Introduction This book is based on a series of five lectures that Carlo s Kenig gave during th e 25th Arkansas Sprin g Lectures Serie s in March 2000 , at th e Universit y o f Arkansas. In these lectures , Keni g describe d hi s joint wor k wit h Tatian a Tor o concernin g end-point analogue s o f the well-know n potentia l theoreti c resul t o f Kellogg , whic h says that th e density k of the harmonic measur e of a C 1 '" domain , ha s logarithm i n Ca an d of the 'converse' o f this result, the free boundary regularity theorem of Alt- Caffarelli [2] , which says that unde r (necessary ) mil d hypothesis, if log k is Ca, the n the domai n mus t b e of class C 1,a . Th e potentia l theoreti c result s ar e extensions of the classica l functio n theoreti c wor k o f Lavrentiev [53 ] and Pommerenk e [61] , and the highe r dimensiona l result s o f Dahlberg [16 ] an d Jerison-Keni g [34] . The free boundary results , on the one hand, giv e a geometric measur e theoreti c characterization o f the support set s of measures which are " asymptotically optimally doubling" i n term s o f "flatness" condition s o n th e support , an d exten d th e Alt - Caffarelli highe r dimensiona l versio n [2 ] of the "converse" resul t o f Pommerenke' s [61], to the end-point VM O case. Thi s type of end-point versio n of the Alt-Caffarell i result wa s first introduce d b y Davi d Jeriso n [32] . The boo k follow s closel y th e forma t o f th e lectures . I n particular , fo r eac h o f the mai n Theorem s i n Chapte r 6 and i n th e first sectio n o f Chapte r 7 , we presen t a shor t "sketc h o f th e proo f whic h i s a n almos t verbati m cop y o f th e argumen t described i n th e lectures . Thes e brie f sketche s ar e followe d b y detaile d proofs . I n this wa y w e hop e t o communicat e th e mai n idea s an d conve y th e enthusias m an d the intuitiv e insigh t whic h mad e th e lecture s s o lively an d exciting . We break thi s patter n i n the proo f o f the las t tw o theorems (Section s tw o an d three i n Chapte r 7) , fo r whic h th e sketc h o f th e proo f alon e i s alread y quit e lon g and technicall y involved . Th e intereste d reade r wil l find detail s fo r thes e theorem s in [45 ] an d [46] . W e hop e tha t ou r presentatio n wil l provid e a "readin g key " t o help navigat e throug h thes e papers . In orde r t o mak e th e presentatio n mor e self-containe d an d comprehensive , a review o f th e classica l result s fo r plana r domain s ha s bee n adde d i n Chapte r 2 , where conforma l mappin g i s the mai n too l to approac h th e problems . ix
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