Abstract
We address the effect of extreme geometry on a non-convex variational problem, motivated by studies on magnetic domain walls trapped by thin necks. The recent analytical results of Kohn and Slastikov (Calc. Var. Partial Differ. Equ. 28:33–57, 2007) revealed a variety of magnetic structures in three-dimensional ferromagnets depending on the size of the constriction. The main purpose of this paper is to study geometrically constrained walls in two dimensions. The analysis turns out to be significantly more challenging and requires the use of different techniques. In particular, the purely variational point of view of Kohn and Slastikov (loc. cit.) cannot be adopted in the present setting and is here replaced by a PDE approach. The existence of local minimizers representing geometrically constrained walls is proven under suitable symmetry assumptions on the domains and an asymptotic characterization of the wall profile is given. The limiting behavior, which depends critically on the scaling of length and height of the neck, turns out to be more complex than in the higher-dimensional case and a richer variety of regimes is shown to exist.
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References
Abramowitz M., Stegun I.: Handbook of Mathematical Functions. Dover Publications, New York (1965)
Bruno P.: Geometrically constrained magnetic wall. Phys. Rev. Lett. 83, 2425–2428 (1999)
Casten R., Holland C.: Instability results for reaction–diffusion equations with Neumann boundary conditions. J. Differ. Equ. 27, 266–273 (1978)
Casado-Daz J., Luna-Laynez M., Murat F.: The diffusion equation in a notched beam. Calc. Var. Partial Differ. Equ. 31, 297–323 (2008)
Chambolle A., Doveri F.: Continuity of Neumann linear elliptic problems on varying two-dimensional bounded open sets. Commun. Partial Differ. Equ. 22, 811–840 (1997)
Dal Maso G., Ebobisse F., Ponsiglione M.: A stability result for nonlinear Neumann problems under boundary variations. J. Math. Pures Appl. 82, 503–532 (2003)
Evans, L.C.: Partial differential equations. Graduate Studies in Mathematics, Vol. 19. American Mathematical Society, Providence, 1998
Hale J.K., Vegas J.: A nonlinear parabolic equation with varying domain. Arch. Ration. Mech. Anal. 86, 99–123 (1984)
Chopra H.D., Hua S.Z.: Ballistic magnetoresistance over 3000% in Ni nanocontacts at room temperature. Phys. Rev. B 66, 020403(R) (2002)
Jimbo S.: Singular perturbation of domains and semilinear elliptic equation. J. Fac. Sci. Univ. Tokyo 35, 27–76 (1988)
Jimbo S.: Singular perturbation of domains and semilinear elliptic equation 2. J. Differ. Equ. 75, 264–289 (1988)
Jimbo S.: Singular perturbation of domains and semilinear elliptic equation 3. Hokkaido Math. J. 33, 11–45 (2004)
Jubert P.-O., Allenspach R., Bischof A.: Magnetic domain walls in constrained geometries. Phys. Rev. B 69, 220410(R) (2004)
Kläui M.: Head-to-head domain walls in magnetic nanostructures. J. Phys. Condens. Matter 20, 313001 (2008)
Kohn R.V., Slastikov V.: Geometrically constrained walls. Calc. Var. Partial Differ. Equ. 28, 33–57 (2007)
Molyneux V.A., Osipov V.V., Ponizovskaya E.V.: Stable two- and three-dimensional geometrically constrained magnetic structures: the action of magnetic fields. Phys. Rev. B 65, 184425 (2002)
Pommerenke, C.: Boundary behaviour of conformal maps. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin, 1992
Rudin W.: Real and Complex Analysis. McGraw-Hill, New York (1987)
Rubistein J., Schatzman M., Sternberg P.: Ginzburg-Landau model in thin loops with narrow constrictions. SIAM J. Appl. Math. 64, 2186–2204 (2004)
Serrin J.: A Harnack inequality for nonlinear equations. Bull. Am. Math. Soc. 69, 481–486 (1963)
Tatara G., Zhao Y.-W., Munoz M., Garcia N.: Domain wall scattering explains 300% ballistic magnetoconductance of nanocontacts. Phys. Rev. Lett 83, 2030–2033 (1999)
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Morini, M., Slastikov, V. Geometrically Constrained Walls in Two Dimensions. Arch Rational Mech Anal 203, 621–692 (2012). https://doi.org/10.1007/s00205-011-0458-3
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DOI: https://doi.org/10.1007/s00205-011-0458-3