Fractional Korn-Type Inequalities and Applications

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-Slobodeckij spaces. The equivalence can be considered a Korn-type characterization of said Sobolev spaces. For vector fields defined on various classes of domains, we obtain a relevant form of the inequality. As an application, we consider variational problems associated to strongly coupled systems of nonlocal equations motivated by a continuum mechanics model known as peridynamics. We use the fractional Korn-type inequalities to characterize vector fields in associated energy spaces and obtain existence and uniqueness of solutions in fractional Sobolev spaces.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdifferential geometryfunctional analysismetric geometrynumerical analysisoptimization and control

Audience: researchers in the topic

Online Seminar "Geometric Analysis"

Series comments: We discuss recent trends related to geometric analysis in a broad sense. The general idea is to solve geometric problems by means of advanced tools in analysis. We will include a wide range of topics such as geometric flows, curvature functionals, discrete differential geometry, and numerical simulation.

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