In this thesis we study three different problems: convex ancient solutions to the power-of-mean curvature flow; Sharp inequalities; regularity results in some applications of optimal transportation. The second chapter is devoted to the power-of-mean curvature flow; We prove some estimates for convex ancient solutions (the existence time for the solution starts from -\infty) to the power-of-mean curvature flow, when the power is strictly greater than \frac{1}{2}. As an application, we prove that in two dimension, the blow-down of an entire convex translating solution, namely u_{h}=\frac{1}{h}u(h^{\frac{1}{1+\alpha}}x), locally uniformly converges to \frac{1}{1+\alpha}|x|^{1+\alpha} as h\rightarrow\infty. The second application is that for generalized curve shortening flow (convex curve evolving in its normal direction with speed equal to a power of its curvature), if the convex compact ancient solution sweeps the whole space \mathbb{R}^{2}, it must be a shrinking circle. Otherwise the solution must be defined in a strip region. In the first section of the third chapter, we prove a one-parameter family of sharp conformally invariant integral inequalities for functions on the $n$-dimensional unit ball. As a limiting case, we obtain an inequality that generalizes Carleman's inequality for harmonic functions in the plane to poly-harmonic functions in higher dimensions. The second section represents joint work with Tobias Weth and Rupert Frank; the main result is that, one can always put a sharp remainder term on the righthand side of the sharp fractional sobolev inequality. In the first section of the final chapter, under some suitable condition, we prove that the solution to the principal-agent problem must be C^{1}. The proof is based on a perturbation argument. The second section represents joint work with Emanuel Indrei; the main result is that, under (A3S) condition on the cost and c-convexity condition on the domains, the free boundary in the optimal partial transport problem is C^{1,\alpha}.

In this thesis we study three different problems: convex ancient solutions to the power-of-mean curvature flow; Sharp inequalities; regularity results in some applications of optimal transportation. The second chapter is devoted to the power-of-mean curvature flow; We prove some estimates for convex ancient solutions (the existence time for the solution starts from -\infty) to the power-of-mean curvature flow, when the power is strictly greater than \frac{1}{2}. As an application, we prove that in two dimension, the blow-down of an entire convex translating solution, namely u_{h}=\frac{1}{h}u(h^{\frac{1}{1+\alpha}}x), locally uniformly converges to \frac{1}{1+\alpha}|x|^{1+\alpha} as h\rightarrow\infty. The second application is that for generalized curve shortening flow (convex curve evolving in its normal direction with speed equal to a power of its curvature), if the convex compact ancient solution sweeps the whole space \mathbb{R}^{2}, it must be a shrinking circle. Otherwise the solution must be defined in a strip region. In the first section of the third chapter, we prove a one-parameter family of sharp conformally invariant integral inequalities for functions on the $n$-dimensional unit ball. As a limiting case, we obtain an inequality that generalizes Carleman's inequality for harmonic functions in the plane to poly-harmonic functions in higher dimensions. The second section represents joint work with Tobias Weth and Rupert Frank; the main result is that, one can always put a sharp remainder term on the righthand side of the sharp fractional sobolev inequality. In the first section of the final chapter, under some suitable condition, we prove that the solution to the principal-agent problem must be C^{1}. The proof is based on a perturbation argument. The second section represents joint work with Emanuel Indrei; the main result is that, under (A3S) condition on the cost and c-convexity condition on the domains, the free boundary in the optimal partial transport problem is C^{1,\alpha}.