A Note on Some Properties of Unbounded Bilinear Forms Associated with Skew-Symmetric L q (Ω) -Matrices

. We study the bilinear forms on the space of measurable p -integrable functions which are generated by skew-symmetric matrices with unbounded coe(cid:30)cients. We give an example showing that if a skew-symmetric matrix contains a locally unbounded L q -elements, then the corresponding quadratic forms can be alternating. These questions are closely related to the existence issues of the Nuemann boundary value problem for p -Laplace elliptic equations with non-symmetric and locally unbounded anisotropic di(cid:27)usion matrices.

Êëþ÷îâi ñëîâà: êîñî-ñèìåòðè÷íà ìàòðèöÿ, íåîáìåaeåíà áiëiíiéíà ôîðìà MSC2020: Pri 15A63, Sec 11E16, 11E39 IF sntrodution his xote hs een minly motivted y rillint pper of FF hikov TD where the uthor gve n exmple of the hirihlet prolem for seondEorder ellipti equtions with lowerEorder terms @the soElled inompressile di'usion equtionA whose solution is nonEuniqueF he priniple point of this exmple is the ft tht suh eqution my hve two sorts of solutionsF xmelyD the (rst type indites solutions tht n e otined vi orresponding pproximtion proedures wheres solutions of the seond type lk this propertyF e typil F uyq exmple of the oundry vlue prolem with the ove mentioned properties is the following one where the mtrix funtion A(x) is mesurle nd skewEsymmetriF sn ftD s it ws shown lter this phenomenon plys ruil role in investiE gtion of wide lss of optiml ontrol prolems ssoited with the hirihlet oundry vlue prolem for the equtions like @IFIA @seeD for instneD I!S nd the referenes thereinAF ht9s why the min intention in this pper is to exmine the key properties of the iliner forms on the L p Espe of mesurle funtionsD whih re generted y skewEsymmetri mtries with unounded oe0ients nd n e ssoited with xuemnn oundry vlue prolems for pEvple equtionsF sn prtiulrD we show tht there exists skewEsymmetri mtrix A ∈ L q (Ω; R N ×N ) nd funtion y * ∈ W 1,p (Ω)D with p −1 + q −1 = 1 nd p ≥ 2D suh tht for given vlue α > 0 the following equlity holds true

PF reliminries
vet Ω e ounded open onneted suset of R N @N ≥ 2A with vipshitz oundry ∂ΩF vet M N e the set of ll N × N rel mtriesF e denote y sym e the set of ll N × N symmetri mtriesF fy mtrix norm in M N @nd for funtions with vlues in S N sym nd S N skew s wellA we men suEmultiplitive norm A := sup |Aξ| : ξ ∈ R N with |ξ| = 1 F sn prtiulrD it is ler tht st is worth lso to note thtD in the se of iuliden norm is the lrgest eigenvlue of the positiveEsemide(nite mtrix A t AF vet p > 2 e given rel numer nd 1 < q < ∞ e its onjugte suh tht 1 p + 1 q = 1F vet L q Ω; S N skew e the normed spe of mesurle integrle with the power q funtions whose vlues re skewEsymmetri mtriesF vet A skew ∈ L q Ω; S N skew e given mtrixF sn wht followsD we ssoite with A skew the iliner skewEsymmetri form st is esy to seeD tht the form Φ A skew (y, v) is unounded on W 1,p (Ω)D in generlF roweverD if we temporry ssume tht A skew ∈ L ∞ (Ω; S N skew )D then the iliner form Φ A skew (•, •) eomes ounded on W 1,p (Ω)F sndeedD in view of the r¤ older inequlityD we get woreoverD in this seD if ξ = η then R N is not integrle on Ω provided ϕ, y ∈ W 1,p (Ω)F his motivtes to introdue of the following setF e sy tht vetorEvlued funtion ξ ∈ L p (Ω; R N ) elongs to the set with some onstnt c depending on ξ nd A skew F es resultD we oserve tht the iliner form Φ A skew (ξ, η) n e de(ned for ll η ∈ L p (Ω; R N ) using the stndrd rule where R N need not to e integrleD in generlF his ft leds us to the onlusion
st is esy to see tht the rnge of atan2(y, with respet to the spheril oordintesF reneD v ∈ C ∞ (∂Ω)D ndD s immediE tely follows from @QFQAD it provides tht fy diret lultion we get F uyq es resultD we infer tht ∇y * ∈ L 2 (Ω; R 3 )D ndD thereforeD y * ∈ H 1 (Ω)F woreoverD we mke use of the following oservtionF ine where ν stnds for the unit outwrd norml to ∂ΩF tep QF vet us show tht the funtion y * D tht hs een introdued eforeD elongs to the set D(A skew )F o do soD we hve to prove the existene of onstnt C(y * ) depending on y * suh tht the estimte ) e n ritrry funtion stisfying the oundry ondition (A skew ∇ψ, ν) R 3 = 0 on ∂Ω.@QFUA henD tking into ount the ft tht div A skew ∈ L 1 (Ω; R 3 )D we n perform the following trnsformtions whih re oviously true for ll ψ, ϕ ∈ C ∞ c (R 3 ) with property @QFUAF rereiE nfterD we understnd the expression div A skew ∇ψ in the sense of the following slr produt div it follows tht the iliner form Ω ∇ϕ, A skew ∇ψ R 3 dx n e de(ned for ll ψ ∈ L 2 (Ω; R 3 ) using the stndrd ontinution prinipleF es resultD we n extend the equlity with respet to ψ with property @QFUA to the following one whih mens tht the y * elongs to the set D(A skew )F sndeedD s immeditely follows from @QFRAD we hve the equlity @QFWA husD the grdient of the funtion v( x |x| ) is orthogonl to the vetor (eld Q = x/|x| 3 outside the originF hereforeD where I 2 = 0 y @QFWAF ine nd v is smooth funtionD it follows tht there exists onstnt C 0 > 0 suh tht |div A skew ∇y * | ≤ C 0 lmost everywhere in ΩF husD div A skew ∇y * ∈ L ∞ (Ω) ndD thereforeD we n suppose tht the element y * elongs to the set D(A skew )F tep RF sing results of the previous stepsD let us show tht the nnouned equlity Φ A skew (∇y * , ∇y * ) = − α 2 < 0 holdsF sndeedD let {ϕ ε } ε→0 ⊂ C ∞ c (R 3 ) e sequene suh tht ϕ ε → y * strongly in H 1 (Ω) nd eh element of this sequene stis(es the oundry ondition (A skew ∇ϕ ε , ν) R 3 = 0 on ∂ΩF hen y ontinuityD we hve ine div A skew ∇y * ∈ L ∞ (Ω) nd ϕ ε → y * strongly in H 1 (Ω)D we n pss to the limit in the rightEhnd side of this reltionF es resultD we get where n ε = −ν stnds to the unit outwrd norml to Γ ε D it follows tht J 1 = 0 euse y * = 0 on ∂ΩD nd J 3 = 0 in view of the following property whereD in the terms of spheril oordintesD .
st remins to omine this ft with @QFIPA nd to notie tht

Ω
div A skew ∇ (y * ) 2 dx = lim ε→0 Ωε div A skew ∇ (y * ) 2 dx.for ll x ∈ ΩD nd hoosing the skewEsymmetri mtrix A skew (x) in the form @QFPAD we dedue from representtion @QFIIA thtΦ A skew (∇y * , ∇y * ) = −α < 0for ny ritrry given positive vlue αF Remark I. yne of the hrteristi fetures of the given exmple is the ft tht the determinnt of the skewEsymmetri mtrix A skew (x) vnishes lmost everywhere in ΩD whih is the typil property of ny skewEsymmetri mtrix in R N ×N with n odd order N F oD we leve the onstrution of similr exmple in the se of even dimensionlity x s n open prolemF