The thermoelastic stresses created in a solid phase domain in the course of solidification of a molten ingot are investigated. A nonlinear behaviour of the solid phase is admitted, too. This problem, obtained from a real situation by many simplifications, contains a moving boundary between the solid and the liquid phase domains. To make the usage of standard numerical packages possible, we propose here a fixed-domain approximation by means of including the liquid phase domain into the problem (in...

Asymptotics of solutions to Schrödinger equations with singular magnetic and electric potentials is investigated. By using a Almgren type monotonicity formula, separation of variables, and an iterative Brezis–Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with a homogeneity of order −1.

We prove that minimizers $u\in W^{1,n}$ of the functional $E_{}\left(u\right)=1/n{\int }_{|}{\nabla u|}^{n}dx+1/\left(4^n\right){\int }_{(}1-{\left|u\right|}^2{)}^{2}dx$, ⊂ ${ℝ}^n$, n ≥ 3, which satisfy the Dirichlet boundary condition $u_{}=g$ on for g: → $S^{n-1}$ with zero topological degree, converge in $W^{1,n}$ and $C_{loc}^{\alpha }$ for any α<1 - upon passing to a subsequence ${}_k\to 0$ - to some minimizing n-harmonic map. This is a generalization of an earlier result obtained for n=2 by Bethuel, Brezis, and Hélein. An example of nonunique asymptotic behaviour (which cannot occur in two dimensions if deg g = 0) is presented.