Stampacchia Lemma And It's Applications

The Stampacchia lemma plays an important role in investigating the regularity of weak solutions of elliptic partial differential equations and integral functionals.The research of these theories can enrich the theoretical results of elliptic partial differential equations and provide theoretical basis for engineering practice.The paper consists of three chapters.Chapters 1 and 3 are introduction and conclusion,re-spectively.In the second chapter,we mainly devote to the Stampacchia lemma and its deduc-tions,and provide the application of this lemma in the regularity of entropy solution and the global boundedness for vector valued minimizers of some anisotropic integral.In the study of the regularity of entropy solution,we focus on studying two kinds of entropy solutions of elliptic equations.One is the boundary value problem as follows where Caratheodory functions ai(x,z):?×RN?R with x?ai(x,z)measurable and z?ai(x,z)continuous.We assume that and where b?1 is a constant.For m?1 we assume that f?Lweakm(?),and By virtue of Stampacchia lemma and its deductions,we obtain the regularity of an entropy solution:We investigate the other entropy solution regularity of boundary value problem as follows and the Caratheodry functions ai(x,z):?×RN?R with x?ai(x,z)measurable and z?ai(x,z)continuous,satisfies and|ai(x,z)|??|zi|pi-1.for almost all x? ? and all z,z?RN where ??1 is a constant.For m?1 we assume that f(x)?Leweakmpi'(?),u? W1,1(?)and Diu*?Lweakmpi(?).By virtue of the Stampacchia lemma and its deductions,we receive the regularity of an entropy solution:We deal with the global boundedness for vector valued minimizer of some anisotropic integral of the typeThe Caratheodry function f(x,z):?×RN×n?R satisfies with M(x)? Lr(?),and r?1,?>0 are constants,which guarantees global boundedness of minimizer u.The density function f(x,z)satisfies a monotonicity inequality with weight.In addition,we add the gradient constraint problem.In the end,we obtain the global boundedness of minimizer u.