In this paper,we construct a power type functional which is the approximation functional of the Singular Trudinger-Moser functional.Moreover,we obtain the concentration level of the functional and show it converges to...In this paper,we construct a power type functional which is the approximation functional of the Singular Trudinger-Moser functional.Moreover,we obtain the concentration level of the functional and show it converges to the concentration level of singular Trudinger-Moser functional on the unit ball.展开更多
This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽...This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽u)+(▽ω+F)·▽u+f in B^(4),under the smallest regularity assumptions of V,ω,ω,F,where f belongs to some Morrey spaces.This work was motivated by many geometrical problems such as the flow of biharmonic mappings.Our results deepens the Lp type regularity theory of[10],and generalizes the work of Du,Kang and Wang[4]on a second order problem to our fourth order problems.展开更多
In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogene...In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogeneous boundary data problem:-div((s^(2)+|▽u|^(2)p-2/2)▽u)=-div(|f|^(p-2)f)+g inΩ,u=h in■Ω,with the(sub-elliptic)degeneracy condition s∈[0,1]and with mixed data f∈L^(p)(Q;R^(n)),g∈Lp/(p-1)(Ω;R^(n))for p∈(1,n).This problem naturally arises in various applications such as dynamics of non-Newtonian fluid theory,electro-rheology,radiation of heat,plastic moulding and many others.Building on the idea of level-set inequality on fractional maximal distribution functions,it enables us to carry out a global regularity result of the solution via fractional maximal operators.Due to the significance of M_(α)and its relation with Riesz potential,estimates via fractional maximal functions allow us to bound oscillations not only for solution but also its fractional derivatives of orderα.Our approach therefore has its own interest.展开更多
We study equations in divergence form with piecewise Cαcoefficients.The domains contain corners and the discontinuity surfaces are attached to the edges of the corners.We obtain piecewise C^(1,α) estimates across th...We study equations in divergence form with piecewise Cαcoefficients.The domains contain corners and the discontinuity surfaces are attached to the edges of the corners.We obtain piecewise C^(1,α) estimates across the discontinuity surfaces and provide an example to illustrate the issue regarding the regularity at the corners.展开更多
In this paper,we consider the semilinear elliptic equation systems{△u+u=αQ_(n)(x)|u|^(α-2)|v|^(β)u in R^(N),-△v+v=βQ(x)|u|^(α)|v|^(β-2)v in R^(N),where N≥3,α,β>1,α+β<2^(*),2^(*)=2N/N-2 and Q_(n) are...In this paper,we consider the semilinear elliptic equation systems{△u+u=αQ_(n)(x)|u|^(α-2)|v|^(β)u in R^(N),-△v+v=βQ(x)|u|^(α)|v|^(β-2)v in R^(N),where N≥3,α,β>1,α+β<2^(*),2^(*)=2N/N-2 and Q_(n) are bounded given functions whose self-focusing cores{x∈R^(N)|Q_(n)(x)>0} shrink to a set with finitely many points as n→∞.Motivated by the work of Fang and Wang[13],we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points,and we build the localized concentrated bound state solutions for the above equation systems.展开更多
In this paper,by choosing some appropriate test functions,we prove the Weyl’s lemma for triharmonic functions based on the new type of mean value formulas.
In this paper,we study the reiterated homogenization operators Lε=-div(A(x/ε,x/ε^(2))∇).We establish the homogenized problem and representation equa-tion by introducing the two correctors.As a consequence,we obtain...In this paper,we study the reiterated homogenization operators Lε=-div(A(x/ε,x/ε^(2))∇).We establish the homogenized problem and representation equa-tion by introducing the two correctors.As a consequence,we obtain the H10 and L2 convergence estimates of solutions.展开更多
文摘In this paper,we construct a power type functional which is the approximation functional of the Singular Trudinger-Moser functional.Moreover,we obtain the concentration level of the functional and show it converges to the concentration level of singular Trudinger-Moser functional on the unit ball.
基金supported by the National Natural Science Foundation of China(12271296,12271195).
文摘This paper is a continuation of recent work by Guo-Xiang-Zheng[10].We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation △^{2}u=△(V▽u)+div(w▽u)+(▽ω+F)·▽u+f in B^(4),under the smallest regularity assumptions of V,ω,ω,F,where f belongs to some Morrey spaces.This work was motivated by many geometrical problems such as the flow of biharmonic mappings.Our results deepens the Lp type regularity theory of[10],and generalizes the work of Du,Kang and Wang[4]on a second order problem to our fourth order problems.
基金supported by Ministry of Education and Training(Vietnam),under grant number B2023-SPS-01。
文摘In this paper,the study of gradient regularity for solutions of a class of elliptic problems of p-Laplace type is offered.In particular,we prove a global result concerning Lorentz-Morrey regularity of the non-homogeneous boundary data problem:-div((s^(2)+|▽u|^(2)p-2/2)▽u)=-div(|f|^(p-2)f)+g inΩ,u=h in■Ω,with the(sub-elliptic)degeneracy condition s∈[0,1]and with mixed data f∈L^(p)(Q;R^(n)),g∈Lp/(p-1)(Ω;R^(n))for p∈(1,n).This problem naturally arises in various applications such as dynamics of non-Newtonian fluid theory,electro-rheology,radiation of heat,plastic moulding and many others.Building on the idea of level-set inequality on fractional maximal distribution functions,it enables us to carry out a global regularity result of the solution via fractional maximal operators.Due to the significance of M_(α)and its relation with Riesz potential,estimates via fractional maximal functions allow us to bound oscillations not only for solution but also its fractional derivatives of orderα.Our approach therefore has its own interest.
基金supported by National Natural Science Foundation of China(12061080,12161087 and 12261093)the Science and Technology Project of the Education Department of Jiangxi Province(GJJ211601)supported by National Natural Science Foundation of China(11871305).
文摘We study equations in divergence form with piecewise Cαcoefficients.The domains contain corners and the discontinuity surfaces are attached to the edges of the corners.We obtain piecewise C^(1,α) estimates across the discontinuity surfaces and provide an example to illustrate the issue regarding the regularity at the corners.
基金supported by the NSFC (12071438)supported by the NSFC (12201232)
文摘In this paper,we consider the semilinear elliptic equation systems{△u+u=αQ_(n)(x)|u|^(α-2)|v|^(β)u in R^(N),-△v+v=βQ(x)|u|^(α)|v|^(β-2)v in R^(N),where N≥3,α,β>1,α+β<2^(*),2^(*)=2N/N-2 and Q_(n) are bounded given functions whose self-focusing cores{x∈R^(N)|Q_(n)(x)>0} shrink to a set with finitely many points as n→∞.Motivated by the work of Fang and Wang[13],we use variational methods to study the limiting profile of ground state solutions which are concentrated at one point of the set with finitely many points,and we build the localized concentrated bound state solutions for the above equation systems.
基金Supported by National Natural Science Foundation of China(Grant Nos.11801006 and 12071489).
文摘In this paper,by choosing some appropriate test functions,we prove the Weyl’s lemma for triharmonic functions based on the new type of mean value formulas.
基金Supported by National Natural Science Foundation of China(Grant Nos.11861045,11626239 and 11701449)China Scholarship Council(Grant No.201708410483)Education Department of Henan Province(Grant No.18A110037).
文摘In this paper,we study the reiterated homogenization operators Lε=-div(A(x/ε,x/ε^(2))∇).We establish the homogenized problem and representation equa-tion by introducing the two correctors.As a consequence,we obtain the H10 and L2 convergence estimates of solutions.
