Regularity For Minimizer Of Integral Functional And Solutions To Elliptic Equations And Systems

In this paper,we mainly study the global regularity for integral functional minimizer and weak solution or entropy solution to elliptic equations and systems.Firstly,we deal with integral functionals ??f(x,Du(x))dx defined on vector valued map-pings u=(u1,…,uN):?(?)Rn?RN,where n,N?2.We show that a suitable "mono-tonicity" inequality,on the density f,guarantees global boundedness for every component uj of the minimizer u,and we also deal with anisotropic variational integrals whose prototype is(?)The aim of this chapter is to derive global regularity results for minimizers (?) under suitable assumptions on P?,q? and r.Secondly,we derive estimates for entropy solutions to degenerate elliptic equations of the form(?)where the Caratheodory function A:?× R × Rn?Rn satisfies degenerate coercivity condition (?)and controllable growth condition (?)and 0<???<?,and f lies in Marcinkiewicz spaces.We use generalized Stampacchia Lemma to prove the main theorem.Counterexample shows that some of the results are optimal.As an application of our results,we use a bootstrap argument to show that the entropy solutions of(?)are bounded provided that 0?a<n(p-1-?)/n-p.Furthermore,we deal with regularity for solutions to boundary value problem of the form(?)Under some standard growth assumptions,we derive some regularity results.Finally,we deal with global integrability for solutions to Dirichlet problem for quasilinear elliptic systems involving N equations of the form(?)where ??{1,…,N}is the equation index,? is an open bounded subset of Rn,n>2,u=(U1,...,uN):??RN and f? Lm(?),2n/n+2?m?n/2.Under ellipticity condition of diagonal coefficients,smallness and staircase support conditions of off-diagonal coefficients,we derive some global integrability results.Meanwhile,we present two generalizations of the classical Stampacchia Lemma.As ap-plications,we consider Dirichlet problem for quasilinear elliptic systems of the form(?)Under ellipticity and degenerate ellipticity conditions of the diagonal coefficients and propor-tional conditions of the off-diagonal coefficients,we derive some global regularity results.