Properties Of Solutions To Dirac-harmonic Equation And Norm Estimates Of Related Operators

In the last several decades,due to the rapid development of nonlinear elastic theory and quasiconformal mapping,elliptic equations for differential forms have achieved sig-nificant progress,typically,which are generalized from Laplace equation into A-harmonic equation for differential forms.Hodge-Dirac operator,which originated from theoretical physics,not only plays a dispensable role in quantum mechanics and general relativity,also provides a powerful tool for the study of geometry and algebra.In 2015,by combin-ing Hodge-Dirac operator with homogenous A-harmonic equation,Ding and Liu initiated homogenous Dirac-harmonic equation and gave the definition of its weak solution,which greatly promotes the development of A-harmonic equation,and meanwhile makes Hodge-Dirac operator applied more widely.Homogenous Dirac-harmonic equation is regarded as an extension of A-harmonic equation.However,its research is still in the beginning,and its mathematical meaning and physical meaning has not be fully investigated yet.Therefore,in this dissertation,we proceed to the study for the properties of solutions to Dirac-harmonic equation and their theoretical applications on the related operators.The main contents of this dissertation are as follows:Firstly,to explore the practical value of Dirac-harmonic equation on the composite operator theory,it investigates the boundedness of two composite operators based on ho-mogenous Dirac-harmonic equation.As is well known,Poincare-type inequalities and Orlicz-Sobolev inequalities perform the decisive role on establishing the theory of the the boundedness for the operators.So,by making use of the proven results of homogenous Dirac-harmonic equation and the properties of L?-averaging domain,along with choosing a special Young function with some restrictions,Poincare-type inequalities and Orlicz-Sobolev inequalities of these two composite operators are derived.Then,it turns out that these two composite operators are bounded in terms of Orlicz norm and Orlicz-Sobolev norm,respectively.Secondly,inspired by the operator D2G in Possion equation,it introduces two types of new iterated operators,namely,D^kG^k and D^k+1G^k,and then discusses the higher order integrability and boundedness for the iterated operators in terms of BMO norms and local Lipschitz norms.Although the composition for Green's operators still keeps the good integrability,Hodge-Dirac operator is related with the exterior differential operator,which results in that it is very difficult to determine and improve the degree of integrability for the composition of Hodge-Dirac operator and Green's operator.To overcome this difficulty,Poincare-Sobolev inequality is employed as a critical tool.Then,by constructing a new parameter with respect to the exponent p and the space dimension n,the higher order integrability is established with two cases,that is,1<p<n and p>n.Moreover,with aid of P-Hodge decomposition theorem,iterated operators are simplified by taking k as an odd number and an even number,respectively.Indeed,the representation of the iterated operator is a key point for the argument of norm comparison theorems.Finally,under some basic assumptions,non-homogenous Dirac-harmonic equation is introduced and studied systematically.The substantial difference between non-homogenous Dirac-harmonic equation and homogenous Dirac-harmonic equation lies on the right part of the equation,which is normally called the non-homogenous term.In fact,just this term leads to our argument more complicatedly.To solve this difficulty,some addi-tional structure restrictions are imposed on the operators A and B.Then,the conver-gence and the basic inequalities of the solutions to non-homogenous Dirac-harmonic e-quation are established,including weak reverse Holder inequality,Caccioppoli inequal-ity and Orlicz-Sobolev imbedding inequalities.At the end this dissertation,by applying Hodge-decomposition theorem and the analytical treatment,a nonlinear bounded oper-ator is constructed skillfully.Moreover,with this auxiliary operator in mind,by using Minty-Browder theorem,the existence and uniqueness theorem of the solutions to a spe-cial non-homogenous Dirac-harmonic equation is proven rigorously.