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Hamiltonian identities for elliptic partial differential equations. (English) Zbl 1148.35023
This paper deals with Hamiltonian identities for elliptic partial differential equations. Let \(x=(x^\prime, x_n) \in {\mathbb{R}}^n\) and \(u\in C^2(\mathbb{R}^n, \mathbb{R}^m)\) be an entire solution of the system of partial differential equations: \[ -\Delta u +\nabla_u H(u(x)) =0, \quad x \in \mathbb{R}^n, \] where \(H\) is a \(C^{1, \alpha}\) potential function. Under certain conditions, the author shows that \(u\) satisfies the following Hamiltonian identity: \[ \int_{\mathbb{R}^{n-1}} \left ( {\frac 12 } (| \nabla_{x^\prime} u| ^2 -| u_{x_n} | ^2 ) +H(u(x)) \right ) dx^\prime =C, \quad \forall x_n \in \mathbb{R}. \] Several applications of the above identity are then discussed in the paper. Particularly, Young’s law for the contact angles in triple junction formation is proven rigorously, and the structure of level curves of saddle solutions to Allen-Cahn equation is analyzed.

MSC:
35J60 Nonlinear elliptic equations
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
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