Abstract
Chapter 1 gathers fundamental facts from functional analysis.
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R.A. Adams, J.L. Fournier, Sobolev Spaces, 2nd edn. (Elsevier, Oxford, 2003)
D.R. Adams, L.I. Hedberg, Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften, vol. 314 (Springer, Berlin, Heidelberg, 1996)
G.P. Akilov, L.V. Kantorovich, Funkcionalnyi Analiz (Nauka, Moskva, 1977)
H. Amann, J. Escher, Analysis III (Birkhäuser, Basel, Boston, Berlin, 2009)
C. Bennett, R. Sharpley, Interpolation of Operators (Academic, Orlando, FL, 1988)
J. Berg, J. Löström, Interpolation Spaces. An Introduction (Springer, Berlin, Heidelberg, New York, 1976)
N. Bourbaki, Integrirovanie. Mery, integrirovanie mer (Nauka, Moskva, 1967)
G. Chen, J. Zhou, Boundary Element Methods (Academic, London, 1992)
M. Chipot, Elliptic Equations: An Introductory Course (Birkhäuser, Basel, Boston, Berlin, 2009)
R.A. Devore, R.C. Sharpley, Besov spaces on domains in R d. Trans. Math. Soc. 335(2), 843–864 (1993)
M. Dobrowolski, Angewandte Functionanalysis. Functionanalysis, Sobolev-Räume und elliptische Differentialgleichungen (Springer, Berlin, Heidelberg, 2006)
R.G. Douglas, Banach Algebra Techniques in Operator Theory (Academic, New York and London, 1972)
J.J. Duistermaat, J.A.C. Kolk, Distributions. Theory and Applications (Birkhäuser, New York, Dordrecht, Heidelberg, London, 2010)
N. Dunford, J.T. Schwartz, Linear Operators I, General Theory (Interscience Publishers, New York, London, 1958)
H. Federer, Geometric Measure Theory (Springer, Berlin, Heidelberg, New York, 1969)
M. Giaquinta, G. Modica, Mathematical Analysis. Linear and Metric Structures and Continuity (Birkhäuser, Boston, 2007)
S. Goldberg, Unbounded Linear Operators. Theory and Applications (McGraw-Hill Book Company, New York, 1966)
P. Grisvard, Elliptic Problems in Nonsmooth Domains (SIAM, Philadelphia, 2011)
W. Hackbush, Integral Equations. Theory and Numerical Treatment (Bikhäuser, Bassel, 1995)
P.R. Halmos, Measure Theory (D. van Nostrand, New York, 1950)
L. Hörmander, The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis (Springer, Berlin, Heidelberg, 2003)
C.E. Kenig, Weighted H p spaces on Lipschitz domains. Am. J. Math. 102, 129–163 (1980)
S.G. Krantz, H.R. Parks, Geometric Integration Theory (Birkhäuser, Boston, 2008)
M.A. Krasnosel’skii, P.P. Zabreiko, E.I. Pustyl’nik, P.E. Sobolevskii, Integral Operators in Spaces of Summable Functions (Izdat. Nauka, Moskva, 1966) (Russian)
A. Kufner, O. John, S. Fučík, Function Spaces (Academia, Prague, 1977)
E.H. Lieb, M. Loss, Analysis (American Mathematical Society, Providence, RI, 1997)
J. Malý, W.P. Ziemer, Fine Regularity of Solutions of Elliptic Partial Differential Equations (American Mathematical Society, Providence, RI, 1997)
V.G. Maz’ya, S.V. Poborchi, Differentiable Functions on Bad Domains (World Scientific, Singapore, 1997)
Y. Meyer, R. Coifman, Wavelets. Calderón–Zygmund and Multilinear Operators (Cambridge University Press, Cambridge, 1997)
M. Mitrea, M. Wright, Boundary value problems for the Stokes system in arbitrary Lipschitz domains. Astérisque 344, Paris (2012)
V. Müller, Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Operator Theory Advances and Applications, vol. 139 (Birkhäuser, Basel, 2007)
J. Nečas, Les méthodes directes en théorie des équations élliptiques (Academia, Prague, 1967)
J. Peetre, New Thoughts on Besov Spaces. Duke University Mathematics Series (Duke University, Durham, 1976)
W. Rudin, Real and Complex Analysis, 2nd edn. (McGraw-Hill, New York, 1974)
M. Schechter, Principles of Functional Analysis (Academic, London, 1971)
M. Schechter, Principles of Functional Analysis (American Mathematical Society, Providence, RI, 2002)
G.E. Shilov, Mathematical Analysis. Second special course (Nauka, Moskva, 1965) (Russian)
E.M. Stein, Singular Integrals and Differentiability of Functions (Princeton University Press, Princeton, NJ, 1970)
E.M. Stein, Harmonic Analysis. Real-Variable Methods, Orthogonality, and Oscilatory Integrals (Princeton University Press, Princeton, NJ, 1993)
L. Tartar, An Introduction to Sobolev Spaces and Interpolation Spaces (Springer, Berlin, Heidelberg, 2007)
A.E. Taylor, Introduction to Functional Analysis (Wiley, New York, 1967)
W.F. Trench, Introduction to Real Analysis (Prentice Hall, Princeton, NJ, 2002)
H. Triebel, Höhere Analysis (VEB Deutscher Verlag der Wissenschaften, Berlin, 1972)
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (VEB Deutscher Verlag der Wissenschaften, Berlin, 1978)
H. Triebel, Theory of Function Spaces (Birkhäuser, Basel, Boston, Stuttgart, 1983)
H. Triebel, Theory of Function Spaces III. Monographs in Mathematics, vol. 100 (Birkhäuser, Basel, 2006)
G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59, 572–611 (1984)
K. Yosida, Functional Analysis (Springer, Berlin, 1965)
W.P. Ziemer, Weakly Differentiable Functions (Springer, New York, 1989)
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Medková, D. (2018). Preliminaries. In: The Laplace Equation. Springer, Cham. https://doi.org/10.1007/978-3-319-74307-3_1
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