Direct Methods in the Calculus of Variat by Enrico Giusti

By Enrico Giusti

This e-book offers a entire dialogue at the life and regularity of minima of normal integrals within the calculus of diversifications and of suggestions to elliptic partial differential equations and platforms of the second one order. whereas direct equipment for the life of options are popular and feature been popular within the final century, the regularity of the minima was once continuously bought by way of the Euler equation as part of the final conception of partial differential equations. during this publication, utilizing the suggestion of the quasi-minimum brought via Giaquinta and the writer, the direct equipment are prolonged to the regularity of the minima of functionals within the calculus of diversifications, and of strategies to partial differential equations. This unified therapy deals a considerable economic system within the assumptions, and allows a deeper knowing of the character of the regularity and singularities of the suggestions. The booklet is largely self-contained, and calls for just a common wisdom of the weather of Lebesgue integration idea.

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C. 6 remains valid only if we assume that the boundary datum U is of class Cl>a, with a > 0, but not in general if U is only Lipschitz-continuous (GlUSTl [4]). On the other hand it can be proved, always under the hypothesis of non-negative mean curvature of the boundary d£l, that for every continuous boundary datum U the DlRICHLET problem has asolution u of class C2(fl)C\ c°(n). The main ingredient in the proof of the last result is the a priori inequality for the gradient: m , ,, , \Du(x0)\ .

We denote by Lp(fl,HN) the space of measurable functions f : fi —> RN such that ||/IU= j^|/| p dz} P <+oo. 2) Moreover, by L°°(Q,,'RN) we indicate the space of bounded measurable functions in Cl. When no possible confusion might arise, we shall write simply Lp(Cl) or even Lp, without explicit mention of the codomain HN. e. in f2} . e. such that fk —> / in Ll(K) for every compact set K C 0) we can extract a subsequence converging to f almost everywhere in CI. If/ is inL p (ft), the set Ft = {x G n : |/(x)| > t} is measurable, and since [ \f\pdx> [ \f\pdx>tp\Ft\, Jn JFt 4 We recall that if V(Q) is a space of functions in fi, V\oc(Q) is the space of functions belonging to V(A) for every open set A CC fi (that is such that A is a compact set contained in Q).

On the other hand it can be proved, always under the hypothesis of non-negative mean curvature of the boundary d£l, that for every continuous boundary datum U the DlRICHLET problem has asolution u of class C2(fl)C\ c°(n). The main ingredient in the proof of the last result is the a priori inequality for the gradient: m , ,, , \Du(x0)\ . 57) in which OSC(U,BR(XQ)) indicates the oscillation of u in the ball of radius R centered at x0 (BOMBIERI, D E GlORGI and MIRANDA [1]).

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