On some Hardy-Sobolev s type variable exponent inequality and its application

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1 Trnsctions of NAS of Azerbijn Issue Mthemtics Series of Phsicl-Technicl nd Mthemticl Sciences On some Hrd-Sobolev s tpe vrible exponent inequlit nd its ppliction Frmn I. Mmedov Sli M. Mmmdli Yusuf Zeren Received: / Revised: 9..27/ Accepted: Abstrct. In this pper it hs been proved Sobolev s tpe vrible exponent inequlit ux C u x px;l u xl x px;l l Ẇ p. l where the exponent function p : l is monotone incresing ner little neighborhood of origin nd monotone decresing ner l stisfing the conditions: t p t dt t C 2 p nd t p l t dt t C p l for < < l. Appling this inequlit nd Browder-Mint theor methods it hs been proved n existence result of solution for some vrible exponent eqution. Kewords. vrible exponent spces inequlit solvbilit Dirichlet problem Mthemtics Subject Clssifiction 2: 26D 42B37 34L3 Introduction One of the results of this pper is the following ssertion on vrible exponent boundedness of the conjugte Hrd opertor ft dt on finite intervl l. F.I. Mmedov Mthemtics nd Mechnics Institute of Ntionl Acdem of Sciences Bku Azerbijn Oil nd Gs Scientific Reserch Project Inst. SOCAR Bku Azerbijn E-mil: frmn-m@mil.ru S.M. Mmmdli Mthemtics nd Mechnics Institute of Ntionl Acdem of Sciences Bku Azerbijn E-mil: sk-426@mil.ru Y.Zeren Yildiz Technicl Universit Istnbul Turke E-mil: usufzeren@hotmil.com x

2 F.I. Mmedov S.M. Mmmdli Y. Zeren 3 Theorem. Let p : l be mesurble function such tht < px p + <. Assume tht p monotone decreses on little neighborhood l ε l of the right end of intervl l. Then it holds n inequlit l x x ftdt C f. p.;l p.;l. for ll mesurble positive functions f : l if nd onl if t dt p l t t C p l < < l.2 where positive constnt C depends on p + C ε l nd p l ε >. This ssertion looks s following for n bsolutel continuous functions. Theorem.2 Let p : l be mesurble function such tht < px p + <. Assume tht p monotone decreses on little neighborhood of right hnd side of the intervl l i.e. on l ε l b some ε >. Then it holds n inequlit ux C u x l x px;l px;l.3 for ll bsolutel continuous functions u : l R with ul = if nd onl if t dt p l t t C p l < < l.4 where positive constnt C depends on p + C C 2 ε l nd p l ε >. The proof of this ssertions m be composed using the following boundedness result for the Hrd opertor Hfx = x x ftdt in Lp. l from [3]. Theorem.3 Let p : l be mesurble function such tht < px p + <. Assume tht p monotone increses on little neighborhood ε of the origin. Then it holds n inequlit x ftdt C f. p.;l.5 x p.;l for ll mesurble positive functions f : l if nd onl if t dt p t t C 2 p < < l.6 where positive constnt C depends on p + C 2 ε l nd p εl >. Combining Theorems. nd.2 one gets the following result.

3 4 On some Hrd-Sobolev s tpe vrible exponent inequlit nd its ppliction Theorem.4 Let p : l be mesurble function such tht < px p + <. Assume tht p monotone increses ner origin decreses ner l on little neighborhood i.e. p on ε nd p on l ε l for some ε >. Then it holds the inequlit ux C xl x px;l l u x px;l.7 for ll bsolutel continuous functions u : l R with u = ul = if nd onl if.6 nd.4 is stisfied; moreover positive constnt C in.7 depends on p + C C 2 ε nd p εl ε > For proof of Theorem.4 let us note the inequlit ux xl x 2 ux l x + ux. l x To prove Theorem. we shll use the inequlit x ft dt x f L p. l L.8 p. l for mesurble function p : l which is monotone incresing ner the origin ε nd be such tht < px p + <. According to the resent works see e.g. [3] [4] [5] for the inequlit.5 it is necessr nd sufficientl tht the condition x p x x C p < < l.9 be fulfilled. In our setting concerning Theorem. we hve the condition.6 nd the monotone decresing of p ner the right end of intervl l in l ε l b some ε >. Insert new exponent function px = pl x. Then we re in region of ppliction of inequlit.5 for the exponent p intervl l spce L p. l nd the opertor H. Indeed check the condition.9: x p x x inserting the expression of the function px using condition.6 = x p l x x C p l = C p tht mens the condition.9 is fulfilled for function px. In order to show.3 it suffices to prove the inequlit l gt dt l x C g. L p. l p.;l. x

4 F.I. Mmedov S.M. Mmmdli Y. Zeren 5 for n mesurble function g L p. l. The monotone incresing ner origin of the function px follows from its expression px = pl x nd the monotone decresing of p ner l. Let f : l be mesurble function nd the condition.4 be stisfied. Using the cited bove result we get the inequlit.8 x x ft dt C f L p. l L p. l. Using the definition of vrible exponent norm clculte both hnd sides of this inequlit seprtel. In this w x x ft dt > : L p. l inserting the expression of the function px > : x x x x pl x } ft dt mking the chnge of vrible z = l x in the exterior integrl > : chnging the limits of integrtion > : l l z l z z z chnging the vrible t = l in the interior integrl > : px } ft dt pzdz } ft dt pzdz } ft dt z pzdz } fl d l z l inserting g = f nd chnging the limits of integrtion in the interior integrl > : l z using the definition of vrible exponent norm = l z z z pzdz } g d gt dt. L p. l Appling the inequlit.8 we get tht is exceeded: C f L p. l using the definition of vrible exponent norm

5 6 On some Hrd-Sobolev s tpe vrible exponent inequlit nd its ppliction > : inserting the expression of the function px > : fx px } fx pl x } mking the chnge of vrible z = l x in the exterior integrl > : chnging the limits of integrtion b nottion gz = fl z > : l > : fl z fl z using the definition of vrible exponent norm = C g. p.;l. pzdz } pzdz } gz pzdz } Therefore the inequlit. hs been estblished which completes the proof of inequlit.5 since g is rbitrr. Define Lip l clss of Lipshitsz continuous functions f : l R such tht f = fl =. Define norm f = f L l + f L p. l in this clss nd close it in this norm. The obtined vrible exponent spce denote s Ẇ p. l. Spce Ẇ p. l. is reflexive Bnch spce if < p p + < see e.g. []. Consider the Dirichlet problem d d px 2 d + xl x = l = px xl x = F x. where > is prmeter F = f x xl x + df with f f L p l. Let px be mesurble function stisfing Hrd s tpe inequlit. to hold e.g. it is stisfied conditions.6.9 nd px increses ner origin decreses ner l.

6 F.I. Mmedov S.M. Mmmdli Y. Zeren 7 To prove the existence of solution of problem. we shll use monotone opertor method. To crr out its insert n opertor A : X X where X = Ẇp. l X its dul spce. Insert n opertor A : u X Au X. We define it s following < Au ϕ > = d px 2 d dϕ + px ϕ ϕ X. xl x.2 We s = x is solution of problem. if i.e. d = < A ϕ > = F ϕ ϕ X px 2 d dϕ + f x ϕx xl x px xl x f xϕ x. In order to crr out the monotone opertor method pproch we hve to show tht the opertor A is monotone propert tht is A A 2 2 <A> 2 coercivit tht is s 3 semi continuit tht is < A + k 2 ϕ > < A + 2 ϕ > s k. Verif : for n 2 X on bse of inequlit for ll b R it follows < A A 2 2 >= + p 2 b p 2 b b px 2 2 px xl x px px 2 2 px Verif 2. Let k Ẇ p. l. Then it is esil seen tht k L p. l s k. Indeed if k L p. l C b some C > not depending on k N on bse of Holder s inequlit for p.-norms see [2] we hve k L p. l C k xl x mx xl x L p. l <x<l C l 2 k xl x L p. l on bse of Hrd s tpe inequlit C l k L p. l

7 8 On some Hrd-Sobolev s tpe vrible exponent inequlit nd its ppliction which contrdicts the ssumption k L p. l. Therefore from k Ẇ p. l it follows k L p. l. On other hnd b the Holder s nd Hrd s inequlities it follows tht Therefore to verif the convergence k Ẇ p. l = k L p. l + k L p. l [ ] C l + k L p. l. < A k k > k Ẇ p. l s k it suffices to show tht < A k k > k L p. l s k L p. l. We hve < A k k > k L p. l = s k L p. l. We hve used tht [ k k L p. l k p L p. l px + k k L p. l k p L p. l k xl x k L p. l px k k L p. l px = b definition. Therefore < A k k > k k p L p. l px ] s k provided. The coercivit hs been shown. 3 Verif the semi continuit. Let the number sequence k nd 2 ϕ X. Then b using the Lebesgue mjornt theorem nd the convergence k it follows tht < A + k 2 ϕ >= + d + k + k 2 xl x d + d 2 d 2 px 2 d + k px 2 + k 2 ϕ xl x px 2 d + d 2 dϕ d 2 dϕ

8 F.I. Mmedov S.M. Mmmdli Y. Zeren xl x px ϕ xl x s k. Therefore ll conditions -3 hve been verified. B ssertion of the Browder- Mint theor it follows: there exists unique solution Ẇ p. l of the problem.. Verif f X b using Holder s nd Hrd s inequlities. + F ϕ = f xl x ϕ f xϕ x f ϕ xl x f ϕ f L ϕ p. l xl x f L L p. p l. l ϕ L p. l C f L p l + f L p l ϕ L p. l C f L p l + f L p l ϕ Ẇ p. l. Hence F X if f f L p. l. To show the uniqueness ppl the monotone condition on opertor A. Let 2 re two solutions of the problem. from spce Ẇ p. l. Then + d px 2 d d 2 px 2 d 2 d d 2 px 2 2 px =. Tht implies = 2. Hence it hs been proved the following result on the solvbilit of the Dirichlet problem.. Theorem.5 Let p : l be mesurble function tht is monotone incresing ner the origin nd decresing ner l. Moreover re stisfied the conditions.6.4 for the function p. Then there exists unique solution of the Dirichlet problem. from spce l for n mesurble functions f f L p l nd. Ẇ p. References. Diening L. Hrjulehto P. Hsto P. Ruzick M.: Lebesgue nd Sobolev Spces with Vrible Exponents 27 of Lecture Notes in Mthemtics Springer Heidelberg Germn Cruz-Uribe D. Fiorenz A.: Vrible Lebesgue Spces Fondtion nd Hrmonic Anlsis Birkh.user 23.

9 On some Hrd-Sobolev s tpe vrible exponent inequlit nd its ppliction 3. Mmedov F.I. Mmmdov F.: A necessr nd sufficient condition for Hrd s opertor in L p. Mth. Nch Mmedov F.I. Zeren Y.: A necessr nd sufficient condition for Hrds opertor in the vrible Lebesgue spce Abst. Appl. Anl. 5/ pges. 5. Mmedov F.I. Mmmdov F. Aliev M.: Boundedness criterions for the Hrd opertor in weighted L p. l spce J. Conv. Anl