Abstract
In this paper we prove symmetry for solutions to the semi-linear elliptic equation Δ u = f ( u ) in B 1 , 0 ≤ u > M , in B 1 , u = M , on ∂ B 1 , \begin{equation*} \Delta u = f(u) \quad \text { in } B_1, \qquad 0 \leq u > M, \quad \text { in } B_1, \qquad u = M, \quad \text { on } \partial B_1, \end{equation*} where M > 0 M>0 is a constant, and B 1 B_1 is the unit ball. Under certain assumptions on the r.h.s. f ( u ) f (u) , the C 1 C^1 -regularity of the free boundary ∂ { u > 0 } \partial \{u>0\} and a second order asymptotic expansion for u u at free boundary points, we derive the spherical symmetry of solutions. A key tool, in addition to the classical moving plane technique, is a boundary Harnack principle (with r.h.s.) that replaces Serrin’s celebrated boundary point lemma, which is not available in our case due to lack of C 2 C^2 -regularity of solutions.
Highlights
Introduction and the main problemWe use the moving-plane technique to prove symmetry for a free boundary problem, whose solutions lack C2-smoothness up to the free boundary
Our approach applies to more general setting, for clarity of exposition, we have considered the most simple equation, which is the following semilinear PDE
The celebrated Serrin’s boundary point lemma is not accessible anymore. We circumvent this by imposing a weak second-order asymptotic expansion for solutions close to the free boundary points; see the statement of our theorem for more details
Summary
We use the moving-plane technique to prove symmetry for a free boundary problem, whose solutions lack C2-smoothness up to the free boundary. We aim at proving spherical symmetry of solutions to the problem, by employing the celebrated moving plane technique originated in A.D. Alexandrov’s work [1], and cleverly used and promoted by James Serrin [15] through a new (Serrin’s) boundary point lemma.
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