We are concerned with a quasi-steady Stefan type problem with Gibbs-Thomson relation and the mobility term which is a model for a crystal growing from supersaturated vapor. The evolving crystal and the Wulff shape of the interfacial energy are assumed to be (right-circular) cylinders. In pattern formation deciding what are the conditions which guarantee that the speed in the normal direction is constant over each facet, so that the facet does not break, is an important question. We formulate such a condition with the aid of a convex variational problem with a convex obstacle type constraint. We derive necessary and sufficient conditions for the nonbreaking of facets in terms of the size and the supersaturation at space infinity when the motion is self-similar.

We consider the porous medium equation (PME) $$ u_t = \Delta (|u|^{m-1}u) \quad \mbox{in ${{\mathbb R}^N} \times {{\mathbb R}^N}_+$}, \quad m>1, $$ with continuous, compactly-supported initial data $\hat{u}$. The PME admits various similarity solutions of the form $$ u_k(x,t)= t^{-{\alpha}_k} \psi_k(x/t^{{\beta}_k}), \quad k=0,1,..., $$ where each $\psi_k \in C_0({{\mathbb R}^N})$ satisfies a quasilinear elliptic equation in ${{\mathbb R}^N}$ and the exponents $\{{\alpha}_k,{\beta}_k\}$ are determined from the solubility of such a nonlinear eigenvalue problem. The nonlinear eigenfunction subset $\Phi=\{\psi_k\}$ consisting of a countable number of continuous families is rather complicated and is known in one dimension and in the radial setting in ${{\mathbb R}^N}$ (in ${{\mathbb R}^N},$ for $N \ge 2$, only the first two eigenfunctions from $\Phi$ are known). We show that the eigenfunction subset $\Phi$ can be evolutionary complete, i.e., describes the asymptotics of arbitrary global $C_0$-solutions of the PME. We prove such a completeness in one dimension and in the radial ${{\mathbb R}^N}$ geometry. The analysis uses Sturm's theorem on zero sets for parabolic equations, scaling techniques, and the theory of gradient dynamical systems. For $m=1$, i.e., for the linear heat equation, the evolution completeness is directly associated with the completeness and closure of the eigenfunction subset for the linear self-adjoint operator $\Delta + \frac 12 y \cdot \nabla + \frac N2 I$ in a weighted $L^2$-space. These linear eigenfunctions are used as branching points of nonlinear ones for $m \approx 1^+$.

We study the relations between symmetry and degree for the Dirichlet problem for harmonic maps from the disc into the $2$-sphere. This allows us to exhibit multiple solutions in many homotopy classes for a wide range of boundary values with symmetries.

We discuss $L^p$-$L^q$ type estimates of the Stokes semigroup and their application to the Navier-Stokes equation in a perturbed half-space. Especially, we have the $L^p$-$L^q$ type estimate of the gradient of the Stokes semigroup for any $p$ and $q$ with $1 < p \leqq q < \infty$, while the same estimate holds only for the exponents $p$ and $q$ with $1 < p\leqq q \leqq n$ in the exterior domain case, where $n$ denotes the space dimension, and, therefore, we can get better results concerning the asymptotic behaviour of solutions to the Navier-Stokes equations compared with the exterior domain case. Our proof of the $L^p$-$L^q$ type estimate of the Stokes semigroup is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the Stokes resolvent near the origin. The order of asymptotic expansion of the Stokes resolvent near the origin is one half better compared with the exterior domain case, because we have the reflection principle on the boundary in the half-space case unlike the whole space case, and, then, such better asymptotics near the boundary are also obtained in the perturbed half-space by a perturbation argument. This is one of the reasons why the result in the perturbed half-space case is essentially better compared with the exterior domain case.