Boundary behaviour for a singular perturbation problem

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• 作者：A.L. Karakhanyan^a ; ^{aram6k@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; H. Shahgholian^b ; ^{henriksh@math.kth.se" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 35R35
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：138
• 期：Complete
• 页码：176-188
• 全文大小：622 K

文摘

In this paper we study the boundary behaviour of the family of solutions {u^ε}$\left\{u^{\epsilon }\right\}$ to singular perturbation problem Δu^ε=β_ε(u^ε),∣u^ε∣≤1$\Delta u^{\epsilon }={\beta }_{\epsilon }\left(u^{\epsilon }\right),\mid u^{\epsilon }\mid \le 1$ in $B_1^{+}=\{x_n>0\}\cap \{\mid x\mid <1\}$, where a smooth boundary data f$f$ is prescribed on the flat portion of $\partial B_1^{+}$. Here ${\beta }_{\epsilon }(\cdot )=\frac{1}{\epsilon }\beta \left(\frac{\cdot }{\epsilon }\right),\beta \in C_0^{\infty }(0,1),\beta \ge 0,{\int }_0^1\beta \left(t\right)=M>0$ is an approximation of identity. If ∇f(z)=0$\nabla f\left(z\right)=0$ whenever f(z)=0$f\left(z\right)=0$ then the level sets edaa0dcd88c73235cb6eff248" title="Click to view the MathML source">∂{u^ε>0}$\partial \{u^{\epsilon }>0\}$ approach the fixed boundary in tangential fashion with uniform speed. The methods we employ here use delicate analysis of local solutions, along with elaborated version of the so-called monotonicity formulas and classification of global profiles.