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

TL;DR

This paper develops a fractional Sobolev space framework, analyzes the regularity of solutions, and describes the spectral structure of a fractional differential operator, generalizing classical results for integer-order operators.

Contribution

Introduces a fundamental fractional Sobolev space theory, studies solution regularity, and characterizes the spectral structure of a fractional operator with Dirichlet boundary conditions.

Findings

Spectral structure of the fractional operator ${_t}D_T^eta {_0}D_t^eta$ characterized.

Regularity results for weak solutions established.

Generalization of classical results for integer-order operators achieved.

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.