Let $D$ be a connected bounded domain in $\mathbb{R}^n$. Let $0<\mu_1\leq\mu_2\leq\dots\leq\mu_k\leq\cdots$ be the eigenvalues of the following Dirichlet problem: $$ \begin{cases}\Delta^2u(x)+V(x)u(x)=\mu\rho(x)u(x),\quad x\in D u|_{\partial D}=\frac{\partial u}{\partial n}|_{\partial D}=0, \end{cases} $$ where $V(x)$ is a nonnegative potential, and $\rho(x)\in C(\bar{D})$ is positive. We prove the following inequalities: $$\mu_{k+1}\leq\frac{1}{k}\sum_{i=1}^k\mu_i+\Bigl[\frac{8(n+2)}{n^2}\Bigl(\frac{\rho_{\max}} {\rho_{\min}}\Bigr)^2\Bigr]^{1/2}\times \frac{1}{k}\sum_{i=1}^k[\mu_i(\mu_{k+1}-\mu_i)]^{1/2}, $$ $$\frac{n^2k^2}{8(n+2)}\leq \Bigl(\frac{\rho_{\max}}{\rho_{\min}}\Bigr)^2\Bigl[\sum_{i=1}^k\frac{\mu_i^{1/2}}{\mu_{k+1}-\mu_i}\Bigr] \times\sum_{i=1}^k\mu_i^{1/2}. $$

Let $M$ be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue $\lambda$. We give upper and lower bounds on the inner radius of the type $C/\lambda^\alpha(\log\lambda)^\beta$. Our proof is based on a local behavior of eigenfunctions discovered by Donnelly and Fefferman and a Poincar\'{e} type inequality proved by Maz'ya. Sharp lower bounds are known only in dimension two. We give an account of this case too.

A sharp upper bound on the first $S^{1}$ invariant eigenvalue of the Laplacian for $S^1$ invariant metrics on $S^2$ is used to find obstructions to the existence of $S^1$ equivariant isometric embeddings of such metrics in $(\R^3,\can)$. As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the metric cannot be equivariantly, isometrically embedded in $(\R^3,\can)$. This leads to generalizations of some classical results in the theory of surfaces.