ON INDIRECT APPROACH TO THE SOLVABILITY OF QUASI-LINEAR DIRICHLET ELLIPTIC BOUNDARY VALUE PROBLEM WITH BMO-ANISOTROPIC P-LAPLACIAN

We study here Dirichlet boundary value problem for a quasi-linear elliptic equation with anisotropic p-Laplace operator in its principle part and L1-control in coe cient of the low-order term. As characteristic feature of such problem is a speci cation of the matrix of anisotropy A = Asym + Askew in BMO-space. Since we cannot expect to have a solution of the state equation in the classical Sobolev space W 1,p 0 (Ω), we specify a suitable functional class in which we look for solutions and prove existence of weak solutions in the sense of Minty using a non standard approximation procedure and compactness arguments in variable spaces.

1. Introduction sn this pper we del with the following oundry vlue prolem −∆ p (A, y) + |y| p−2 yu = − div f in Ω, y = 0 on ∂Ω, u ∈ L 1 (Ω), u(x) ≥ 0 FeF in Ω , @IFIA where −∆ p (A, y) = − div |(∇y, A∇y)| p−2 2 A∇y @IFPA is the nisotropi pEvplinD 2 ≤ p < +∞D A is the mtrix of nisotropyD y d ∈ L 2 (Ω) nd f ∈ L ∞ (Ω; R N ) re given distriutionsF he interest to ellipti equtions whose prinipl prt is n nisotropi pE vple opertor rises from vrious pplied ontexts relted to omposite mteE rils suh s nonliner dieletri ompositesD whose nonliner ehvior is modeled y the soElled powerElow @seeD for instneD ID PI nd referenes thereinAF prom mthemtil point of viewD the interest of nisotropi pEvplin lies on its * heprtment of hi'erentil iqutionsD yles ronhr hnipro xtionl niversityD UPD qgrin vFD hniproD RWHIHD krineD p.kogut@i.ua † heprtment of ystem enlysis nd gontrolD xtionl wining niversityD IWD vornitskii vFD RWHHS hnipro ‡ snstitute of epplied ystem enlysisD xtionl edemy of ienes nd winistry of idution nd iene of krineD QUGQSD eremogy vFD seeD HQHST uyivD krineD kupenko.olga@gmail.com F sF uogutD yF F uupenkoD PHIVF nonlinerity nd n e'et of degeneryD whih turns out to e the mjor di'erene from the stndrd vplin on R N F es hrteristi feture of oundry vlue prolem @IFIA is spei(tion of the mtrix of nisotropy A = B + DD where B := A sym = (A+A t )/2 nd D := A skew = (A−A t )/2D nd the ontrol u ∈ L 1 (Ω)F sn prtiulrD we ssume tht the mtrix A is suh tht where α, β ∈ L 1 (Ω)D β(x) ≥ α(x) ≥ 0 lmost everywhere in ΩD α ∈ L ∞ (Ω)D α −1 ∈ L 1 (Ω)D nd αD α −1 D nd β extended y zero outside of Ω re in BM O(R N )F e note tht these ssumptions on the lss of dmissile mtries re essenE tilly weker thn they usully re in the literture @seeD for instneD VD WD IID IWD PHAF sn ftD we del with the hirihlet oundry vlue prolem @fA for degenerte nisotropi ellipti eqution with unounded oe0ients in its prinipl prt nd with L 1 Eounded ontrol in the oe0ient of the lowEorder termF st is wellEknown tht suh f n exhiit the soElled vvrentie' phenoE menonD nonEuniqueness of the wek solutions s well s other surprising onsequenE es @seeD for instneD PDRAF es resultD the existeneD uniquenessD nd vritionl properties of the wek solution to the ove f usully re drstilly di'erent from the orresponding properties of solutions to the ellipti equtions with oerE ive L ∞ Emtries of nisotropy @we refer to TD PT!PVD QI for the detils nd other results in this (eldAF enother distinguishing feture of the oundry vlue prolem @QFIA!@QFPA is the ft tht the skewEsymmetri prt D of the mtrix A is merely mesurle nd its suEmultiplitive norm elongs to the BM OEspe @rther thn the spe L ∞ Ω AF his irumstne n entil numer of pthologies with respet to the stndrd properties of fs for ellipti equtions with nisotropi pEvplin even with 9 good9 symmetri prt A nd smooth rightEhnd side f F sn prtiulrD the unoundedness of the skewEsymmetri prt of mtrix A ∈ M ad n hve re)etion in nonEuniqueness of wek solutions to the orresponding oundry vlue prolemF por more detils nd other types of solutions to ellipti equtions with unounded oe0ients we refer to UD IR!ITD QQF oD in ontrst to the pper QPD where the uthor onsider the se of wellEposed hirihlet oundry vlue prolem for qusiEliner ellipti eqution with unounded oe0ients in its prinipl prtD we del with n illEposed oundry vlue prolemF e introdue speil funtionl spe X u,B relted to given ontrol u nd symmetri prt B of mtrix AD nd prove @see heorem RFIA tht the originl oundry vlue prolem dmits wek solutions in the sense of wintyF woreoverD we show tht for every ontrol u ∈ L 1 (Ω)D wek solutions @in the sense of wintyA to the orresponding f n e otined s the limit of solutions to oerive prolems with ounded oe0ientsD using ny L ∞ Epproximtion of BM OEmtrix AF uh solutions re lled pproximtion solutions in QQF heir hrteristi feture is the ft tht they ly in vrile spe X u,B ndD in generlD do not stisfy the energy equlity ut rther some energy inequlityF e lso derive priori estimtes for suh solutions tht do not depend on the skewEsymmetri prt D of mtrix AF es iEprodut of our pprohD we derive the onditions gurnteeing the equlity H 1,p 0,B (Ω) = W 1,p 0,B (Ω)D iFeF we estlish the density of smooth omptly supported funtions in W 1,p 0,B (Ω)F

Notation and Preliminaries
vet Ω e ounded open suset of R N @N ≥ 1A with vipshitz oundryF vet p e rel numer suh tht 2 ≤ p < ∞D nd let q = p/(p − 1) e the onjugte of pF vet M N e the set of ll N × N rel mtriesF e denote y S N skew nd S N sym the set of ll skewEsymmetri nd symmetri mtriesD respetivelyF e lwys identify eh mtrix A ∈ M N with the deomposition A = B + DD where B := 1 2 A + A t ∈ S N sym nd D := 1 2 A − A t ∈ S N skew F woreoverD pplying the gholesky method to the symmetri prt of mtrix A @see ssson nd ueller QHAD we dedue the existene of lower tringulr mtrix L suh tht B( is ll entered t x nd of rdius (Q) = rD nd the supremum is tken over ll lls Q ⊂ R N F yviouslyD L ∞ (R N ) ⊂ BM O(R N )F es n exmple of unounded funtion in BM O(R N )D one n tke ln |x|F por our further nlysisD we mke use of the following resultX if g ∈ BM O(R N ) then the tohnExirenerg estimte = L 1 Ω; S N sym e the spe of mesurle solutely integE rle funtions whose vlues re symmetri mtriesF fy BM O(Ω; S N skew ) we denote the spe of ll skewEsymmetri mtries D = [d ij ] @theEsoElled mtries of ounded men osilltionA suh tht D ∈ L 1 (Ω; S N skew ) nd their suEmultipliE tive norm extended y zero to the entire R N is in BM O(R N )F he similr spei(tion holds for the spe BM O(Ω; M N )F Matrices with Degenerate EigenvaluesF vet αD β e given elements of L 1 (Ω) stisfying the onditions . @PFQA Remark PFI. es immeditely follows from the tohnExirenerg estimte @PFIA nd ssumption @PFQAD we hve hold true for α nd βF oD we n suppose tht α, α −1 , β ∈ L r (Ω) for ll r ≥ 1 provided the onditions @PFPA!@PFQA hold trueF e de(ne the lss of mtries M ad s follows

Setting of the Boundary Value Problem
where we dopt u s given ontrol funtionF st is worth to notie thtD in view of the de(nition of the set M ad D we del with oundry vlue prolem for degenerte qusiEliner ellipti eqution with singulr oe0ientsF st mens tht even for symmetri mtries of oe0ients A ∈ M ad this prolem n exhiit the vvrentie' phenomenon @iFeF W 1,p 0,B (Ω) = H 1,p 0,B (Ω)A ndD s onsequeneD nonEuniqueness of the wek solutionsF husD the originl oundry vlue prolem @QFIA!@QFPA is illEposedD in generlF he nother distinguishing feture of the oundry vlue prolem @QFIA! @QFPA is the ft tht the skewEsymmetri prt D of the mtrix A ∈ M ad is merely mesurle nd elongs to the spe BM O Ω; M N @rther thn the spe of ounded mtries L ∞ Ω; M N AF his irumstne n entil numer of pthologies with respet to the stndrd properties of fs for ellipti equtions with nisotropi pEvplin even with 9 good9 symmetri prt B of A nd smooth rightEhnd side f F sn prtiulrD the unoundedness of the skewEsymmetri prt of mtrix A ∈ M ad n hve re)etion in nonEuniqueness of wek solutions to the orresponding oundry vlue prolemF por more detils nd other types of solutions to ellipti equtions with unounded oe0ients we refer to UDIR!ITDQQF e ssoite to the oundry vlue prolem @QFIA!@QFPA the following spe Denition 3.1. e sy thtD for (xed ontrol u nd given distriutions A ∈ M ad D nd f ∈ L ∞ (Ω) N D funtion y = y(A, u, f ) is wek solution @in the sense of wintyA to oundry vlue prolem @QFIA!@QFPA if y ∈ X u,B nd the inequlitŷ holds for ny ϕ ∈ C ∞ 0 (Ω)F o egin withD let us show tht this de(nition mkes senseF sndeedD y the initil ssumptions nd r¤ older9s inequlityD we hvê es for the (rst term in @QFRAD we oserve tht husD the well posedness of eh term in the vritionl inequlity @QFRA ndD heneD the onsisteny of the de(nition of the wek solution in the sense of winty to the onsidered oundry vlue prolemD oviously follows from the estimtes @QFSAE@QFTAD @QFVAF Remark QFI. he estimte @QFVA nd the ft tht (∇ϕ(x), D(x)∇ϕ(x)) = 0 FeF in Ω y the skewEsymmetry property of DD imply tht the vritionl inequlity @QFRA n e rewritten s followŝ qetting inspired y thisD we ll funtion y ∈ X u,B wek solution @in the sense of wintyA to oundry vlue prolem @QFIA!@QFPA if it stis(es the inequlity @QFWA for every test funtion ϕ ∈ C ∞ 0 (Ω)F king this remrk into ountD it is resonle to onsider nother de(nition of the wek solution to the given oundry vlue prolemD in the terms of distriuE tionsD whih ppers more nturlX sn spite of the ft tht the reltions etween these de(nitions re very intrite for generl mtrix A ∈ M ad @for n exmple when these de(nitions led to the di'erent solutions even for liner equtionsD we refer to PSAD we n leverge the integrl identity @QFIHA for the following estimte ˆΩ |(∇y, B∇y)| p−2 2 (A∇y, ∇ϕ) dx where {ϕ k } k∈N ⊂ C ∞ 0 (Ω) nd ϕ k → z strongly in X u,B @it is the se when we essentilly use the ft tht C ∞ 0 (Ω) is dense in H 1,p 0,B (Ω) ∩ L p (Ω, u dx)AF sn prtiulrD if y ∈ D(X u,B )D then we n de(ne the vlue [y, y] A nd this one is (nite for every y ∈ D(X u,B )D lthough the 4integrnd4 |(∇y, B∇y)| needs not e integrle on ΩD in generlF es resultD we n derive form @QFIHA the energy equlity for distriutionl solutions Remark RFI. he simplest wy to onstrut sequene {A k } k∈N ⊂ M ad (Ω)D possessing the properties @RFIA!@RFQAD is to set or pply the proedure of the diret teklov smoothing to given mtrix A ∈ M ad (Ω) with some positive omptly supported smooth kernel @seeD for instneD ISAF Proof. he onditions @RFIA!@RFQA ensure tht B −1 k ∈ L ∞ (Ω; S N sym ) for ll k ∈ N nd @up to susequeneA where the matrices T k and T are dened by @RFSA.
Proof. sndeedD y de(nition of the spe L q (Ω, y @PFRA ≤ const < +∞. @RFVA yn epproximtion of yg in goe0ients for pEfihrmoni iqution PS reneD the sequene {v k } k∈N is ounded in L q (Ω, B k dx) N F purther we notie thtD y the initil ssumption @RFPAD vemm RFID nd BM OE properties of the mtries LD L −1 D nd DD we see tht the sequene for ny ϕ ∈ C ∞ 0 (Ω)F reneD y veesgue9s heoremD we hve the strong onvergene is equiEintegrleF yn the other hndD property @RFPA nd vemm RFI imply thtD within susequeneD hereforeD the equlity @RFIIA is diret onsequene of veesgue hominted heoremF husD the strong onvergene in vrile spe L q (Ω, B k dx) N of the sequene {v k } k∈N is estlishedF he property @RFUA n e proved following the sme rgumentsF por our further nlysisD we mke use of the following oneptF Denition 4.1. e sy tht ounded sequene sn order to motivte this de(nitionD we give the following resultF k∈N be a sequence with the following properties: sup k∈NˆΩ u|y k | p + (∇y k , B k ∇y k ) p 2 dx < +∞; @RFIQA Then, within a subsequence, the original sequence is w-convergent. Moreover, each w-limit pair (A, y) belongs to the set M ad (Ω) × H 1,p 0,B (Ω) ∩ L p (Ω, u dx) .
Proof. o egin withD we note tht the onditions @iA!@iiA nd estimtes @PFIPA! @PFIQA immeditely imply the oundedness of the sequene in W 1,1 (Ω; M N ) nd in vrile spes H 1,p 0,B k (Ω) nd L p (Ω, u dx)F woreoverD due to the inequlities @PFIRA!@PFISAD we hve the ompt emedding ine p * s = N ps N −ps > p provided s > N p D it follows tht the sequene {y k } k∈N is ompt with respet to the norm topology of L p (Ω)F husD omining this ft with the omptness riterium for the wek onverE gene in vrile spes @see roposition PFIAD we n dedue the existene of pir (y, z) ∈ L p (Ω) × L p (Ω, u dx) × L p (Ω, B dx) N suh thtD within susequene of {y k } k∈N D we hve yur im is to show tht y = zD v = ∇yD nd s onsequene y ∈ H 1,p 0,B (Ω) ∩ L p (Ω, u dx)F ith tht in mindD we note tht for every mesurle suset K ⊂ ΩD the estimtê implies equiEintegrility of the fmily {|∇y k | R N }F gomining this ft with esE timte @PFIQA nd property @iiAD we dedue tht the sequene {|∇y k |} k∈N is wekly ompt in L 1 (Ω)F ineD for n ritrry ξ ∈ C ∞ 0 (Ω) N D we hve husD in view of the wek omptness property of {∇y k } k∈N in L 1 (Ω) N D we onlude ∇y k v in L 1 (Ω; R N ) s n → ∞. @RFIVA ine y k ∈ W 1,1 (Ω) for ll k ∈ N nd the oolev spe W 1,1 (Ω) is ompleteD @RFIRA nd @RFIVA imply ∇y = vD nd onsequently y ∈ H 1,p 0,B (Ω)F o end the proofD it remins to estlish the equlity y = z FeF in ΩF ine the sequene {y k ∈ L p (Ω, u dx)} k∈N is ounded nd for ny mesurle set K ⊆ ΩD we hveˆK it follows tht the sequene {y k u} k∈N is equiEintegrle nd wekly ompt in L 1 (Ω) ndD heneD the wek onvergene @RFISA is equivlent to the wek onvergene y k u zu in L 1 (Ω). @RFIWA purtherD we note tht y wz9y inequlity @PFPQAF ine the set C ∞ 0 (Ω) is dense in H 1,p 0,B (Ω)D it follows tht the fmily {u(y k − y)} k∈N is wekly ompt in L 1 (Ω)F king into ount the omptness of the emedding H 1,p 0,B (Ω) → L p (Ω) nd the wek onvergene y k y in L p (Ω)D we n suppose tht y k → y lmost everywhere in ΩF reneD u(y k − y) → 0 FeF in ΩF hen the strong onvergene u(y k − y) → 0 in L 1 (Ω) immeditely follows from the veesgue heoremF husD in order to onlude the desired equlity y = zD it is enough to omine this inferene with the property @RFIWAF he proof is ompleteF e re now in position to prove the min result of this setionF xmelyD we show tht the oundry vlue prolem @QFIA!@QFQA dmits wek solutionF Theorem 4.1. For given f ∈ L ∞ (Ω) N , u ∈ L 1 (Ω), u ≥ 0 a.e. in Ω, γ > 0, and for an arbitrary matrix A ∈ M ad , there exists a weak solution y ∈ X u,B (in the sense of Minty) to boundary value problem @QFIA@QFPA with an a priori estimate 1 p @RFPHA and the energy relation Ω |(∇y, B∇y)| p 2 dx +ˆΩ |y| p u dx ≤ˆΩ(f, ∇y) dx. @RFPIA Proof. vet u ∈ U ad e n ritrry dmissile ontrolF por given mtrix A ∈ M ad let us onsider n pproximtion {A k } k∈N ⊂ M ad (Ω) with properties @RFIA!@RFQAD nd the orresponding vritionl prolem pind y k ∈ W 1,p 0 (Ω) suh tht @RFPPA ine A k ∈ L ∞ (Ω; M N )D it follows tht (∇y k , A k ∇y k ) = (∇y k , B k ∇y k )F reneD y the wellEknown result of qusiEliner ellipti equtions @see PWD heorem PFIRAD for every k ∈ ND the prolem @RFPPA dmits unique wek solution y k ∈ W 1,p 0 (Ω) suh thtˆΩ @RFPRA st is ler tht the energy equlity @RFPQA leds to the following estimte ndD y vemm RFQD we n suppose the existene of n element y ∈ X u,B suh tht @within susequeneA y is sujeted to the estimte @RFPHA nd y k y in L p (Ω, u dx), @RFPTA ∇y k ∇y in the vrile spe L p (Ω, B k dx) N . @RFPUA e re now in position to pss to the limit in @RFPRA s k → ∞F ith tht in mind we mke use of vemm RFPF sn prtiulrD we utilize the properties @RFTA! @RFUAF henD it follows from he(nition PFP nd @RFPTA!@RFPUA tht y @RFPTA nd de(nition of the wek onvergene in L p (Ω, u dx)D we n pss to the limit in @RFPRA s k → ∞ nd redily otin the desired reltion @QFWAF husD y is wek solution to the oundry vlue prolem @QFIA!@QFQAF es for the energy inequlity @RFPIAD it follows from @RFPQA nd the wek onvergene properties @RFPTA!@RFPUAF Remark RFP. es follows from pproximtion proedure tht ws used in the proof of heorem RFID it lwys leds to some wek solution of the originl oundry vlue prolemF uh solutions re lled pproximtion solutions in QQF he hrteristi feture of suh solutions is the ft tht they stisfy energy inequlity @RFPIA nd their priori estimte @RFPHA does not depend on the skewE symmetri prt D ∈ BM O(Ω; S N skew ) of mtrix A ∈ M ad (Ω)F woreoverD it is unknown in generl whether pproximtion solutions re the wek solutions to the oundry vlue prolem @QFIA!@QFPA in the sense of distriutions nd elong to the set D(X u,B )F