Some Studies Of Nonlinear Partial Differential Equations

This paper contains three related studies of nonlinear PDEs:the Neumann problem of fully nonlinear elliptic equations,the classification of blow-ups for nonlinear half-heat equations,and the quantitative stability of Sobolev inequality.In the first part,we study Neumann problems for a class of Hessian type equation,which is fully nonlinear and elliptic.By the continuity method and LiebermanTrudinger's theory,the solvability can be reduced to the a priori C2 estimate.Following the method of Ma-Qiu which solve the Hessian equation,we can obtain the existence and uniqueness of the solution of the equation being studied.Then,we also consider the Neumann problem and prescribed contact angle problem of k-curvature equations,and use maximum principle to get global gradient estimates.In the second part,we classify the blow-ups for nonlinear half-heat equations with sub-critical exponent.We first classify all self-similar solutions for the half-heat equation when dimension n?4,which in turn can classify all blow-ups of type ?.We also estimate the blow-up rate to prove that all blow-ups are of type ?.For the Cauchy problem,we also classify its blow-ups and get a non-degeneracy result.Our proof is based on two key ingredients:the Pohozaev type inequality and the Giga-Kohn type monotonicity formula for half-heat equations.For the first time,our work extends Giga-Kohn's series of results on nonlinear heat equations to fractional heat equations.In the third part,we study the stability of the Sobolev inequality with exponent 2 in view of the Euler-Lagrange equation,and obtain a sharp quantitative estimate in high dimension(n?6).Due to the important applications to problems in the calculus of variations and PDEs,a lot of work has been done to study the stability of functional and geometric inequalities.It is well-known that the positive solution of the Euler-Lagrange equation must be the so-called Talenti bubbles.Struwe qualitatively proves that if a nonnegative function u almost solves the Euler-Lagrange equation,then the Hilbert distance between the u and the sum of weakly interacted Talenti bubbles must be small.FigalliGlaudo first obtain a sharp quantitative estimate for 3?n?5,a linear control,and prove that the linear control is no longer true when n?6 by counter examples.By finite dimension reduction,we get a fine estimate of the leading term of error,so as to obtain the sharp quantitative estimate in high dimension,some nonlinear controls,and we completely solve this problem.