Strict Convexity Of Solutions To A Class Of Semilinear Elliptic Equations

In this thesis for Master's degree, we mainly consider two kinds of equations. Firstly, we consider a class of semilinear elliptic equations whereΩis a convex domain in R2. If the solution of the equation (1) is convex, we show it is strictly convex, under certain conditions on a(x), G(w), H(w) andΩ. Secondly, we consider another class of semilinear elliptic equations whereΩis a bounded convex domain in R2 and M is either a real constant or+∞. We are interested in establishing, for a suitable monotone function g(t), that for any solution u of (2), g(u)is strictly convex inΩ.The organization of this thesis is as the following:The chapter 1 is the introduction. We mainly consider the background, the recent development of the problem and some preliminary knowledge.In chapter 2, we work with a class of elliptic equations which have the form of (1). Under suitable assumptions on a(x), G and H, if w is convex, then w is strictly convex inΩ, whereΩis a convex domain in R2.In chapter 3, by applying the strict convexity theorem, we study the strict con-vexity of some function of solutions of (2). Under suitable assumptions on f, h, a(x) and g(u), we can get g(u) is strict convex, where u is solution of (2).