In this paper, the following result is given by using Hodge decomposition and weak reverse Holder inequality： For every r1 with P-（2^n＋1 100^n^2 p（2^3＋n/（P-1）＋1）b/a）^-1〈r1〈p,there exists the exponent r2 =...In this paper, the following result is given by using Hodge decomposition and weak reverse Holder inequality： For every r1 with P-（2^n＋1 100^n^2 p（2^3＋n/（P-1）＋1）b/a）^-1〈r1〈p,there exists the exponent r2 = r2（n, r1,p） 〉 p, such that for every very weak solution u∈W^1r1,loc（Ω） to A-harmonic equation, u also belongs to W^1r2,loc（Ω） . In particular, u is the weak solution to A-harmonic equation in the usual sense.展开更多
For Ω a bounded subset of R n,n 2,ψ any function in Ω with values in R∪{±∞}andθ∈W1,(q i)(Ω),let K(q i)ψ,θ(Ω)={v∈W1,(q i)(Ω):vψ,a.e.and v-θ∈W1,(q i)0(Ω}.This paper deals with solutions to K(q i)ψ...For Ω a bounded subset of R n,n 2,ψ any function in Ω with values in R∪{±∞}andθ∈W1,(q i)(Ω),let K(q i)ψ,θ(Ω)={v∈W1,(q i)(Ω):vψ,a.e.and v-θ∈W1,(q i)0(Ω}.This paper deals with solutions to K(q i)ψ,θ-obstacle problems for the A-harmonic equation-divA(x,u(x),u(x))=-divf(x)as well as the integral functional I(u;Ω)=Ωf(x,u(x),u(x))dx.Local regularity and local boundedness results are obtained under some coercive and controllable growth conditions on the operator A and some growth conditions on the integrand f.展开更多
This paper deals with the very weak solutions of A-harmonic equation divA（x, u（x））=0 （＊）where the operator A satisfies the monotonicity inequality, the controllable growth condition and the homogeneity conditio...This paper deals with the very weak solutions of A-harmonic equation divA（x, u（x））=0 （＊）where the operator A satisfies the monotonicity inequality, the controllable growth condition and the homogeneity condition. The extremum principle for very weak solutions of A-harmonic equation is derived by using the stability result of Iwaniec-Hodge decomposition： There exists an integrable exponent r1=r1（p,n,β/α）=1/2[p-α/100n^2β＋√（p＋α/100n^2β）^2-4α/100n^2β] such that if u（x） ∈ W^1,r（Ω）is a very weak solution of the A-harmonic equation （＊）, and m ≤ u（x） ≤ M on ЭΩ in the Sobolev sense, then m ≤u（x） 〈 M almost everywhere in Ω, provided that r 〉 r1. As a corollary, we prove that the O-Dirichlet boundary value problem {div_A（x, u（x））=0,u∈W0^1,r（Ω）of the A-harmonic equation has only zero solution if r 〉 r1.展开更多
DenoteκψØ(Ω)={υ∈w1,p(Ω):υ≥ψ,a,e.andυ-Ø∈w1,po(Ω)},where is any function in Q C R^(N),N≥2,with values in RU[±∞]and e is a measurable function.This paper deals with global integrability for u...DenoteκψØ(Ω)={υ∈w1,p(Ω):υ≥ψ,a,e.andυ-Ø∈w1,po(Ω)},where is any function in Q C R^(N),N≥2,with values in RU[±∞]and e is a measurable function.This paper deals with global integrability for u E Kμ,e such that∫Ω﹤Α(χ,▽υ),▽(w-u)﹥dx≥∫Ω﹤f,▽(w-u)dx,■w∈■ψØ(Ω),with/A■≈|■|^(p-1),1<p<N.Some global integrability results are obtained.展开更多
We prove two-Ar^λ（Ω）-weighted imbedding theorems for differential forms. These results can be used to study the weighted norms of the homotopy operator T from the Banach space LV（D, ∧^l） to the Sobolev space W^...We prove two-Ar^λ（Ω）-weighted imbedding theorems for differential forms. These results can be used to study the weighted norms of the homotopy operator T from the Banach space LV（D, ∧^l） to the Sobolev space W^1,p（D, ∧^l-1）, l = 0, 1,..., n, and to establish the weighted L^p-estimates for differential forms. Finally, we give some applications of the above results to quasiregular mappings.展开更多
文摘In this paper, the following result is given by using Hodge decomposition and weak reverse Holder inequality： For every r1 with P-（2^n＋1 100^n^2 p（2^3＋n/（P-1）＋1）b/a）^-1〈r1〈p,there exists the exponent r2 = r2（n, r1,p） 〉 p, such that for every very weak solution u∈W^1r1,loc（Ω） to A-harmonic equation, u also belongs to W^1r2,loc（Ω） . In particular, u is the weak solution to A-harmonic equation in the usual sense.
基金supported by National Natural Science Foundation of China (Grant No. 10971224)Natural Science Foundation of Hebei Province (Grant No. A2011201011)
文摘For Ω a bounded subset of R n,n 2,ψ any function in Ω with values in R∪{±∞}andθ∈W1,(q i)(Ω),let K(q i)ψ,θ(Ω)={v∈W1,(q i)(Ω):vψ,a.e.and v-θ∈W1,(q i)0(Ω}.This paper deals with solutions to K(q i)ψ,θ-obstacle problems for the A-harmonic equation-divA(x,u(x),u(x))=-divf(x)as well as the integral functional I(u;Ω)=Ωf(x,u(x),u(x))dx.Local regularity and local boundedness results are obtained under some coercive and controllable growth conditions on the operator A and some growth conditions on the integrand f.
文摘This paper deals with the very weak solutions of A-harmonic equation divA（x, u（x））=0 （＊）where the operator A satisfies the monotonicity inequality, the controllable growth condition and the homogeneity condition. The extremum principle for very weak solutions of A-harmonic equation is derived by using the stability result of Iwaniec-Hodge decomposition： There exists an integrable exponent r1=r1（p,n,β/α）=1/2[p-α/100n^2β＋√（p＋α/100n^2β）^2-4α/100n^2β] such that if u（x） ∈ W^1,r（Ω）is a very weak solution of the A-harmonic equation （＊）, and m ≤ u（x） ≤ M on ЭΩ in the Sobolev sense, then m ≤u（x） 〈 M almost everywhere in Ω, provided that r 〉 r1. As a corollary, we prove that the O-Dirichlet boundary value problem {div_A（x, u（x））=0,u∈W0^1,r（Ω）of the A-harmonic equation has only zero solution if r 〉 r1.
基金supported by the Postgraduate Innovation Project of Hebei Province(No.CXZZSS2020005)the second author was supported by NSFC(No.12071021),NSF of Hebei Province(No.A2019201120).
文摘DenoteκψØ(Ω)={υ∈w1,p(Ω):υ≥ψ,a,e.andυ-Ø∈w1,po(Ω)},where is any function in Q C R^(N),N≥2,with values in RU[±∞]and e is a measurable function.This paper deals with global integrability for u E Kμ,e such that∫Ω﹤Α(χ,▽υ),▽(w-u)﹥dx≥∫Ω﹤f,▽(w-u)dx,■w∈■ψØ(Ω),with/A■≈|■|^(p-1),1<p<N.Some global integrability results are obtained.
基金The research supported by National Natural Science Foundation of China (A0324610)Scientific Research Foundation of Hebei Polytechnic University (200520).
文摘We prove two-Ar^λ（Ω）-weighted imbedding theorems for differential forms. These results can be used to study the weighted norms of the homotopy operator T from the Banach space LV（D, ∧^l） to the Sobolev space W^1,p（D, ∧^l-1）, l = 0, 1,..., n, and to establish the weighted L^p-estimates for differential forms. Finally, we give some applications of the above results to quasiregular mappings.
