Abstract
In this chapter we shall occupy ourselves with the study of certain functions represented by either domain or surface integrals, with the purpose of establishing under what conditions such functions turn out to be integrable, or continuous, Hölder continuous, differentiable, etc. Among others, we shall study certain integrals which we can consider a natural extension of the ordinary potential either of a domain or of a single or double layer. This fact shows the importance of the contents of this chapter to the end of a systematic treatment of the theory of elliptic equations.
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References
Evans, G.C. and E.R.C. Miles: Potential of general masses in single and double layers. The relative boundary value problems. Amer. J. Math. 53 (1931) 493–516. For other indications of previous works of these authors see also: G.C. Evans: Complements of potential theory. Amer. J. Math. 54 (1932) 213-234; 55 (1933) 29-49; 57 (1935) 623-626.
This result and its application to the case of ordinary potentials are contained in Calderon, A. P. and Zygmund, A., On the existence of certain singular integrals, Acta. Math. 88 (1952) 85–139. For application to the general case see for example C. Miranda [19] § 4.
H. Whitney: Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63–89. Functions differentiable on the boundaries of regions. Ann. of. Math. 35 (1934), 482-485.
E. Gagliardo: Caratterizzazione delle traccie sulla frontiera relative ad alcune classi di funzioni in n variabili. Rend. Sem. Mat. Padova 27 (1957), 284–305.
S. M. Nikol’skii: On imbedding, continuation and approximation theorems for differentiate functions of several variables. Usp. Mat. Nauk 16 no. 6, (1961) 63–144 (Russian); transl. in Russian Math. Surveys 16 no. 5., (1961), 55-104.
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© 1970 Springer-Verlag Berlin Heidelberg
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Miranda, C. (1970). Functions represented by integrals. In: Partial Differential Equations of Elliptic Type. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 2 . Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-35147-5_2
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