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# Download A Singular Dark Marking on the Sky (1915)(en)(3s) by Barnard E.E. PDF

By Barnard E.E.

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33. Let k,m e No anrfp > 1. 9) is sharp. If (k — m — l)p = N, then the imbedding is continuous for any (3 e (0,1). 34. Let u e Wk'p(G) with k e N, p e E, kp > N and p > 1. Then u € Cm(G) for 0 < m < k — N/p and there exists a constant c, independent of u, such that THEOREM sup \Dau(x)\ < c\\u\\Wk,PiG) xeG for all \a\ < k - N/p. 35. Let G be a lipschitzian domain and let Ts c G be piecewise Ck-smooth s-dimensional manifold. Let k > 1, p > 1, kp < N, N -kp < s < N and I

3) I'l-p-p-Q #uv) dQ.

Wk'p(G), 1 < p < oo : the Sobolev space equipped with the norm VP \G s I PIS * n e closure of CQ°(G) with respect to the norm || P(G)- (G\O) = Wk'p(G\B£(0)), For p = 2 we use the notation Wk(G) = Wk'2(G), Ve > 0. W\$(G) = W\$'2(G). 26. Let us say that u e Wk'p(G) satisfies u < 0 on dG in the sense of traces, if its positive part u+ = max{u, 0} € Wo >P(G). 6 HOLDER AND SOBOLEV SPACES 27 is continuous in a neighborhood of dG, then u satisfies u < 0 on dG, if the inequality holds in the classical pointwise sense.