# 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. doi: 10.1007/s11118-010-9186-9. Google Scholar [2] W. F. Brown, Micromagnetics,, John Wiley and Sons, (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. 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. doi: 10.1142/S0218202513500073. Google Scholar [5] R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials,, Continuum Mech. Thermodyn, 2 (1990), 215. 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, (1984). Google Scholar [7] E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970). Google Scholar

show all references

##### References:
 [1] C. Amrouche, H. Bouzit and U. Razafison, On the two and three dimensional Oseen potentials,, Potential Anal., 34 (2011), 163. doi: 10.1007/s11118-010-9186-9. Google Scholar [2] W. F. Brown, Micromagnetics,, John Wiley and Sons, (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. 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. doi: 10.1142/S0218202513500073. Google Scholar [5] R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials,, Continuum Mech. Thermodyn, 2 (1990), 215. 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, (1984). Google Scholar [7] E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970). Google Scholar
 [1] Luciano Pandolfi. Riesz systems and moment method in the study of viscoelasticity in one space dimension. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1487-1510. doi: 10.3934/dcdsb.2010.14.1487 [2] Jiankai Xu, Song Jiang, Huoxiong Wu. Some properties of positive solutions for an integral system with the double weighted Riesz potentials. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2117-2134. doi: 10.3934/cpaa.2016030 [3] Scott W. Hansen, Rajeev Rajaram. Riesz basis property and related results for a Rao-Nakra sandwich beam. Conference Publications, 2005, 2005 (Special) : 365-375. doi: 10.3934/proc.2005.2005.365 [4] Luciano Pandolfi. Riesz systems, spectral controllability and a source identification problem for heat equations with memory. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 745-759. doi: 10.3934/dcdss.2011.4.745 [5] Sergei Avdonin, Julian Edward. Controllability for a string with attached masses and Riesz bases for asymmetric spaces. Mathematical Control & Related Fields, 2019, 9 (3) : 453-494. doi: 10.3934/mcrf.2019021 [6] Gen Qi Xu, Siu Pang Yung. Stability and Riesz basis property of a star-shaped network of Euler-Bernoulli beams with joint damping. Networks & Heterogeneous Media, 2008, 3 (4) : 723-747. doi: 10.3934/nhm.2008.3.723 [7] Jun Cao, Der-Chen Chang, Dachun Yang, Sibei Yang. Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1435-1463. doi: 10.3934/cpaa.2014.13.1435 [8] Kolade M. Owolabi, Abdon Atangana. High-order solvers for space-fractional differential equations with Riesz derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 567-590. doi: 10.3934/dcdss.2019037 [9] Yusheng Jia, Weishi Liu, Mingji Zhang. Qualitative properties of ionic flows via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Ion size effects. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1775-1802. doi: 10.3934/dcdsb.2016022 [10] Manh Hong Duong, Hoang Minh Tran. On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3407-3438. doi: 10.3934/dcds.2018146 [11] Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119 [12] Hong Lu, Ji Li, Joseph Shackelford, Jeremy Vorenberg, Mingji Zhang. Ion size effects on individual fluxes via Poisson-Nernst-Planck systems with Bikerman's local hard-sphere potential: Analysis without electroneutrality boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1623-1643. doi: 10.3934/dcdsb.2018064 [13] Reinhard Farwig, Ronald B. Guenther, Enrique A. Thomann, Šárka Nečasová. The fundamental solution of linearized nonstationary Navier-Stokes equations of motion around a rotating and translating body. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 511-529. doi: 10.3934/dcds.2014.34.511 [14] Thomas Chen, Ryan Denlinger, Nataša Pavlović. Moments and regularity for a Boltzmann equation via Wigner transform. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 4979-5015. doi: 10.3934/dcds.2019204 [15] Hans-Otto Walther. On Poisson's state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 365-379. doi: 10.3934/dcds.2013.33.365 [16] Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068 [17] Zifei Shen, Fashun Gao, Minbo Yang. On critical Choquard equation with potential well. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3567-3593. doi: 10.3934/dcds.2018151 [18] Karen Yagdjian, Anahit Galstian. Fundamental solutions for wave equation in Robertson-Walker model of universe and $L^p-L^q$ -decay estimates. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 483-502. doi: 10.3934/dcdss.2009.2.483 [19] Anat Amir. Sharpness of Zapolsky's inequality for quasi-states and Poisson brackets. Electronic Research Announcements, 2011, 18: 61-68. doi: 10.3934/era.2011.18.61 [20] David Mumford, Peter W. Michor. On Euler's equation and 'EPDiff'. Journal of Geometric Mechanics, 2013, 5 (3) : 319-344. doi: 10.3934/jgm.2013.5.319

2018 Impact Factor: 0.545