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Holder continuity results for a class of functionals with non standard growth

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(144) 

(145)   <  *  = ™ 7 ¸ 4 , ziœ  “ Š~0z|{D “ • ƒ‚{ ¿ ¸ 4 *-, ziœ  “ €~9zn{H “ •7–l—N˜ i‚]{ À  ¸ 4 , “  zn{D ~9zn{H “ ‚{ ,* < ,   ™ 7 5¸ 4 , ziœ  “ Š~0z|{D “ • = ƒ‚{ ‘ ¿   

(146)   < * ¸ 4 *-, ziœ  “ Š~0z|{D “ •7–l—#˜ i‚]{  

(147)  = À ¸ 4 *., ziœ  “ €~0z|{D “ • i‚]{ ¸ 4 , “  zn{H ~9zn{H “ ‚{ ™ 7 ¸ 4 *-, zƒœ  “ €~9zn{H “ • = i‚]{  ¸ 4 , “  z|{D ~9zn{D “ ‚]{ ¼}. . . ?D@. 9. J!. &. !.

(148). . . !. . . !. /. &. . .

(149). . *. !. !. . &. (.

(150). !. (. (. (.

(151). . . .  . (.

(152). (. . *. ".

(153). . *.

(154). . *. K&. (. . !. (. . *. . . . #. *.

(155).

(156). *. . *.

(157). &.

(158). .

(159). K&. !. (. *. K&. !.

(160).

(161). /. . .

(162). . .

(163). . !. . .

(164). . (. 4. . . .

(165). . .

(166). !.

(167). œ#ç. (.

(168). *.

(169). . *. .  .

(170) ,.   

(171)   < * )  ) 6. [Z]k elpbp aluhV Z]p ‘  ¿ } } } de k‡»DZ `bpp€„bVNf[„FsbV$z|cZ&Y[f^]sb`bV_V Y®csk‡c&p elp­„+Ž;fcsb^]Vcöelp€YUki`fŸk[kƒeqf[pbVN^ X½fZ&sbm9VŸf[»DsZV_`bZp YV_„†o VN„†pwVNkk Œcá½pZ„Yf[V_sZ7\]V$V_uY Z]7pu_„†ctVN\wX elfiVNh p„bZk m ’ ’ , ™ 7 ¿  5¸ 4, “  zn{H ~0z|{D “ ‚]{ } ‹jà sbZ]oVNY[»bVeqpbc^]eqpbc&^yeqpf[Ž sb7 V-XbY7V_z \w) eq’Z}`) k®m|} c]u} fU kjc&£ p„ `k[elp^ë9Z]elp u_c&YV‡Å eqpbV#Ð]` c&aqegfihŠ‹®Vsct\]V ™ 7 ¸ 4+*.,SziœŒ “ €~9zn{D “ • = i‚]{  ¸ 4, “  zn{D ~9zn{H “ ‚{ ™ 7 &œ  “ €~ “  • = = – 4 *., ˜ ¸ 4, “   zn{H €~0z|{D “ ‚]{ = == = = ’ i. • ™ 7 &œ  “ €~ “  • – 4 *., ˜ • ¸ 4 , “   zn{D Š~0z|{D “ • ‚]{ = == = = ’ i. • ™ 7 &œ  “ €~ “  • – 4+*., ˜ • ¸ 4 , zƒœ  “ €~9zn{H “ • i‚]{ = == = ’ i. • ™ 7 &œ  “ €~ “  • – 4+*., ˜ • ¸ 4 , zƒœ  “ €~9zn{H “ •7–l—N˜‰–d’ " > ˜ i‚]{ } ‹fjsbsbV VNY[VyZeqadp-„†V_f[Y²suVyZ]adpckƒkƒff²c&p;elpbf,V#Z&Ð;mž`c&~ÏaqegelfipÓh­´Ñ‹®Vc&p` „(kƒV#„%Y[V#u_z calbaf[ s(c&fNh­Ž+f[»wsh­V_ZZY`bV_Y²o uUsbéZ eduŽ+V]~͎Hegf,Æ mnZaqalÈ Z71D‹‡z5´jk‡ f[s‹®c&fVy8 k[Vf€Ò ~1Õ 1 f[£HZ TW» VV k[îVc]f0u_u- eqZXµè¶X Z])Äale+ eqpb)+VNÐ;8 `cal8ÄelfihŠcp‹®„(V²‹®^VVf Y[VNo€c&Yà­f[sc&f  - œ£ ê Z¼ö Ykƒf`kƒeqpb^ f[sV_ZYV_o -’Uc‘ p1 „-fsbV_p = =. ’ i• • ¸ 4, ziœ  “ €~0z|{D “ •7–l—#˜Ì–d’ "> ˜ i‚]{  = d – ’ ˜  ‰. • ™ 7 ¸ 4, zƒœ  “ €~9zn{D “ •7–l—#˜ ƒ‚{ "> = = d – ’ ˜  i. • / ~9zn{D z|~H ) / •    " > / ™ 7  ¸ 4 *., œ  /// /  ‚{ / / / = = = = d – ’ ˜ ’. i. •   " > ™ 7  • ¸4 *., œ  Ò ~Õ •1 ‡• = • 1  ‚{  = = = = ¶ 7 Òqz • ÁÒ ~ Õ •1 –Û’ "> ˜ • ’ "%1"%1 > ‘+’‘ >  ’ i• Õ ™ 7  z

(172) Œ£ ê Z Ž†öpcalaqh ™ 7 z

(173)  ¸ 4+*-,žziœ  “ Š~0z|{D “ • = ƒ‚{(} ‹jõŸsbZ7‹VNY[:V ‹ 7V$XbY[Z†7&zu_) V_’VN}„Š) VNkƒ} f[ eqo} c&} f[6 eqpb^ ’ f sbV,Y[VNocelpelpb^ f[VNY[o€k®kifUc&Y[f[eqpb^ »wh  0TÏV²u_c&pÄ`kƒVyz w c&p„Wz w cp„­c&^]celpÄf[sbV²m|c]ufjfsc7f‡~½edk Z]aq„†VNY‡uZ]p]felpw`bZ]`kz|k[V_V€z †ƒµ  ™ñ ¸54 , z “ ~9zn{H zn~+ “    “   zn{D “ •t–q—N˜ ‚]{ ™ 7 z 1  ¸ 4 , zƒœ  “ €~9zn{H “ • = †‚]{-£ œNé D&. =. . !. &. 4. !. (.

(174).

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(176).

(177). *. .

(178). &. (. *.

(179). (.

(180). &. !. (.

(181).  . &. (. !. . !. *.

(182). &. . *. .

(183). *. . . .. ?A@. +<. DE<.

(184). <. . . !. (. !. &.

(185). &. (. *.

(186). . *. . !. #.

(187). &.

(188). &. . . &. !. &.

(189). &. . !. '&. !. . '&. !. 4. . &. '&. . . . ?D@. !. !. . 6&.

(190). #.

(191). .

(192) ‰p½ckƒeqoeladc&Yj‹Œcth€‹®V^VfŒfsbVVNkƒf[eqo€c7f[V²Zm  µ  ™› ¸ 4 , z “ ~9zn{D z|~H “   “ Š~0z|{D “ •7–l—#˜ ‚]{ ™ 7 z 1 ¸ 4,z[œ  “ €~9zn{D “ • = †‚{(£ TÏz

(193) ) VŠ} ki) fY[} V#k[k,}  fs} 6 c7ffŒsbcVÄaqk[u_ZZ\wpedkƒc­fcÒ p]~ fUÕ k€z|z5„†kƒVNVNV pbZ&c&f^;VNc&„ÏeqpŠ»wf[h sV_7_Z$YmnV_Z]o `bpéb„ӏ ]eqpW fsbV€XbYV_\weqZ`k²eqpbVNÐ;`calelf[eqVNk„bV_XDV_p„ÓZ]p ° ZŠ’ ^V_f‡f[sV VNkƒf[eq’o€c7f[V#k‡Zm Äc1 p„  ‹®V u_cp-c&Y^`bVV_Îbcufalh ckjeqp%Ò P0åyىÕ0»b`†f,`kƒeqpb^-z †jc&p „­Z`bY sbeq^sV_Yjeqp]fV_^]Yc»belaqelfih€f[sbVNZYV_o STÏVZ»bfc&eqp œ ¸ 4+*., “ €~0z|{D “ • = ‚]{yx 7 ’ z  ¿ }  ™ 7 ’ z walZ]^ œ ¸ 4 =-, “ €~0z|{D “ • = ‚{yx 7 ’ z  ¿ }  ™ 7 ’ z waqZ^ ‹jà Z]sbalVNaqY[VNV,uf[feqsbpbV ^ŸufsbZ]pVŒkiXbfUYc&V_p;\w|f elZ]7,`pbkZ7» ‹3Z]`b„†pVN„bX kSVNpc„bpk‡„ ck[`baqk[oZyo`bX elpbZ]^²p `b6 Xy‹®£ V ^]VfjznàVNV_Xbeqpb^,elp;fZ,c]u_uZ]`bp;ffsc7f z 1w ™ 7 z

(194) [µ = œ

(195) “ “ • ™   z ƒ ¸ 4 *-, ziœ  €~9zn{H ƒ‚{-£z qœ ] ²$,º 7 ’ z ²†alZ]^  'HbD ÷

(196) %D %('VH%5 Hb % 'VHbD H 'JM TÏV kƒV_fjmnZY‡k[eloXbaqeduelfih  z ®µè¶ ’ z waqZ^ œ   z

(197) ®£ ° sbV c]k[k[`boX†felZ]pÏz wŒc&aqalZ7‹‡kŒ`k®fZŠk[cthfsc7f aleqo  z ²'¶ y£ õŸZ7‹Ž†»;hÄf[sbVoeqpbeqocalelfihŠZm  }†mnYZo z †Œc&p„Ïz wŽb‹®VZ»†fUc&eqp  zn~+ ™ elp†m 4 ,   Íz ®}  "    – ˜ ‹jsbVNY[V²‹®V kƒV_f = Íz ®µè¶ 7  z ¸ 4 *., zƒœ  “ €~9zn{H “ • i‚]{-£ õfŸsbZ7V ‹ „†ed‹ kifUVyc&c&pYu_V V elpÏc€XDZ]k[egfelZ]p½f[Z­c&XbXalhFÒ Ö _Õ=ŽDfsb= V_Z]Y[VNo œ]$øVf  ¶ ~€›Ç ’È ’ bz , ŸVNÐ;`beqXbXDVN„-‹jelf[s ‚ z  ’ }  Œµ· ¶  ‘  = ‘ ¿ =  ¸ 4 , “   ’ zn{D   zn{H “ ‚{W£ Ìeqpfj„†edk `u_V#VNc„Ók[hy»wf[h-ZfkƒsbVNVŠVŸf„†sedkic7fUfŒc&pf[su_V$VŠmn‚` pu° felsbZ]V_ppFc&a »wh%Ò

(198) Ö eqk® ÕÌaqZ7Ž0‹ f[sbVNVNY®Zk[YV_V_oo"eduœZp;elff elpwmnZ`baqZ]aqZ7`‹‡kžk,‹jf[elf[ss c&f YVNf[k[sbX VNV#Y[uV€f®V_f[ΆZeqkƒf[fsbk V, f[Z]X Æ Z]al~¼Z]^ h Ç ’È ’ zb, Œk[`uUs¼f[sc&f = ==  z i²¸ 4, “ €~9zn{H   z|{D “ ‚]{ ™ Ò Íz ÌÕ ¿ } z  ‰  z   ™  z|~Hj} ==  Íz  z   i  edkjcalZ†uNc&a+oelpeloeqr_V_YjZmfsbV²mn`bpuf[eqZpca (  z $0 ¸ 4, “     w“ ‚]{-£ ¿ . !. !. #.

(199). '&. .. .

(200). . . !. 6&. +(. !. !. '&. '&. &. *. &. )(. !. '&. &. &"*. !. !. . 6&. !. (. 6&. !. *. &. '&. >!. '&. 6&. )(. K!. &+*. '&. . . '. . . '&. . 6&. . #%$. . . . . . 6&. 6&. '&. . . . . &. .

(201). #. . . . . #. .

(202). . . . 6&. . &. . . . . . '&. . œ. &. . . . .

(203). .

(204) = hŠfsbVoelpelo€c&aqelfihŠZ&m  ‹ V²sct\]V,f[sc&f®mnZ]Y‡V_\]V_= YhywÍÆ-Ç ’È • z, ®µ = %  z    z  } ,  ™  z  w®} ,  ¸4 , “   z|{D%ywŒzn{H   zn{D “ ‚]{ ¿ ™  z   w®} ,  ç œ  ¸ 4, “   zn{DÍywŒz|{D “ • = ‚]{ =  ç œ  ¸ 4 , “   zn{H “ • ‚]{y 7 Íz ®} kƒeqpb^€^YZ7‹Œf[s ckkƒ`oyXbf[eqZpk= z w®egfjmnZ]alaqZ7‹‡kŒelp½ck[eloXbaqV‹®cth= fsc7f ¸ 4, “   z|{D “ • ‚{ ™ 7¸ 4, “   z|{D0Íy wŒz|{D “ • ‚]{yx 7&z %z  ¿ ®} ‹jelf[3s 7 7z ) ’ } ) } }   ° sbedkŒoVNc&p kŒf[sc&f  eqkjyc CŸˆÌoelpbeqoelrNV_YjZmfsbV²mn`bpuf[eqZpca = Íz  • “ “ ( ¸ 4,    ¿ ºœ ‚{(}  ‹jYUc7sbfVNVNY[„ŠV eqp %z  z ) ’° } sb) VNpÄ} elfŒ}  edk ‡VN¡ ck[œ]hy f[Z»kƒVNk[V_VŸYf\sV²c7f[fs z|c7k[fŸV_V'fsbÒ VÕ̎w„bf[V_sbXDVNV_ZpY„†V_VNo pubVy &`b'XDf[Zsbp VNY[6 V$V’ Άc&edkipfU„ k®c6 pÄV_ÎwedkŸXDeqZppuV_Z]p;Y[f XDZ&Zˆm sbeq^sV_Yjeqp]fV_^]Yc»belaqelfih¼š²ÆFz· }| zƒœ "=8  ;ƒŒcp„ cyu_Zpkƒfcp;f 7,¡ kƒ`uUs¼f[s c7f = •    ¸5: ,3<-= “   z|{D “ ‚]{  ™ 7 ¸54 , “   zn{H “ • ‚]{y7 œ  Íz ¿  £ Ò Ö

(205) pF Õ̎bfsbf[sVÄV_ZZf[YsbV_oVNYœ,s c&‹®pV²„ÓsmnctY\ZV,o fsfc7sbf VÄ= ^]Y[Z7‹ŒfsÍckkƒ`boX†felZ]pºz † c&p„FmnY[Z]o XY[Z]X VNYƒfih = z  i „†V#„†`uV#„Í»wh  ‘+’ ¸ 4 , “   zn{H “ • ‚]{ ™  z   ™  zn~+ ™ñ ¸ 4 , zƒœ  “ €~9zn{H “ • †‚]{-} =  k[Z = •  ¸ : “   z|{D “  ‚]{ ™ 7 ¸4+*-,'ziœ  “ €~9zn{H “ • ƒ‚{-£ z qœœ# (. . . . '&. . #. . . . . #. . . . &. . #. . .

(206). . #. . '&. . . 4. . '&. &. . . '&. (. *. &. 4. . . '&. ;.. . . .

(207). :9. 6&. .

(208). (. *. &. . . .

(209). 2J . . ,3<-=. . . . '.

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