American Institute of Mathematical Sciences

• 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

 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 $\lambda\geq 0$.
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] Probability Theory and Related Fields, 144 (2009), 633-667. doi: 10.1007/s00440-008-0157-7.  Google Scholar [2] Stochastic Processes and their Applications, 120 (2010), 1535-1562. Google Scholar [3] Archive for Rational Mechanics and Analysis, 195 (2009), 409-439. doi: 10.1007/s00205-008-0206-5.  Google Scholar [4] Annals of Probability, 39 (2011), 176-223. doi: 10.1214/10-AOP554.  Google Scholar [5] Math. Nachr., 152 (1991), 229-245. Google Scholar [6] Springer-Verlag, Berlin, 1968.  Google Scholar [7] 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] Ann. Inst. H. Poincaré Probab. Statist, 48 (2012), 188-211. Google Scholar

show all references

References:
 [1] Probability Theory and Related Fields, 144 (2009), 633-667. doi: 10.1007/s00440-008-0157-7.  Google Scholar [2] Stochastic Processes and their Applications, 120 (2010), 1535-1562. Google Scholar [3] Archive for Rational Mechanics and Analysis, 195 (2009), 409-439. doi: 10.1007/s00205-008-0206-5.  Google Scholar [4] Annals of Probability, 39 (2011), 176-223. doi: 10.1214/10-AOP554.  Google Scholar [5] Math. Nachr., 152 (1991), 229-245. Google Scholar [6] Springer-Verlag, Berlin, 1968.  Google Scholar [7] 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] Ann. Inst. H. Poincaré Probab. Statist, 48 (2012), 188-211. Google Scholar
 [1] 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 [2] Simone Fiori, Italo Cervigni, Mattia Ippoliti, Claudio Menotta. Synthetic nonlinear second-order oscillators on Riemannian manifolds and their numerical simulation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021088 [3] Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020136 [4] Li Chu, Bo Wang, Jie Zhang, Hong-Wei Zhang. Convergence analysis of a smoothing SAA method for a stochastic mathematical program with second-order cone complementarity constraints. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1863-1886. doi: 10.3934/jimo.2020050 [5] Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2619-2633. doi: 10.3934/dcds.2020377 [6] Fabio Sperotto Bemfica, Marcelo Mendes Disconzi, Casey Rodriguez, Yuanzhen Shao. Local existence and uniqueness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021069 [7] A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044 [8] Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 [9] Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021017 [10] Andreia Chapouto. A remark on the well-posedness of the modified KdV equation in the Fourier-Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3915-3950. doi: 10.3934/dcds.2021022 [11] Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511 [12] Cheng Wang. Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations. Electronic Research Archive, , () : -. doi: 10.3934/era.2021019 [13] Jinyi Sun, Zunwei Fu, Yue Yin, Minghua Yang. Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3409-3425. doi: 10.3934/dcdsb.2020237 [14] Elimhan N. Mahmudov. Second order discrete time-varying and time-invariant linear continuous systems and Kalman type conditions. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021010 [15] Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037 [16] Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021022 [17] Arunima Bhattacharya, Micah Warren. $C^{2, \alpha}$ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024 [18] Minh-Phuong Tran, Thanh-Nhan Nguyen. Pointwise gradient bounds for a class of very singular quasilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021043 [19] Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019 [20] Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021016

2019 Impact Factor: 1.105