In this thesis we study three problems in the field of geometric analysis: eigenvalue estimate of non-linear operators, existence of minimal surfaces and isoperimetric problems. These problems are more or less related to the topic of geometric calculus of variations, which is the study of extreme points of functionals defined on manifolds.
The first part is devoted to the study of lower bound of the principal eigenvalue of a family of non-linear elliptic operator $L_p$. Using the gradient and maximum comparison technique developed in \cite{Ko18} together with ideas from \cite{LW19eigenvalue2}, we proved that on a compact metric measure space(possibly with convex boundary) $(M,g,m)$ with curvature-dimension condition $BE(\kappa, N) (\kappa \neq 0)$, if $L$ is a elliptic diffusion operator whose invariant measure is $m$, then the principal eigenvalue of $L_p$ is bounded below by the first eigenvalue of a one-dimensional ODE with Neumann boundary condition. We showed that this is sharp result by constructing examples of metric measure space $M$ on which the eigenvalue problem of $L_p$ degenerates into the model equation problem. This work extends the $\kappa = 0$ case proved in \cite{Ko18}.
The second part is devoted to the study of existence of free boundary minimal hypersurfaces in compact manifolds, from a min-max theoretical point of view. Following the ideas from \cite{ambrozio2018min} and \cite{marques2019equidistribution}, we prove that in a simply connected compact manifold $(M,\partial M, g)$ under certain conditions) with its metric that is locally maximising the width of $M$, there is a sequence of equidistributed free boundary minimal hypersurfaces.
The third part is devoted to the study of anisotropic isoperimetric inequality for regions outside of a ball in $\mathbb{R}^n$. Based on Alexandrov-Bakelman-Pucci's Method, we use the concept of generalized normal cone introduced by \cite{LWW20}, to show that for any region outside a Euclidean ball, its isoperimetric ratio has a lower bound that can only be achieved by a half-Wulff shape cut by a half-space.
