Regularities Of Very Weak Solutions For Elliptic Equation Of Divergence Form And Related Obstacle Problems

This doctoral dissertation is mainly to consider the following three problems in-volving the regularities of very weak solutions to nonlinear elliptic equation of diver-gence form and related obstacle problems.Our main content is shown as follows.First,we study various properties of very weak solutions to the equations of A-harmonic form(zero property of gradient,higher integrability of the gradients,a removable singularity of very weak solutions,etc).Second,we investigate the regularity of very weak solu-tions to the Dirichlet problem of nonlinear elliptic systems in divergence form.Third,we are to study a local Holder continuity for A-harmoni equation with variable expo-nent growth and its obstacle problem.More precisely,the specific contents include the following chapters.In chapter 1,let us briefly recall the research background,various related literatures and the recent developments which is involved in our research topic.In chapter 2,we are to attain the zero-point property for the gradients of very weak solutions to the so-called A-harmonic equation in differential form.By establishing the Caccioppoli estimate of very weak solutions,we obtain the weak inverse Holder inequality for the gradients of very weak solutions.Furthermore,combining with the definition of intrinsic zeros,then we get the zeros of infinite orders vanishing for the gradient of very weak solutions.In chapter 3,we are to prove a higher integrability for the gradients of very weak solutions to the A-harmonic equation of differential form.While the integrable exponent r of the gradients of very weak solutions is near to the natural integrable exponent p of the gradients of weak solutions,we establish the weak inverse Holder inequality of the gradients by way of taking a special test function in the very weak sense due to the so-called Hodge decomposition.According to the ideas of the papers by Iwaniec and his collaborators,we get that the integrable exponent continues to increase such that it attain the gradient integrability of weak solutions.So,our result leads to that the very weak solution of A-harmonic equation is actually a classical weak solution.In chapter 4,we are to show a removable singularity of very weak solutions for nonlinear elliptic equation of differential form.To this end,we apply the so-called Hodge decomposition of the perturbed vector fields to take an appropriate test func-tion in a very weak sense.Here,the Caccioppoli estimate of very weak solutions is established and the capacity argument is employed.Finally,the result of this removable singularity of very weak solutions is extended to more general elliptic equations with controllable growths in the weighted form by way of a similar argument.In chapter 5,we are to give the regularity of very weak solution reflected by the regularity of boundary value functions to the Dirichlet problem of a class of nonlin-ear elliptic systems.Also,we take an appropriate test functions in the sense of very weak solutions by the so-called Hodge decomposition.Finally,we discuss various reg-ularities of very weak solution reflected by the different regularities of boundary value functions on the basis of using the Sobolev embedding theorem and the Stampacchia lemma.In chapter 6,we are to prove a local Holder continuity for the gradients of weak solutions to a class of A-harmonic equation with variable exponent growth.By using the log-Holder continuity of variable exponential p(x),we first obtain a local Holder continuity by establishing the approximation relationship between the gradients of weak solution u and the gradients of weak solutions v of the reference equation with constant exponent growth.Further,we combine with the reverse Holder inequality of Du and use iterating argument to attain the desired result.In chapter 7,we prove a local Holder continuity of the gradients of weak solution to a class of nonlinear elliptic obstacle problems with variable exponents.Here,we still give a series of approximating relations between ?u and ?v many times except the method what we used is similar to Chapter six.By combining with the reverse Holder inequality of Du and the iterating argument,we get a local Holder continuity of Du.