Abstract
Consider a material homogeneous body occupying a region Ω ⊂ R N.We assume that ∂Ω is of class C 1 and let n denote its outward unit normal. We identify the body with Ω and let k > 0 be its dimensionless conductivity.
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References
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See Problem 7.1.
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This method is a particular case of the Duhamel principle. See Section 2.3 of the Complements of Chapter VI.
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The eigenfunctions v; are Hölder continuous in S2. This is the content of Corollary 12.1 of Chapter III. By the Schauder estimates of Section 9 of Chapter II, v; E C2+’ (S2). Therefore, by a bootstrap argument v; E C°°(12). Actually v; are of class C2+v up to 852, so that (a = v; can be taken as a testing function in (10.14). Such C estimates up to the boundary have been indicated in Section 9 of the complements of Chapter II.
Vladimir Andreevich Steklov, 1864–1926.
For a discussion on logarithmic convexity methods in ill-posed problems we refer to L.E. Payne, Improperly Posed Problems in Partial Differential Equations, SIAM, Vol. 22, Philadelphia PA, 1975 and references therein.
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Equation (2.1 c) arises in the filtration of a fluid in a porous medium. See A.E. Scheidegger, footnote 2 of the Preliminaries. The similarity solution (2.2) was derived independently by Barenblatt and Pattie: G.I. Barenblatt, On some unsteady motions of a liquid or a gas in a porous medium, Prikl. Mat. Meh.. #16 (1952), pp. 67–78; R.E. Pattie, Diffusion from an instantaneous point source with a concentration-dependent coefficient, Quarterly J. of Appl. Math.. #12 (1959) pp. 407–409.
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DiBenedetto, E. (1995). The Heat Equation. In: Partial Differential Equations. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-2840-5_6
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