of these spaces in terms of a suitable Fourier series, and conclude the paper with some results on elliptic regularity of the problem $\lambda u - \Delta_Wu = f,$ for $\lambda\geq 0$.
Keywords: second-order elliptic equations, special Fourier series, compact embedding., $W$-Sobolev spaces, regularity.
Mathematics Subject Classification: Primary: 35J15; Secondary: 46E3.
Citation: Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597
References:
[1]

A. Faggionato, M. Jara and C. Landim, Hydrodynamic behavior of one dimensional subdiffusive exclusion processes with random conductances, Probability Theory and Related Fields, 144 (2009), 633-667. doi: 10.1007/s00440-008-0157-7.  Google Scholar

[2]

J. Farfan, A. B. Simas and F. J. Valentim, Equilibrium fluctuations for exclusion processes with conductances in random environments, Stochastic Processes and their Applications, 120 (2010), 1535-1562. Google Scholar

[3]

T. Franco, C. Landim, Hydrodynamic limit of gradient exclusion processes with conductances, Archive for Rational Mechanics and Analysis, 195 (2009), 409-439. doi: 10.1007/s00205-008-0206-5.  Google Scholar

[4]

M. Jara, C. Landim and A. Teixeira, Quenched scaling limits of trap models, Annals of Probability, 39 (2011), 176-223. doi: 10.1214/10-AOP554.  Google Scholar

[5]

J.-U. Löbus, Generalized second order differential operators, Math. Nachr., 152 (1991), 229-245. Google Scholar

[6]

P. Mandl, Analytical treatment of one-dimensional Markov processes, Grundlehren der mathematischen Wissenschaften, 151, Springer-Verlag, Berlin, 1968.  Google Scholar

[7]

A. B. Simas and F. J. Valentim, $W$-Sobolev spaces, Journal of Mathematical Analysis and Applications, 382 (2011), 214-230. Google Scholar

[8]

A. B. Simas and F. J. Valentim, Homogenization of second-order generalized elliptic operators,, submitted for publication., ().   Google Scholar

[9]

F. J. Valentim, Hydrodynamic limit of a $d$-dimensional exclusion process with conductances, Ann. Inst. H. Poincaré Probab. Statist, 48 (2012), 188-211. Google Scholar

show all references

References:
[1]

A. Faggionato, M. Jara and C. Landim, Hydrodynamic behavior of one dimensional subdiffusive exclusion processes with random conductances, Probability Theory and Related Fields, 144 (2009), 633-667. doi: 10.1007/s00440-008-0157-7.  Google Scholar

[2]

J. Farfan, A. B. Simas and F. J. Valentim, Equilibrium fluctuations for exclusion processes with conductances in random environments, Stochastic Processes and their Applications, 120 (2010), 1535-1562. Google Scholar

[3]

T. Franco, C. Landim, Hydrodynamic limit of gradient exclusion processes with conductances, Archive for Rational Mechanics and Analysis, 195 (2009), 409-439. doi: 10.1007/s00205-008-0206-5.  Google Scholar

[4]

M. Jara, C. Landim and A. Teixeira, Quenched scaling limits of trap models, Annals of Probability, 39 (2011), 176-223. doi: 10.1214/10-AOP554.  Google Scholar

[5]

J.-U. Löbus, Generalized second order differential operators, Math. Nachr., 152 (1991), 229-245. Google Scholar

[6]

P. Mandl, Analytical treatment of one-dimensional Markov processes, Grundlehren der mathematischen Wissenschaften, 151, Springer-Verlag, Berlin, 1968.  Google Scholar

[7]

A. B. Simas and F. J. Valentim, $W$-Sobolev spaces, Journal of Mathematical Analysis and Applications, 382 (2011), 214-230. Google Scholar

[8]

A. B. Simas and F. J. Valentim, Homogenization of second-order generalized elliptic operators,, submitted for publication., ().   Google Scholar

[9]

F. J. Valentim, Hydrodynamic limit of a $d$-dimensional exclusion process with conductances, Ann. Inst. H. Poincaré Probab. Statist, 48 (2012), 188-211. Google Scholar

[1]

Annamaria Canino, Elisa De Giorgio, Berardino Sciunzi. Second order regularity for degenerate nonlinear elliptic equations. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4231-4242. doi: 10.3934/dcds.2018184
[2]

José F. Cariñena, Javier de Lucas Araujo. Superposition rules and second-order Riccati equations. Journal of Geometric Mechanics, 2011, 3 (1) : 1-22. doi: 10.3934/jgm.2011.3.1
[3]

Hongwei Lou, Jiongmin Yong. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Mathematical Control & Related Fields, 2018, 8 (1) : 57-88. doi: 10.3934/mcrf.2018003
[4]

Olivier Druet, Emmanuel Hebey and Frederic Robert. A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth. Electronic Research Announcements, 2003, 9: 19-25.

[5]

Sang-Gyun Youn. On the Sobolev embedding properties for compact matrix quantum groups of Kac type. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3341-3366. doi: 10.3934/cpaa.2020148
[6]

W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209
[7]

José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213
[8]

Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609
[9]

Jaume Llibre, Amar Makhlouf. Periodic solutions of some classes of continuous second-order differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 477-482. doi: 10.3934/dcdsb.2017022
[10]

M. Euler, N. Euler, M. C. Nucci. On nonlocal symmetries generated by recursion operators: Second-order evolution equations. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4239-4247. doi: 10.3934/dcds.2017181
[11]

Xuan Wu, Huafeng Xiao. Periodic solutions for a class of second-order differential delay equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021159
[12]

Tran Hong Thai, Nguyen Anh Dai, Pham Tuan Anh. Global dynamics of some system of second-order difference equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2021077
[13]

Alain Haraux, Mitsuharu Ôtani. Analyticity and regularity for a class of second order evolution equations. Evolution Equations & Control Theory, 2013, 2 (1) : 101-117. doi: 10.3934/eect.2013.2.101
[14]

Angelo Favini, Gisèle Ruiz Goldstein, Jerome A. Goldstein, Silvia Romanelli. Selfadjointness of degenerate elliptic operators on higher order Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 581-593. doi: 10.3934/dcdss.2011.4.581
[15]

Alberto Fiorenza, Anna Mercaldo, Jean Michel Rakotoson. Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 893-906. doi: 10.3934/dcds.2002.8.893
[16]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184
[17]

Doyoon Kim, Seungjin Ryu. The weak maximum principle for second-order elliptic and parabolic conormal derivative problems. Communications on Pure & Applied Analysis, 2020, 19 (1) : 493-510. doi: 10.3934/cpaa.2020024
[18]

Bernd Kawohl, Vasilii Kurta. A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1747-1762. doi: 10.3934/cpaa.2011.10.1747
[19]

Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure & Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761
[20]

Guanrong Li, Yanping Chen, Yunqing Huang. A hybridized weak Galerkin finite element scheme for general second-order elliptic problems. Electronic Research Archive, 2020, 28 (2) : 821-836. doi: 10.3934/era.2020042

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences