Tangential Touch between Free and Fixed Boundaries in a Problem from Superconductivity

On the tangential touch between the free and the fixed boundaries for the two-phase obstacle-like problem

In this paper we consider the following two-phase obstacle-problem-like equation in the unit half-ball $Δu = λ_{+}χ_{\{u>0\}}-λ_{-}χ_{\{u<0\}}, λ_\pm>0$. We prove that the free boundary touches the fixed one in (uniformly) tangential fashion if the boundary data $f$ and its first and second derivatives vanish at the touch-point.

Tangential Touch between the Free and the Fixed Boundary in a Semilinear Free Boundary Problem in Two Dimensions

The main result of this paper concerns the behavior of a free boundary arising from a minimization problem, close to the fixed boundary in two dimensions.

完全非线性 Alt-Phillips 问题中自由边界与固定边界的切向接触

Tangential touch between free and fixed boundaries for the fully nonlinear Alt-Phillips problem

本文研究参数 $γ\in(1,2)$ 的完全非线性 Alt-Phillips 问题，证明自由边界在 Dirichlet 数据为零处与固定边界切向接触，该结论在 Laplacian 情形下同样新颖。

Tangential contact of free boundaries and the fixed boundary for variational solutions to a free transmission Problem

In this article we study functionals of the following type $$ \int_Ω \Big ( \langle A(x,u)\nabla u, \nabla u\rangle + Λ(x,u) \Big )\,dx $$ here $A(x,u)= A_+(x)χ_{\{u>0\}}+A_-(x) χ_{\{u\leq 0\}}$ for some elliptic and bounded matrices $A_{\pm}$ with Hölder continuous entries and $Λ(x,u) = λ_+(x) χ_{\{u>0\}} + λ_-(x) χ_{\{u\le 0\}}$. We prove that the free boundaries of minimizers of the above functional touches the fixed boundary $\partial Ω$ in a tangential fashion, provide the graph of boundary data touches its zeros smoothly. This assumption is reflected in the \eqref{DPT} condition.

Non-transversal intersection of free and fixed boundary for fully nonlinear elliptic operators in two dimensions

In the study of classical obstacle problems, it is well known that in many configurations the free boundary intersects the fixed boundary tangentially. The arguments involved in producing results of this type rely on the linear structure of the operator. In this paper we employ a different approach and prove tangential touch of free and fixed boundary in two dimensions for fully nonlinear elliptic operators. Along the way, several $n$-dimensional results of independent interest are obtained such as BMO-estimates, $C^{1,1}$ regularity up to the fixed boundary, and a description of the behavior …

There are Asymmetric Minimizers for the One-Dimensional Ginzburg-Landau Model of Superconductivity

We study a boundary value problem associated with a system of two second order differential equations with cubic nonlinearity which model a film of superconductor material subjected to a tangential magnetic field. We show that for an appropriate range of parameters there are {\it asymmetric} solutions, and only trivial {\it symmetric} solutions. We then show that the associated energy function is negative for the asymmetric solutions, and zero for the trivial symmetric solution. It follows that a global minimizer of the energy is asymmetric.