Some Researches On The Double Obstacle Problems Associated With Nonhomogeneous A-Harmonic Equation

Partial differential equation is a very wide topic which has very deep influence andimportant applications in our daily life,many physics problems can be attributed toproblems of partial differential equation In the theoretical study of partial differentialequation,the study ofA—harmonic equation takes an importaut role Recently,the theoryofA—harmonic equation has widely applications in potential theory,elasticity theory,quasiconformal analysis,physics and other fieldsThe obstacle problem to A—harmonic equation also has broad and profound impactIn the applications of practical problems,many physical problems can be attributed to thevariational inequalities over obstacle and related optimal control problemsThis paper considers some properties of solutions to the double obstacle problemsabout nonhomogeneous A—harmonic equation As a generalization of the single obstacleproblem associated with homogeneous A—harmonic equation,it has many propertieseither similar or different with them In this paper,vc-e deal with two kinds of obstacleproblems One is the(?)-obstacle problem and another is the(?)-obstacle problemFor (?)-obstacle problem,we give the definition its solution and make use of theproperties of the relevant operator,by choosing appropriate test functions,combined withH61der inequality,Poincar6 inequality and other basic inequalities,we get two propertiesabout solutions The first is the higher integrability result of the derivatives of solutionsThe higher integrability means the Sobolev integrability exponent of the solution largethan the natural integrability exponent P And the second is the convergence of solutionsabout varying integrability exponents For(?)bstacle problem,as a generalization ofobstacle problems,we first extend the definition of weak solutions of(?)bstacleproblem to very weak solutions of (?)stacle problem,then vc-e get its property asquaslmlnlmlzers in r-Dirichlet problems and with the Hodge decomposition theorem weprove the stabilitv of very w-eak solutions about obstacle functions...