In this article, we consider nonlinear elliptic systems of divergence type with Dini continuous coefficients. The authors use a new method introduced by Duzaar and Grotowski, to prove partial regularity for weak solut...In this article, we consider nonlinear elliptic systems of divergence type with Dini continuous coefficients. The authors use a new method introduced by Duzaar and Grotowski, to prove partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation and directly establish the optimal HSlder exponent for the derivative of a weak solution on its regular set.展开更多
We present a new method to determine the optimal regularity of positive solutions u∈C^(4)(Ω\{0})∩C^(0)(Ω) of the Hénon-Hardy equation,i.e.,Δ^(2)u=|x|^(α)u^(p)inΩ,(0,1) where Ω■RN(N≥4) is a bounded smoot...We present a new method to determine the optimal regularity of positive solutions u∈C^(4)(Ω\{0})∩C^(0)(Ω) of the Hénon-Hardy equation,i.e.,Δ^(2)u=|x|^(α)u^(p)inΩ,(0,1) where Ω■RN(N≥4) is a bounded smooth domain with 0∈Ω,α>-4,and p∈R.It is clear that 0 is an isolated singular point of solutions of(0.1) and the optimal regularity of u in Ω relies on the parameter α.It is also important to see that the regularity of u at x=0 determines the regularity of u in Ω.We first establish asymptotic expansions up to arbitrary orders at x=0 of prescribed positive solutions u ∈C^(4)(Ω{0}) ∩ C^(0)(Ω)of(0.1).Then we show that the regularity at x=0 of each positive solution u of(0.1) can be determined by some terms in asymptotic expansions of the related positive radial solution of the equation(0.1) with Ω=B,where B is the unit ball of R^(N).The main idea works for more general equations with singular weights.展开更多
We use the concept of the inside-(a,η,h)domain to construct a subsolution to the Dirichlet problem for minimal graphs over convex domains in hyperbolic space.As an application,we prove that the Hölder exponent m...We use the concept of the inside-(a,η,h)domain to construct a subsolution to the Dirichlet problem for minimal graphs over convex domains in hyperbolic space.As an application,we prove that the Hölder exponent max{1/a,1/(n+1)}for the problem is optimal for any a∈[2,+∞].展开更多
Given f ,g two Cr,a either symplectic forms or volume forms on a bounded open set Ω R nwith 0〈a〈1and r≥0,we give natural conditions for the existence of amap（φ∈Diff＋1,a（ Ω;Ω）satisfying φ＊（g）=finΩ and ...Given f ,g two Cr,a either symplectic forms or volume forms on a bounded open set Ω R nwith 0〈a〈1and r≥0,we give natural conditions for the existence of amap（φ∈Diff＋1,a（ Ω;Ω）satisfying φ＊（g）=finΩ and supp（φ-id） Ω.展开更多
We consider the partial regularity for weak solutions to superquadratic elliptic systems with controllable growth condition, under the assumption of Dini continuous coefficients. The proof relies upon an iteration sch...We consider the partial regularity for weak solutions to superquadratic elliptic systems with controllable growth condition, under the assumption of Dini continuous coefficients. The proof relies upon an iteration scheme of a decay estimate for a new type of excess functional. To establish the decay estimate, we use the technique of A-harmonic approximation and obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particular, the proof yields directly the optimal H¨older exponent for the derivative of the weak solutions on the regular set.展开更多
基金Supported by NSF of China (10531020)the Education Department of Fujian Province(JK2009045)the Program of 985 Innovation Engieering on Information in Xiamen University(2004-2007)
文摘In this article, we consider nonlinear elliptic systems of divergence type with Dini continuous coefficients. The authors use a new method introduced by Duzaar and Grotowski, to prove partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation and directly establish the optimal HSlder exponent for the derivative of a weak solution on its regular set.
基金supported by National Natural Science Foundation of China (Grant No. 11571093)supported by the Fundamental Research Funds for the Central Universities (Grant No. WK0010000064)Anhui Provincial Natural Science Foundation (Grant No. BJ0010000026)。
文摘We present a new method to determine the optimal regularity of positive solutions u∈C^(4)(Ω\{0})∩C^(0)(Ω) of the Hénon-Hardy equation,i.e.,Δ^(2)u=|x|^(α)u^(p)inΩ,(0,1) where Ω■RN(N≥4) is a bounded smooth domain with 0∈Ω,α>-4,and p∈R.It is clear that 0 is an isolated singular point of solutions of(0.1) and the optimal regularity of u in Ω relies on the parameter α.It is also important to see that the regularity of u at x=0 determines the regularity of u in Ω.We first establish asymptotic expansions up to arbitrary orders at x=0 of prescribed positive solutions u ∈C^(4)(Ω{0}) ∩ C^(0)(Ω)of(0.1).Then we show that the regularity at x=0 of each positive solution u of(0.1) can be determined by some terms in asymptotic expansions of the related positive radial solution of the equation(0.1) with Ω=B,where B is the unit ball of R^(N).The main idea works for more general equations with singular weights.
基金This work was supported in part by the Natural Science Foundation of Beijing Municipality(No.1212002)the National Natural Science Foundation of China(Grant Nos.12071017,11871432).
文摘We use the concept of the inside-(a,η,h)domain to construct a subsolution to the Dirichlet problem for minimal graphs over convex domains in hyperbolic space.As an application,we prove that the Hölder exponent max{1/a,1/(n+1)}for the problem is optimal for any a∈[2,+∞].
文摘Given f ,g two Cr,a either symplectic forms or volume forms on a bounded open set Ω R nwith 0〈a〈1and r≥0,we give natural conditions for the existence of amap（φ∈Diff＋1,a（ Ω;Ω）satisfying φ＊（g）=finΩ and supp（φ-id） Ω.
基金Supported by the National Natural Science Foundation of China(No.10976026)Natural Science Foundation of Fujian Province(2012D102)
文摘We consider the partial regularity for weak solutions to superquadratic elliptic systems with controllable growth condition, under the assumption of Dini continuous coefficients. The proof relies upon an iteration scheme of a decay estimate for a new type of excess functional. To establish the decay estimate, we use the technique of A-harmonic approximation and obtain a general criterion for a weak solution to be regular in the neighborhood of a given point. In particular, the proof yields directly the optimal H¨older exponent for the derivative of the weak solutions on the regular set.
