# Die Hamilton-Jacobische Theorie fuer Doppelintegrale by Prange G. By Prange G.

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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 ). 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 ).