A Dirichlet Problem for Distributions and Specifications for by Michael Rockner

April 3, 2017 | Science Mathematics | By admin | 0 Comments

By Michael Rockner

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Let £ € P' . L (U,u) and for Now the assertion immediately follws. denote the space of f € L (U,u) D (classes of) y-integrable functions on U set II f ||p :- J|f I dy . 10. Lemma. There exists the sequence (f Proof. ) - : '< M € 8 C,U x*dn>» such that P(M) = 1 M := M and for every (L (U,u),|| M« € 8 such that || ) . P(M ? ) = 1 0 M„ is the desired set, where M. 9. 11. Proposition. Proof. P(fl _) ^-, converges uniformly on j(U,K)) - 1 . 4 (ii) it follows that ^ i f k, 5 ( x >- f n, C Wl-

H d y ( x ) < 2" E Proof. Let n €U be such that K U supp y c U n k . , x - || w ° * ||E , x e u , sucn tnat DIRICHLET PROBLEM AND RANDOM FIELDS are y - i n t e g r a b l e . By 2 . 9 ( i i ) we have f o r 29 n > nQ K U n -»*"E -<2||^o|| E . 9 (iii) and the theorem of dominated convergence imply the assertion. a We see that the subsequence hence on K. 3 depends on is fixed we set from now on for simplicity (U ) ^_, := (U_ ). Cm . 6). Fix y , (U ) ^ Q (U,K) as before in I , it will satisfy the properties P € Bi .

Then by property (ii) of tension H (£) of *£,(£) to P £ € ft(U) there exists a unique linear ex- P(K ) , which is continuous with respect to T n We define if H^O cp € P(K ) n : P -> m by = U U,n for some n € IN . Since by definition = ^ ( C ) , ^ and because of the uniqueness of is well defined. But clearly Given n,k,£ € IN and U H (£) 5 ( O € V1 tp € P, (K ) for every for every and hence cp € P ] n € IN , the map £ € ft(U) . 13. 9 (iii). Then, since Let P the Dirichlet problem for Remark.

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