Fractional Sobolev Space and Spectral Structure of Fractional Dirichlet Boundary Value Problem

Abstract

Based on the need of studying the fractional boundary value problems by using variational methods, in this paper, we introduce a fundamental theory framework of fractional Sobolev space in one dimension, study the regularity of weak solutions for a fractional boundary value problem with variational structure, give out the spectral structure of operator ${}_t{D}_T^{\alpha }{}_0{D}_t^{\alpha }$ with Dirichlet boundary value conditions. Especially, when $\alpha =1$, the operator ${}_t{D}_T^{\alpha }{}_0{D}_t^{\alpha }=-D^2$. So, the results of this paper are the generalization of corresponding conclusions for integer differential operator to some extent.

The weak fractional derivative has been defined in [D. Idczak, S. Walczak, Fractional Sobolev spaces via Riemann-Liouville derivatives, J. Funct. Spaces Appl. 2013 (2013) Article ID 128043]. So there are some bugs that need to be modified, and we would like to withdraw this paper