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We start by proving that under the nondegeneracy property the multi valued term
generates a continuous operator from Loo(I) into Lq(I), for any q E [1, (0).
Lemma 3 (i) Let 10, wE £,YO(I) and assume that 10 satisfies the strong p-nondegenemcy
property. Then for any q E [1,(0) there exists C > 0 such that for any z,z E Loo(I))
z(x) E ;3(1O(x)), z(x) E ;3(w(x)) a.e. x E I we have
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(ii) If w, W E LOO(/) and satisfy the weak p-nondegenemcy property then
Dz(x) - z(x))(1O(x) - w(x))dx S (aj - aile I I 10 - w Ilioo(I) .
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Proof of Lemma 3. If I I w - w l i LOO(I» EO then
Assume now that I I 10 - w Ilu"(I) S EO. Define the coincidence sets
A = {x E I: 1o(x) = -IO} A = {x E I: w(x) = -10},
as well as the descomposition
where
D+ = {x E I: w(x) > -IO} D_ = {x E I: w(x) < -IO}
and n+JL are defined similarly replacing 10 by W. Let z, z defined as in the statement.
Then
Iz(x) - z(x)1 S (aj - a;)
z(x) = z(x)
Thus as III = 2
But we have
on A U A U (D+ n n_) U (D_ n n+ )
on (D+nn+)U(D_nn_)
Indeed; it is clear that A c B,. Moreover,
w(x)- I I w - w IILoo(I)S w(x) SII 10 - w IILOO(l ) +w(x) a.e. x E I.
Then the inclusion A C B, is obvious. If x E D+ n n_, -10 < 1o( x) S f + w( x) < -10 + f
and so x E B,. Finally if x E D_ n n+, -10 - f S -10 -I1O(:r) - w(x)1 S tu(x) + 1o(x)-