Journal of Fixed Point Theory and Applications

, Volume 19, Issue 4, pp 2697–2731 | Cite as

\({{\varvec{C}}}^\mathbf{1 }\)-regularity for minima of functionals with \({\varvec{p}}({\varvec{x}})\)-growth

  • Jihoon Ok


We prove the \(C^1\)-regularity for local minima to the functionals with p(x)-growth such that \(w\mapsto \int _\Omega \left[ f(x,w,Dw)+gw\right] \, \mathrm{d}x\), where f satisfies a certain p(x)-growth condition. We assume that g belongs to the Lorentz space \(L^{n,1}\), and that \(p(\cdot )\) and \(f(\cdot ,z,\xi )\) satisfy Dini type continuity conditions. Our assumptions are sharp with respect to the type of regularity results obtained.


Functional with variable growth Gradient continuity Dini continuity Lorentz space 

Mathematics Subject Classification

Primary 49N60 Secondary 35j20 


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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsKyung Hee UniversityYonginKorea

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