The Regularity Of A Singular Differential Equation And Its Applications

In this thesis, we study the regularity for a class of singular diferential equationsand the higher regularity for a fully nonlinear singular elliptic equation.In the first part, we investigate a class of singular diferential equations. Firstly, weget the growth estimate for the solution and the properties of the solution near boundaryby using comparison principle and a scaling argument. Then, we introduce an iteration,use the method of freezing coefcients and apply the induction argument to prove ourregularity results. Finally, we extend our methods to study the regularity for a class ofsingular diferential equations with parameters.In the second part, we apply the results which have been obtained in the first partto Ginzburg-Landau equation, harmonic map problem and parabolic Ginzburg-Landauequation, respectively. As a result, we get the corresponding regularity results for theradial solutions to these problems.In the third part, we study the higher regularity for the solution to a class of fullynonlinear elliptic equations with strongly lower terms at the boundary. For the equationwhose singularity happens only at the coefcient of the first order derivative term, we finda compatibility condition and prove the solution is C∞smooth up to the boundary if andonly if the compatibility condition is satisfied.