A fractional Korn-type inequality

James Scott,Tadele Mengesha

doi:10.3934/dcds.2019137

Abstract

We show that a class of spaces of vector fields whose semi-norms involve the magnitude of directional difference quotients is in fact equivalent to the class of fractional Sobolev spaces. The equivalence can be considered a Korn-type characterization of fractional Sobolev spaces. We use the result to understand better the energy space associated to a strongly coupled system of nonlocal equations related to a nonlocal continuum model via peridynamics. Moreover, the equivalence permits us to apply classical space embeddings in proving that weak solutions to the nonlocal system enjoy both improved differentiability and improved integrability.

Highlights

Introduction and statement of main resultsThe main focus of this paper is to study the function space of vector fields given by XK,p(Ω) := v ∈ Lp(Ω; Rd) : [v]pXK,p(Ω) < ∞, [v]pXK,p(Ω) :=Ω (v(y) − v(x)) (y − x) p K(y − x) · dy dx, |y − x| |y − x|where Ω ⊂ Rd is an open subset and the kernel K(z) is a nonnegative function with appropriate integrability
For p = 2, the space XK,2(Ω) has been used in nonlocal continuum mechanics [23,24,25] where it appears as the energy space corresponding to the peridynamic strain energy in small strain linear models
Combining these inequalities with the characterization of W s,p(Rd; Rd) in terms of Poisson integrals we obtain the equivalence of spaces

Summary

Introduction

Korn’s inequality, fractional Sobolev spaces, Poisson integral, coupled nonlocal equations, self-improving properties. Our proof of Theorem 1.1 makes use of the classical characterization of functions in the fractional spaces in terms of their Poisson integrals. Combining these inequalities with the characterization of W s,p(Rd; Rd) in terms of Poisson integrals we obtain the equivalence of spaces.

Results

Conclusion