Jingyi Chen

First, we consider the initial boundary value problem for a class of pseudo-parabolic equationswith combined nonlinearities. Employing the simplified potential well method, we obtain theexistence of a global weak solution and a finite time blow-up phenomenon. By a two-steptrick, we prove that the global weak solution has exponential decay. A conclusion we derivedis that the power-type nonlinearity breaks the infinite time blow-up phenomenon caused bythe logarithmic nonlinearity. In addition, we estimate an upper bound and a lower bound forthe blow-up time under certain conditions.Second, we study the initial boundary value problem for a class of fourth order parabolic equations with combined logarithmic nonlinearities. Using the simplified potential well methodand the logarithmic Sobolev inequality, we prove the existence and blow-up results of weaksolutions. The blow-up phenomenon implies that a class of logarithmic nonlinearity is important for the solutions to blow up in finite time. Applying the concavity method, an upper bound for the blow-up time is estimated. We provide a lower bound for the blow-up time by combining the interpolation inequality with the Sobolev inequality.Finally, we consider the Dirichlet problem for a class of degenerate fourth order elliptic equations with singular potential on conical singular manifolds. We introduce the background of conical singular manifolds and the weighted cone Sobolev spaces. For the singular potential term, we establish the Rellich inequality on manifolds with conical singularities, which is used to get the equivalent norm on a suitable weighted cone Sobolev space. Applying the variational method, we obtain the existence of multiple weak solutions in a suitable weighted cone Sobolev space. Moreover, we show that the domain of the degenerate operator is a direct sum of an infinite-dimensional space and a finite-dimensional space of functions with specific asymptotics near the cone point.

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In this work, we focus on three problems. First, we give a relationship between the number of eigenvalues of the Jacobi operator below a certain threshold and the topology of closed constant mean curvature (CMC) surfaces in three-dimensional Riemannian manifolds. We then obtain that the (weak) Morse index of CMC surfaces in an arbitrary 3-manifold is bounded below by a linear function of the genus when the constant mean curvature is greater than a certain nonnegative value. In particular, this implies that stable CMC surfaces are topological spheres. Corresponding results for CMC surfaces with free boundary in 3-manifolds with boundary are obtained as well. Second, we consider the space of embedded free boundary CMC surfaces with bounded topology, bounded area, and bounded boundary length in a 3-manifold N with boundary. We show that this space is almost compact in the sense that any sequence of surfaces in this space has a convergent subsequence that converges to a free boundary CMC surface, graphically and smoothly except on a finite set of singularities. If in addition Ric_N>0 and the boundary of N is convex, then the convergence is at most 2-sheeted. In particular, it is 1-sheeted if the limiting surface is not a minimal surface. Third, we consider the maximization of Steklov eigenvalues in higher dimensions. We show that for compact manifolds of dimension at least 3 with nonempty boundary, we can modify the manifold by performing surgeries of codimension 2 or higher, while keeping the Steklov spectrum nearly unchanged. This shows that certain changes in the topology of a domain do not have an effect when considering shape optimization questions for Steklov eigenvalues in dimension 3 and higher.

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No abstract available.

In this thesis we investigate some problems on the uniqueness of mean curvatureflow and the existence of minimal surfaces, by geometric and analyticmethods. A summary of the main results is as follows.(i) The special Lagrangian submanifolds form a very important class ofminimal submanifolds, which can be constructed via the method ofmean curvature flow. In the graphical setting, the potential functionfor the Lagrangian mean curvature ow satisfies a fully nonlinearparabolic equation [formula omitted]where the ⋋j's are the eigenvalues of the Hessian D²u.We prove a uniqueness result for unbounded solutions of (1) withoutany growth condition, via the method of viscosity solutions (, ):for any continuous u₀ in ℝn, there is a unique continuous viscositysolution to (1) in ℝn x [0;∞).(ii) Let N be a complete, homogeneously regular Riemannian manifoldof dimN ≥ 3 and let M be a compact submanifold of N. Let Ʃ bea compact Riemann surface with boundary. A branched immersionu : (Σ,∂Σ) → (N,M) is a minimal surface with free boundary in Mif u(Σ) has zero mean curvature and u(Σ) is orthogonal to M alongu(∂Σ)⊑ M.We study the free boundary problem for minimal immersions of compactbordered Riemann surfaces and prove that Σ if is not a disk, then there exists a free boundary minimalimmersion of Σ minimizing area in any given conjugacy class ofa map in C⁰(Σ,∂Σ;N,M) that is incompressible; the kernel of i* : π₁(M) → π₁(N) admits a generating set suchthat each member is freely homotopic to the boundary of an areaminimizing disk that solves the free boundary problem. (iii) Under certain nonnegativity assumptions on the curvature of a 3-manifold N and convexity assumptions on the boundary M=∂N, we investigate controlling topology for free boundary minimal surfaces of low index:• We derive bounds on the genus, number of boundary components;• We prove a rigidity result;• We give area estimates in term of the scalar curvature of N.

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