The New Method Of Regularity Theory And Applications To Nonlinear Partial Differential Systems And Quasi-Convex Integrals

The regularity theory is one of the most challenging problems of modern theory of partial differential equations.It has been paid close attentions to for a long history.As early as in 1900,D.Hilbert posed his 23 well-known open problems on the International Congress of Mathematicians at Paris.Two of them(19th and 20th)are related to the regularity of weak solutions,which gives prominance to the difficulties and the importances of the regualrity the-ory.The classical method of partial regularity theory is the "freezing the co-efficients" method.Precisely,by "freezing the coefficients",we obtain an elliptic system with constant coefficients,and the solution of the Dirichlet problem associated to these coefficients with boundary data u and the solution itself can then be compared.Then we can obtain the important decay esti-mates by iterated and yield the results of partial regularity.This procedure need the complex and troublesome reverse-H(o|¨)lder inequality or the Gehring lemma.What makes things worse is that the H(o|¨)lder exponent of partial regu-larity obtained by this method is not optimal.It means that one just can get a H(o|¨)lder exponent more litter than the one in the H(o|¨)lder continuity condition of the given coefficient function.In addition,the singular set has not been estimated exactly.In this paper,we use a new method—the method of A-harmonic approx-imation,to consider partial regularity theory for weak solutions of nonlinear partial differential systems under controllable growth condition and natural growth condition,respectively.A-harmonic approximation leraraa—the key ingredient of the new method—puts up a bridge between A-harmonic func-tion and nonlinear partial differential systems,which makes us can construct a specified function corresponding with weak solutions u.The A-harmonic approximation lemma reveals that there exists an A-harmonic function closing to the specified function in L~2.Making full use of those known properties of A-harmonic function,one can derive the desired decay estimate and then obtain the partial regularity results.The key difference between the method of "freezing the coefficients" and "A-harmonic approximation" is that the solution is compared not to the solution of the Dirichlet problem for the systems with frozen coefficients, but rather to an A-harmonic function which lies close to a defined specified function in L~2.The new method not only allows one to simplify the proof of partial regularity result,but also to avoid the technique diffculty associated with applying reverse H(o|¨)lder inequality or Gehring's lemma.In particularly, one can directly establish the optimal HSlder exponent for the derivation of weak solutions.Throughtout the paper,m stands for the growth exponent of the derivation of weak solutions.The following are some new ideas in this paper:(1)Weak solutions of partial differential systems under natural growth condition are bounded,i.e.|u|≤M＜∞,which means that we can argue analogously,even more simple than the corresponding systems under control-lable growth condition.What mostly pleased us is that the estimation on singular set we obtained under natural growth condition is much more exactly than the one obtained by the classical method.(2)As the author's best knowledge,there is no any better results on the partial regularity theory of partial differential systems under controllable growth condition.And there is no litter literature on the proof of Cacciop-poli second inequality under controllable growth condition,even for the case m≡2,still less in the case m＞2 or 1＜m＜2.In this paper,we deduce Caccioppoli second inequality under controllable growth condition by a new method.Thus,the proof of the inequality is new and of some independent interest.(3)In this paper,we not only deal successfully with the optimal partial regularity for weak solutions of nonlinear elliptic systems when m＞2,but also for the case 1＜m＜2.It is sure that the result is a breakthrough progression on nonlinear elliptic systems with general structure growth condition during last twenty years.(4)Combining the technique used in nonlinear elliptic systems with the characteristic of almost minimizers of quasiconvex integrals,we successfully derive optimal partial regularity for almost minimizers of quasiconvex integrals.In a word,in this paper,by combining A-harmonic approximation tech-nique or its different types of forms,together with other techniques,we prove optimal partial regularity for weak solutions of nonlinear elliptic systems,non-linear parabolic systems,degenerate elliptic systems and stationary Navier-Stokes systems under controllable growth condition and natural growth condi-tion,respectively.Moreover,we also obtain the regularity properties of almost minimizers of quasiconvex integrals.In precise: 1,Nonlinear Elliptic SystemsIn the case m＞2,we use the method of A-harmonic approximation to get optimal partial regularity.But A-harmonic approximation technique only can deal with the case m≡2,and is absolutely helpless for the case m≠2. In order to overcome the difficulty,we adapt of the method of interpolation.In the case 1＜m＜2,the method of A-harmonic approximation is no longer suitable.Thanks to another A-harmonic approximation method,where the function defined analogous as P-Laplace function,we can proceed to the proof of the optimal partial regularity result.However,a new problem has arisen.That is the exponent of integral function of this situation takes on negative(-1/2＜(m－2)/2＜0),which causes one can't use the technique as usual. In order to remove the hindrance,in light of the paper Acerbi and Fusco considered in 1989 for the partial regularity for minimizers of functional,we use a new method to deduce the desired result.In this section,we prove Caccioppoli second inequality again and obtain the optimal partial regularity.2,Degenerate Elliptic SystemsIn this section,the method of A-harmonic approximation is transformed into the p-harmonic approximation method.But the special property of con-trollable growth condition itself and weak solutions u unbounded let our study fall into dire straits.Benefitting by a paper written by Tan Zhong and Yan Ziqian in 1992,where the authors considered the regularity of weak solutions to some degenerate elliptic equations and obstacle problems,and combining p-harmonic approximation technique with the properties of Sobolev space,we deduce a suitable Caccioppoli inequality,and then derive a special decay es-timate,which made the optimal partial regularity problem for weak solutions of degenerate elliptic systems readily solved.3,Nonlinear Parabolic SystemsIn this section,we use a new method developed on the base of the method of A-harmonic approximation—A-caloric approximation method—to solve the optimal partial regularity problem.However,here weak solutions not only depend on time t but also lake of bounded condition.And there are no ap-propriate Caccioppoli second inequality.All of these make us get the optimal partial regularity result hopeless.At last,motivated by the technique used by Yan Ziqian in 1986 dealt with the partial regularity theory for weak solu-tions of nonlinear parabolic systems,we tap those good properties of Sobolve space,combining A-caloric approximation technique with the technique used in nonlinear elliptic systems,to derive the desired result in the end.4,Stationary Navier-Stokes SystemsThere have had beautiful results for the partial regularity and the esti-mation of singular set of Stationary Navier-Stokes systems with simple type. But,for stationary Navier-Stokes systems with general structure condition, there without any better results than what Giaquinta obtained in 1979.In this paper,by combining A-harmonic approximation with the technique used by Giaquinta in 1979,together with the relation between div and grad of weak solutions,we improve Giaquinta's work and get the optimal partial regularity result.5,Minimizers of Quasiconvex Integrals In the case m＞2 and the integral function f(Du)of quasiconvex integral independing on x and u:combining A-harmonic approximation technique and the method of interpolation inequality,together with the special properties of minimizers of quasiconvex integrals,the optimal partial regularity can be obtained directly.For the case 1＜m＜2 and the integral function f(x,u,Du)of mini-mizers of quasiconvex integrals depending on x and u:by the technique used in nonlinear elliptic systems on x and u,noting that the special properties of minimizers of quasiconvex integral,combining A-harmonic approximation technique,we can get the desired result.