OPTIMAL INTERIOR PARTIAL REGULARITY FOR NONLINEAR ELLIPTIC SYSTEMS UNDER THE NATURAL GROWTH CONDITION:THE METHOD OF A-HARMONIC APPROXIMATION

In this article, the authors consider the nonlinear elliptic systems under the natural growth condition. They use a new method introduced by Duzaar and Grotowski, for proving partial regularity for weak solutions, based on a generalization of the technique of harmonic approximation. And directly establish the optimal Holder exponent for the derivative of a weak solution.

ON VERY WEAK SOLUTIONS OF A-HARMONICEQUATION WITH VERY WEAK BOUNDARYVALUES

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.

Weak WT_2-class of differential forms and weakly A-harmonic tensors

In this paper, we give the definition of weak WT2-class of differential forms, and then obtain its weak reverse Holder inequality. As an application, we give an alternative proof of the higher integrability result of weakly A-harmonic tensors due to B. Stroffolini.

PARTIAL REGULARITY FOR STATIONARY NAVIER-STOKES SYSTEMS BY THE METHOD OF A-HARMONIC APPROXIMATION

In this article,we prove a regularity result for weak solutions away from singular set of stationary Navier-Stokes systems with subquadratic growth under controllable growth condition.The proof is based on the A-harmonic approximation technique.In this article,we extend the result of Shuhong Chen and Zhong Tan[7]and Giaquinta and Modica[18]to the stationary Navier-Stokes system with subquadratic growth.

Some new integral inequalities for conjugate A-harmonic tensors

A new kind of weight-Ar^λ3 （λ1, λ2, Ω）-weight is used to prove the local and global integral inequalities for conjugate A-harmonic tensors, which can be regarded as generalizations of the classical results. Some applications of the above results to quasiregular mappings are given.

REGULARITY FOR VERY WEAK SOLUTIONS TO A-HARMONIC EQUATION

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.

Extremum Principle for Very Weak Solutions of A-Harmonic Equation with Weight

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.

The Method of A-Harmonic Approximation and Boundary Regularity for Nonlinear Elliptic Systems under the Natural Growth Condition

We consider the questions of boundary regularity for weak solutions of second-order nonlinear elliptic systems under the natural growth condition. We obtain a general criterion for a weak solution to be regular in the neighborhood of a given boundary point. The proof yields directly the optimal regularity for the solution in this neighborhood. This result is new for the situation under the natural growth conditions.

Removable Singularities of Solutions of A-harmonic Type Equations

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.

EXTREMUM PRINCIPLE FOR VERY WEAK SOLUTIONS OF A-HARMONIC EQUATION

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.

LOCAL REGULARITY RESULT FOR SOLUTIONS OF OBSTACLE PROBLEMS

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.

PARTIAL REGULARITY FOR WEAK SOLUTIONS OF STATIONARY NAVIER-STOKES SYSTEMS

This article is concerned with the partial regularity for the weak solutions of stationary Navier-Stokes system under the controllable growth condition.By A-harmonic approximation technique,the optimal regularity is obtained.

OPTIMAL INTERIOR PARTIAL REGULARITY FOR NONLINEAR ELLIPTIC SYSTEMS WITH DINI CONTINUOUS COEFFICIENTS

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.

PARTIAL REGULARITY OF STATIONARY NAVIER-STOKES SYSTEMS UNDER NATURAL GROWTH CONDITION

In this article, we consider the partial regularity of stationary Navier-Stokes system under the natural growth condition. Applying the method of A-harmonic approximation,we obtain some results about the partial regularity and establish the optimal Holder exponent for the derivative of a weak solution on its regular set.

LOCAL REGULARITY RESULT IN OBSTACLE PROBLEMS

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）.

ORTHOGONAL PROJECllON OF THE DOMAIN BORNDARY OPERATOR FOR ELLIPITIC PROBLEM BY DOMAIN DECOMPLSITION

Some preconditionere for discretizations of elliptic beundary problenisare studied by J. H. Bramble , J. E. Pasciak and A. H. Schartz. In this paper, thediscrete A-harmonic functions W given in the papers written by the above men-tioned authore are partio

Regularity for solutions to anisotropic obstacle problems

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.

Global Integrability for Solutions to Obstacle Problems

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.

TWO-WEIGHT IMBEDDING THEOREMS IN THE SPACE OF DIFFERENTIAL FORMS

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.

PARTIAL REGULARITY RESULT OF SUPERQUADRATIC ELLIPTIC SYSTEMS WITH DINI CONTINUOUS COEFFICIENTS

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.