In this paper,some existence results for the fourth order nonlinear subelliptic equations on the Heisenberg group are given by means of variational methods.
In this paper,the following result is given by using Hodge decomposition: There exists r(0) = r(0)(n,p,a,b), such that if u is an element of W-loc(1,r)(Omega) is a very weak solution of (1.1),with C max{1,p - 1} < ...In this paper,the following result is given by using Hodge decomposition: There exists r(0) = r(0)(n,p,a,b), such that if u is an element of W-loc(1,r)(Omega) is a very weak solution of (1.1),with C max{1,p - 1} < r < p and u is an element of W-0(1,r)(Omega;partial derivativeOmega\E) where E subset of partial derivativeOmega is a closed set and small in an appropriate capacity sense, then u = 0, a.e. in Omega provided that r(0) < r < p.展开更多
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
Extremum principle for very weak solutions of A-harmonic equation div A(x,▽u)=0 is obtained, where the operator A:Ω × Rn→Rnsatisfies some coercivity and controllable growth conditions with Mucken-houpt weight.
We prove a uniform Harnack μu = 0, where △G is a sublaplacian, μ is scale-invariant Kato condition. inequality for nonnegative solutions of △u - a non-negative Radon measure and satisfying
A Caccioppoli type estimate is established for a class of second order PDEs of divergence type, and its removable singularities of Hausdorff dimension greater than zero is obtained.
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
De Giorgi猜想起源于Bernstain提出的一个著名的几何问题:在小于8维的全空间中,方程△u-u+u^3=0的单调解是否退化成一维方程的解,这就是所谓的解的一维对称性问题.Birindelli关于Heisenberg群上次Laplace方程解的一维对称性做了大量工作...De Giorgi猜想起源于Bernstain提出的一个著名的几何问题:在小于8维的全空间中,方程△u-u+u^3=0的单调解是否退化成一维方程的解,这就是所谓的解的一维对称性问题.Birindelli关于Heisenberg群上次Laplace方程解的一维对称性做了大量工作.利用Heisenberg型群的左平移不变性构造平移参数族,用平移的方法将欧氏空间半线性椭圆方程解的一维对称性结果推广到了Heisenberg型群上.展开更多
This paper gives the local regularity result for solutions to obstacle problems of A-harmonic equation divA(x, ξu(x)) = 0, |A.(x,ξ)|≈|?|p-1, when 1 < p < n and the obstacle function (?)≥0.
In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cut-off.This equation is partially elliptic in the velocity direction and degenerates in the...In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cut-off.This equation is partially elliptic in the velocity direction and degenerates in the spatial variable.We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in the Gevrey class at positive time with the Gevrey index depending on the angular singularity.Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for the Cauchy problem of the linearized Boltzmann operator.展开更多
We obtain a local regularity result for solutions to kφ,θ-obstacle problem of A-harmonic equation divA（x, u（x）, ↓△u（x）） = 0, where .A ： Ω ×R × Rn → Rn is aCarath^odory function satisfying some c...We obtain a local regularity result for solutions to kφ,θ-obstacle problem of A-harmonic equation divA（x, u（x）, ↓△u（x）） = 0, where .A ： Ω ×R × Rn → Rn is aCarath^odory function satisfying some coercivity and growth conditions with the naturalexponent 1 〈 p 〈 n, the obstacle function φ≥ 0, and the boundary data θ ∈ W1mp（Ω）.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
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
文摘In this paper,some existence results for the fourth order nonlinear subelliptic equations on the Heisenberg group are given by means of variational methods.
文摘In this paper,the following result is given by using Hodge decomposition: There exists r(0) = r(0)(n,p,a,b), such that if u is an element of W-loc(1,r)(Omega) is a very weak solution of (1.1),with C max{1,p - 1} < r < p and u is an element of W-0(1,r)(Omega;partial derivativeOmega\E) where E subset of partial derivativeOmega is a closed set and small in an appropriate capacity sense, then u = 0, a.e. in Omega provided that r(0) < r < p.
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.
文摘Extremum principle for very weak solutions of A-harmonic equation div A(x,▽u)=0 is obtained, where the operator A:Ω × Rn→Rnsatisfies some coercivity and controllable growth conditions with Mucken-houpt weight.
文摘We prove a uniform Harnack μu = 0, where △G is a sublaplacian, μ is scale-invariant Kato condition. inequality for nonnegative solutions of △u - a non-negative Radon measure and satisfying
基金Supported by National Natural Science Foundation (No.49805005)partially by Research Foundation of Northern Jiaotong University (2002SM061)
文摘A Caccioppoli type estimate is established for a class of second order PDEs of divergence type, and its removable singularities of Hausdorff dimension greater than zero is obtained.
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.
文摘This paper gives the local regularity result for solutions to obstacle problems of A-harmonic equation divA(x, ξu(x)) = 0, |A.(x,ξ)|≈|?|p-1, when 1 < p < n and the obstacle function (?)≥0.
基金supported by National Natural Science Foundation of China(Grant No.11631011)supported by National Natural Science Foundation of China(Grant Nos.11961160716,11871054 and 11771342)+1 种基金the Natural Science Foundation of Hubei Province(Grant No.2019CFA007)the Fundamental Research Funds for the Central Universities(Grant No.2042020kf0210)。
文摘In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cut-off.This equation is partially elliptic in the velocity direction and degenerates in the spatial variable.We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in the Gevrey class at positive time with the Gevrey index depending on the angular singularity.Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for the Cauchy problem of the linearized Boltzmann operator.
基金supported by NSF of Hebei Province (07M003)supported by NSFC (10771195)NSF of Zhejiang Province(Y607128)
文摘We obtain a local regularity result for solutions to kφ,θ-obstacle problem of A-harmonic equation divA（x, u（x）, ↓△u（x）） = 0, where .A ： Ω ×R × Rn → Rn is aCarath^odory function satisfying some coercivity and growth conditions with the naturalexponent 1 〈 p 〈 n, the obstacle function φ≥ 0, and the boundary data θ ∈ W1mp（Ω）.
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.
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.
