By Bombieri E. (ed.)

W.K. ALLARD: at the first version of zone and generalized suggest curvature.- F.J. ALMGREN Jr.: Geometric degree concept and elliptic variational problems.- E. GIUSTI: minimum surfaces with obstacles.- J. GUCKENHEIMER: Singularities in soap-bubble-like and soap-film-like surfaces.- D. KINDERLEHRER: The analyticity of the twist of fate set in variational inequalities.- M. MIRANDA: limitations of Caciopoli units within the calculus of variations.- L. PICCININI: De Giorgi’s degree and skinny stumbling blocks.

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**Sample text**

6) as the definition of the principal value we obtain reasonable results regardless of the order of the pole. ) §6 Hyperfunction of the form f(ax + b) As has been shown, given a hyperfunction f(x) = H. F. F(z), hyperfunctions of the form f{ -x) and f(x) are defined as fe-x) =- H. F. F{-z), f(x) = - H. F. 2) CHAPTER 3 37 respectively. 1) means that a new hyperfunction is created from the hyperfunction f(x) by replacing the argument x by -x. Generalising this procedure for the replacement of argument x by ax + b, (a and b being real constants), we can use the following definition.

F. {~dnn (_;~)} = H. F. F. dzn+1 -jz = __1_ H. F. e. newly defined hyperfunctions X-I 0 8(n l (x) can be expressed as derivatives of an already known hyperfunction 8 (x). This is analogous to the expression sin x, cos x, tan x, . in terms of exp ix. 10) (a p real), where ¢(z) is an analytic function regular on the x-axis. 11 ) (x _ a)m . x . 11) itself can be expressed in terms of simple hyperfunctions, as will be seen in Example 7. BASIC HYPERFUNCTIONS 40 EXAMPLE 7. l/(x - a) 0 8(x), a¥- O. e.

F. F'(z) = ± H. F. F'( -z), so that f'(x) = =ff'( -x). (ii) ¢(x)f(x) = H. F. ¢(z)F(z). From the parities of ¢(z) and F(z) we obtain the result immediately. • §4 Hyperfunction with generating function F(z) We saw that the flow with complex velocity w = F( -z) is the mirror image of the flow with complex velocity w = F(z) with respect to the origin z = o. Now, instead of the mirror image with respect to the origin, let us take the mirror image with respect to the x-axis. Then again we have an irrotational source-free flow, whose complex velocity is given by w = F(z).