Abstract
The aim of this chapter is to provide a detailed mathematical study of a simplified scalar version of the elastic problem presented in Chap. 1. Since this problem is easier than the elastic case it will be studied in the generic framework of a d-dimensional space, with d ≥ 3.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ammari, H.: An Introduction to Mathematics of Emerging Biomedical Imaging. Mathématiques et Applications, vol. 62. Springer, Berlin (2008)
Ammari, H., Kang, H.: Reconstruction of Small Inhomogeneities from Boundary Measurements. Lecture Notes in Mathematics. Springer, Berlin (2004)
Ammari, H., Kang, H.: Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory. Applied Mathematical Sciences, vol. 162. Springer, Berlin (2007)
Ammari, H., Moskow, S., Vogelius, M.: Boundary integral formulas for the reconstruction of electromagnetic imperfections of small diameter. ESAIM Control Optim. Calc. Var. 9, 49–66 (2003)
Ammari, H., Griesmaier, R., Hanke, M.: Identification of small inhomogeneities: asymptotic factorization. Math. Comput. 76, 1425–1448 (2007)
Ammari, H., Bretin, E., Garnier, J., Kang, H., Lee, H., Wahab, A.: Mathematical Methods in Elasticity Imaging. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2015)
Amrouche, C., Bonzom, F.: Exterior problems in the half-space for the Laplace operator in weighted Sobolev spaces. J. Differ. Equ. 246, 1894–1920 (2009)
Amrouche, C., Nečasová, S.: Laplace equation in the half-space with a nonhomogeneous Dirichlet boundary condition. Math. Boh. 126, 265–274 (2001)
Amrouche, C., Girault, V., Giroire, J.: Weighted Sobolev spaces for Laplace’s equation in \(\mathbb {R}^n\). J. Math. Pures Appl. 73, 579–606 (1994)
Amrouche, C., Dambrine, M., Raudin, Y.: An L p theory of linear elasticity in the half-space. J. Differ. Equ. 253, 906–932 (2012)
Amrouche, C., Meslameni, M., Nečasová, S.: Linearized Navier-Stokes equations in \(\mathbb {R}^3\): an approach in weighted Sobolev spaces. Discrete Contin. Dynam. Syst. 7, 901–916 (2014)
Aspri, A., Beretta, E., Mascia, C.: Asymptotic expansion for harmonic functions in the half-space with a pressurized cavity. Math. Meth. Appl. Sci. 39(10), 2415–2430 (2016)
Beretta, E., Mukherjee, A., Vogelius, M.S.: Asymptotic formulas for steady state voltage potentials in the presence of conductivity imperfections of small area. Z. Angew. Math. Phys. 52, 543–572 (2001)
Beretta, E., Francini, E., Vogelius, M.S.: Asymptotic formulas for steady state voltage potentials in the presence of thin inhomogeneities. A rigorous error analysis. J. Math. Pures Appl. 82, 1277–1301 (2003)
Calderón, A.P.: Cauchy integrals on Lipschitz curves and related operators. Proc. Natl. Acad. Sci. U.S.A. 74, 1324–1327 (1977)
Capdeboscq, Y., Vogelius, M.S.: A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction. Math. Model. Numer. Anal. 37, 159–173 (2003)
Coifman, R.R., McIntosh, A., Meyer, Y.: L’intégrale de Cauchy définit un opérateur bourné sur L 2 pour les courbes lipschitziennes. Ann. Math. 116, 361–387 (1982)
Escauriaza, L., Fabes, E.B., Verchota, G.: On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries. Proc. Am. Math. Soc. 115, 1069–1076 (1992)
Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence (1998)
Fabes, E.B., Jodeit, M., Lewis, J.E.: Double layer potentials for domains with corners and edges. Indiana Univ. Math. J. 26, 95–114 (1977)
Fabes, E.B., Jodeit, M., Riviére, N.M.: Potential techniques for boundary value problems on C 1 domains. Acta Math. 141, 165–186 (1978)
Folland, G.B.: Introduction to Partial Differential Equations. Princeton University Press, Princeton (1995)
Friedman, A., Vogelius, M.: Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence. Arch. Ration. Mech. Anal. 105, 299–326 (1984)
Hanouzet, B.: Espaces de Sobolev avec poids. Application au problème de Dirichlet dans un demi-espace. Rendiconti del Seminario Matematico della Università di Padova 46, 227–272 (1971)
Hein Hoernig, R.O.: Green’s functions and integral equations for the Laplace and Helmholtz operators in impedance half-spaces. Mathematics, Ecole Polytechnique X (2010)
Kesavan, S.: Topics in Functional Analysis and Applications. New Age International (P) Limited, Publishers, New Delhi (1989)
Kondrat’ev, V.A., Oleinik, O.A.: Boundary-value problems for the system of elasticity theory in unbounded domains. Korn’s inequalities. Russ. Math. Surv. 43, 65–119 (1988)
Kress, R.: Linear Integral Equations. Springer, Berlin (1989)
McLean, W.C.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)
Salsa, S.: Equazioni a derivate parziali metodi, modelli e applicazioni. Springer, Berlin (2010)
Verchota, G.C.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59, 572–611 (1984)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Aspri, A. (2019). A Scalar Model in the Half-Space. In: An Elastic Model for Volcanology. Lecture Notes in Geosystems Mathematics and Computing. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-31475-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-31475-0_2
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-31474-3
Online ISBN: 978-3-030-31475-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)