$W$-Sobolev spaces: Higher order and regularity

Alexandre B. Simas; Fábio J. Valentim

doi:10.3934/cpaa.2015.14.597

Article Contents

2015, Volume 14, Issue 2: 597-607. Doi: 10.3934/cpaa.2015.14.597

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$W$-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
2.
Departamento de Matemática, Universidade Federal do Espírito Santo, Av. Fernando Ferrari, 514, Goiabeiras, 29075-910, Vitória - ES

Received: June 2014

Revised: September 2014

Published: March 2015

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Abstract

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 $\lambda\geq 0$.

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Mathematics Subject Classification: Primary: 35J15; Secondary: 46E35.

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References

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