Strongly Nonlocal Dislocation Dynamics in Crystals

References

• Cabré , X. , Sire , Y. Nonlinear equations for fractional Laplacians II: Existence, uniqueness, and qualitative properties of solutions . Trans. Amer. Math. Soc. , to appear.
   Web of Science ®Google Scholar
• Caffarelli , L.A. , Cabré , X. ( 1995 ). Fully Nonlinear Elliptic Equations . Providence , RI : American Mathematical Society .
   Google Scholar
• Di Nezza , E. , Palatucci , G. , Valdinoci , E. ( 2012 ). Hitchhiker's guide to the fractional Sobolev spaces . Bull. Sci. math. 136 : 521 – 573 .
   Web of Science ®Google Scholar
• Dipierro , S. , Palatucci , G. , Valdinoci , E. Dislocation dynamics in crystals: A macroscopic theory in a fractional Laplace setting . Comm. Math. Phys. , to appear.
   Web of Science ®Google Scholar
• Forcadel , N. , Imbert , C. , Monneau , R. ( 2009 ). Homogenization of some particle systems with two-body interactions and of the dislocation dynamics . Discrete Contin. Dyn. Syst. 23 : 785 – 826 .
   Web of Science ®Google Scholar
• González , M.d.M. , Monneau , R. ( 2012 ). Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one . Discrete Contin. Dyn. Syst. 32 : 1255 – 1286 .
   Web of Science ®Google Scholar
• Monneau , R. , Patrizi , S. ( 2012 ). Homogenization of the Peierls-Nabarro model for dislocation dynamics . J. Diff. Eqs. 253 : 2064 – 2105 .
   Web of Science ®Google Scholar
• Nabarro , F.R.N. ( 1997 ). Fifty-year study of the Peierls–Nabarro stress . Mat. Sci. Eng. A 234 – 236 .
   Web of Science ®Google Scholar
• Palatucci , G. , Savin , O. , Valdinoci , E. (2013). Local and global minimizers for a variational energy involving a fractional norm. Ann. Mat. Pura Appl. (4) 192:673–718.
   Web of Science ®Google Scholar
• Servadei , R. , Valdinoci , E. ( 2014 ). Weak and viscosity solutions of the fractional Laplace equation . Publ. Mat. 58 : 133 – 154 .
   Web of Science ®Google Scholar
• Silvestre , L. ( 2005 ). Regularity of the obstacle problem for a fractional power of the Laplace operator. Ph.D. Thesis, University of Texas at Austin, Austin, Texas, USA .
   Google Scholar
• Silvestre , L. ( 2007 ). Regularity of the obstacle problem for a fractional power of the Laplace operator . Comm. Pure Appl. Math. 60 : 67 – 112 .
   Web of Science ®Google Scholar