[Paper Review] A free boundary problem for semi-linear elliptic equation and its applications

Jianfeng Cheng, Lili Du|arXiv (Cornell University)|Jun 3, 2020

Nonlinear Partial Differential Equations30 references4 citations

TL;DR

This paper establishes the existence, Lipschitz continuity, non-degeneracy, and regularity of solutions and the free boundary for a semilinear nonhomogeneous elliptic equation with a Bernoulli-type free boundary condition. Using a variational approach, it proves the free boundary is $ C^{1,eta} $-regular and applies the results to prove the well-posedness of steady, incompressible, inviscid jet and cavitational flows with general vorticity.

ABSTRACT

In this paper, we consider a free boundary problem of a semilinear nonhomogeneous elliptic equation with Bernoulli's type free boundary. The existence and regularity of the solution to the free boundary problem are established by use of the variational approach. In particular, we establish the Lipschitz continuity and non-degeneracy of a minimum, and regularity of the free boundary. As a direct and important application, the well-posedness results on the steady, incompressible inviscid jet and cavitational flow with general vorticity are also obtained in this paper.

Motivation & Objective

• To establish the existence and regularity of solutions to a free boundary problem governed by a semilinear nonhomogeneous elliptic equation with a Bernoulli-type condition.
• To extend classical results from the homogeneous case (e.g., Alt and Caffarelli) to the nonhomogeneous, semilinear setting with variable vorticity.
• To prove the Lipschitz continuity and non-degeneracy of the minimizer, and the $ C^{1,eta} $-regularity of the free boundary.
• To apply the theoretical results to the well-posedness of steady, incompressible, inviscid jet and cavitational flows with general vorticity.
• To establish uniqueness and monotonicity of the solution and the free boundary in the context of physical flow models.

Proposed method

• Formulates the problem as a variational minimization of the functional $ J( heta) = ∫_\Omega (|\nabla\theta|^2 + F(\theta) + \lambda^2(X) \chi_{\{\theta>0\}}) \, dX $, where $ F(t) = -2\int_0^t f(s)\,ds $.
• Uses the direct method of calculus of variations to prove existence of a minimizer in a suitable Sobolev space.
• Applies a blow-up analysis and measure estimates to study the behavior of the solution near the free boundary.
• Employs the strong maximum principle and Hopf’s lemma to compare solutions and establish uniqueness of the free boundary and the boundary value $ \lambda $.
• Establishes flatness and linear growth estimates near the free boundary to deduce its regularity.
• Uses the asymptotic behavior of solutions in the downstream region to prove uniqueness of the flow profile.

Experimental results

Research questions

• RQ1 Does a minimizer of the energy functional exist for the semilinear elliptic free boundary problem with nonhomogeneous term $ f(\psi) \neq 0 $?
• RQ2 Can the Lipschitz continuity and non-degeneracy of the minimizer be established under general vorticity conditions?
• RQ3 Is the free boundary $ \partial\{\psi > 0\} \cap \Omega $ regular (specifically $ C^{1,\beta} $) when $ f(\psi) \neq 0 $?
• RQ4 Can the theoretical results be applied to prove the existence and uniqueness of steady, incompressible, inviscid jet and cavitational flows with general vorticity?
• RQ5 Is the boundary value $ \lambda $ unique in the context of the free boundary problem for physical flow models?

Key findings

• The minimizer $ \psi $ of the energy functional is Lipschitz continuous in $ \Omega $, as established in Theorem 2.4.
• The minimizer satisfies a non-degeneracy condition: $ \sup_{B_r(x_0)} \psi \geq c r $ for some $ c > 0 $, uniformly in $ r $, when $ x_0 \in \partial\{\psi > 0\} $.
• The free boundary $ \Gamma = \partial\{\psi > 0\} \cap \Omega $ is $ C^{1,\beta} $-regular, as proven in Theorem 3.15.
• The solution $ \psi $ to the free boundary problem is unique in the class of minimizers satisfying the given boundary and free boundary conditions.
• The boundary value $ \lambda $ is uniquely determined by the asymptotic behavior of the solution in the downstream region.
• The well-posedness of steady, incompressible, inviscid jet and cavitational flows with general vorticity is established via the variational framework.