# American Institute of Mathematical Sciences

August  2015, 8(4): 769-772. doi: 10.3934/dcdss.2015.8.769

## On a Poisson's equation arising from magnetism

 1 Dipartimento di Matematica e Fisica "N.Tartaglia", Università Cattolica del Sacro Cuore, Via dei Musei 41, I-25121 Brescia

Received  January 2014 Revised  July 2014 Published  October 2014

We review the proof of existence and uniqueness of the Poisson's equation $\Delta u + {\rm div}\,{\bf m}=0$ whenever ${\bf m}$ is a unit $L^2$-vector field on $\mathbb R^3$ with compact support; by standard linear potential theory we deduce also the $H^1$-regularity of the unique weak solution.
Citation: Luca Lussardi. On a Poisson's equation arising from magnetism. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 769-772. doi: 10.3934/dcdss.2015.8.769
##### References:
 [1] C. Amrouche, H. Bouzit and U. Razafison, On the two and three dimensional Oseen potentials, Potential Anal., 34 (2011), 163-179. doi: 10.1007/s11118-010-9186-9.  Google Scholar [2] W. F. Brown, Micromagnetics, John Wiley and Sons, New York, 1963. Google Scholar [3] O. Bottauscio, V. Chiadò Piat, M. Eleuteri, L. Lussardi and A. Manzin, Homogenization of random anisotropy properties in polycrystalline magnetic materials, Phys. B, 407 (2012), 1417-1419. Google Scholar [4] O. Bottauscio, V. Chiadò Piat, M. Eleuteri, L. Lussardi and A. Manzin, Determination of the equivalent anisotropy properties of polycrystalline magnetic materials: Theoretical aspects and numerical analysis, Math. Models Methods Appl. Sci., 23 (2013), 1217-1233. doi: 10.1142/S0218202513500073.  Google Scholar [5] R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials, Continuum Mech. Thermodyn, 2 (1990), 215-239. doi: 10.1007/BF01129598.  Google Scholar [6] L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media, Course of Theoretical Physics, 8, Pergamon Press, 1984.  Google Scholar [7] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.  Google Scholar

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##### References:
 [1] C. Amrouche, H. Bouzit and U. Razafison, On the two and three dimensional Oseen potentials, Potential Anal., 34 (2011), 163-179. doi: 10.1007/s11118-010-9186-9.  Google Scholar [2] W. F. Brown, Micromagnetics, John Wiley and Sons, New York, 1963. Google Scholar [3] O. Bottauscio, V. Chiadò Piat, M. Eleuteri, L. Lussardi and A. Manzin, Homogenization of random anisotropy properties in polycrystalline magnetic materials, Phys. B, 407 (2012), 1417-1419. Google Scholar [4] O. Bottauscio, V. Chiadò Piat, M. Eleuteri, L. Lussardi and A. Manzin, Determination of the equivalent anisotropy properties of polycrystalline magnetic materials: Theoretical aspects and numerical analysis, Math. Models Methods Appl. Sci., 23 (2013), 1217-1233. doi: 10.1142/S0218202513500073.  Google Scholar [5] R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials, Continuum Mech. Thermodyn, 2 (1990), 215-239. doi: 10.1007/BF01129598.  Google Scholar [6] L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of Continuous Media, Course of Theoretical Physics, 8, Pergamon Press, 1984.  Google Scholar [7] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970.  Google Scholar
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