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Creators/Authors contains: "Scott, James M."

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  1. ; (, SIAM Journal on Mathematical Analysis)
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  2. ; ; (, Nonlinear Analysis)
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  3. ; (, Continuum Mechanics and Thermodynamics)
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  4. ; (, SIAM Journal on Mathematical Analysis)
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  5. ; (, Nonlinearity)
    Abstract We study the existence and uniqueness of solutions to the vector field Peierls–Nabarro (PN) model for curved dislocations in a transversely isotropic medium. Under suitable assumptions for the misfit potential on the slip plane, we reduce the 3D PN model to a nonlocal scalar Ginzburg–Landau equation. For a particular range of elastic coefficients, the nonlocal scalar equation with explicit nonlocal positive kernel is derived. We prove that any stable steady solution has a one-dimensional profile. As a result, we obtain that solutions to the scalar equation, as well as the original 3D system, are characterized as a one-parameter family of straight dislocations. This paper generalizes results found previously for the full isotropic case to an anisotropic setting. 
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  6. ; ; ; (, Mathematics and Mechanics of Complex Systems)
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  7. ; (, Studia Mathematica)
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  8. ; ; ; ; (, Journal of Geophysical Research: Solid Earth)
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  9. ; (, Communications in Mathematical Sciences)
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  10. ; (, Communications on Pure & Applied Analysis)
    We prove higher Sobolev regularity for bounded weak solutions to a class of nonlinear nonlocal integro-differential equations. The leading operator exhibits nonuniform growth, switching between two different fractional elliptic "phases" that are determined by the zero set of a modulating coefficient. Solutions are shown to improve both in integrability and differentiability. These results apply to operators with rough kernels and modulating coefficients. To obtain these results we adapt a particular fractional version of the Gehring lemma developed by Kuusi, Mingione, and Sire in their work "Nonlocal self-improving properties" Analysis & PDE, 8(1):57–114 for the specific nonlinear setting under investigation in this manuscript. 
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