Free Boundary Problems by Darya Apushkinskaya

Author:Darya Apushkinskaya
Language: eng
Format: epub
ISBN: 9783319970790
Publisher: Springer International Publishing

Lemma 2.19

Let be a degree 2 parabolic homogeneous function with respect to the origin, and let (0, 0) ∈ Γ(u). Then for some we have

(2.77)

Proof

Let e be a unit spatial direction orthogonal to e 1. We claim that the function v := D eu does not change its sign. To prove this, first we extend v by zero across the plane Π to the entire space and keep the notation v for the extension. From homogeneity of u with respect to the origin it follows that

(2.78)

where the functional Φ is defined by the formula (A.2). However, in our case the equality (2.78) is possible only if C(e) = 0, see Sect. A.1 in Appendix A. This implies or for all points of . So, we have proved that D eu preserves its sign. Since this is true for all spatial directions e orthogonal to e 1, it follows that u(x, t) is two-space dimensional, i.e., in suitable spatial coordinate

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