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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] A. Faggionato, M. Jara and C. Landim, Hydrodynamic behavior of one dimensional subdiffusive exclusion processes with random conductances,, \emph{Probability Theory and Related Fields}, 144 (2009), 633. doi: 10.1007/s00440-008-0157-7. [2] J. Farfan, A. B. Simas and F. J. Valentim, Equilibrium fluctuations for exclusion processes with conductances in random environments,, \emph{Stochastic Processes and their Applications}, 120 (2010), 1535. [3] T. Franco, C. Landim, Hydrodynamic limit of gradient exclusion processes with conductances,, \emph{Archive for Rational Mechanics and Analysis}, 195 (2009), 409. doi: 10.1007/s00205-008-0206-5. [4] M. Jara, C. Landim and A. Teixeira, Quenched scaling limits of trap models,, \emph{Annals of Probability}, 39 (2011), 176. doi: 10.1214/10-AOP554. [5] J.-U. Löbus, Generalized second order differential operators,, \emph{Math. Nachr.}, 152 (1991), 229. [6] P. Mandl, Analytical treatment of one-dimensional Markov processes, Grundlehren der mathematischen Wissenschaften, 151,, Springer-Verlag, (1968). [7] A. B. Simas and F. J. Valentim, $W$-Sobolev spaces,, \emph{Journal of Mathematical Analysis and Applications}, 382 (2011), 214. [8] A. B. Simas and F. J. Valentim, Homogenization of second-order generalized elliptic operators,, submitted for publication., (). [9] F. J. Valentim, Hydrodynamic limit of a $d$-dimensional exclusion process with conductances,, \emph{Ann. Inst. H. Poincar\'e Probab. Statist}, 48 (2012), 188.

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##### References:
 [1] A. Faggionato, M. Jara and C. Landim, Hydrodynamic behavior of one dimensional subdiffusive exclusion processes with random conductances,, \emph{Probability Theory and Related Fields}, 144 (2009), 633. doi: 10.1007/s00440-008-0157-7. [2] J. Farfan, A. B. Simas and F. J. Valentim, Equilibrium fluctuations for exclusion processes with conductances in random environments,, \emph{Stochastic Processes and their Applications}, 120 (2010), 1535. [3] T. Franco, C. Landim, Hydrodynamic limit of gradient exclusion processes with conductances,, \emph{Archive for Rational Mechanics and Analysis}, 195 (2009), 409. doi: 10.1007/s00205-008-0206-5. [4] M. Jara, C. Landim and A. Teixeira, Quenched scaling limits of trap models,, \emph{Annals of Probability}, 39 (2011), 176. doi: 10.1214/10-AOP554. [5] J.-U. Löbus, Generalized second order differential operators,, \emph{Math. Nachr.}, 152 (1991), 229. [6] P. Mandl, Analytical treatment of one-dimensional Markov processes, Grundlehren der mathematischen Wissenschaften, 151,, Springer-Verlag, (1968). [7] A. B. Simas and F. J. Valentim, $W$-Sobolev spaces,, \emph{Journal of Mathematical Analysis and Applications}, 382 (2011), 214. [8] A. B. Simas and F. J. Valentim, Homogenization of second-order generalized elliptic operators,, submitted for publication., (). [9] F. J. Valentim, Hydrodynamic limit of a $d$-dimensional exclusion process with conductances,, \emph{Ann. Inst. H. Poincar\'e Probab. Statist}, 48 (2012), 188.
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