[Paper Review] On the uniqueness and monotonicity of solutions of free boundary problems

Daniele Bartolucci, Aleks Jevnikar|arXiv (Cornell University)|Jan 1, 2022

Nonlinear Partial Differential Equations52 references9 citations

TL;DR

This paper establishes the uniqueness and monotonicity of positive solutions to a class of free boundary problems arising in plasma physics, using a dual variational formulation. It proves that for any smooth, bounded domain Ω⊂ℝᴺ, solutions are unique and exhibit universal monotonicity in boundary density and energy up to a sharp threshold determined by the Sobolev embedding constant. The result resolves a long-standing open problem on uniqueness of variational solutions and provides a complete description of the solution branch on the two-dimensional ball.

ABSTRACT

For any smooth and bounded domain Ω⊂RN, we prove uniqueness of positive solutions of free boundary problems arising in plasma physics on Ω in a neat interval depending only by the best constant of the Sobolev embedding H01(Ω)↪L2p(Ω), [Formula presented] and show that the boundary density and a suitably defined energy share a universal monotonic behavior. At least to our knowledge, for p>1, this is the first result about the uniqueness for a domain which is not a two-dimensional ball and in particular the very first result about the monotonicity of solutions, which seems to be new even for p=1. The threshold, which is sharp for p=1, yields a new condition which guarantees that there is no free boundary inside Ω. As a corollary, in the same range, we solve a long-standing open problem (dating back to the work of Berestycki-Brezis in 1980) about the uniqueness of variational solutions. Moreover, on a two-dimensional ball we describe the full branch of positive solutions, that is, we prove the monotonicity along the curve of positive solutions until the boundary density vanishes.

Motivation & Objective

• To resolve the long-standing open problem of uniqueness for variational solutions of free boundary problems in plasma physics.
• To establish monotonic behavior of boundary density and energy along the solution curve.
• To characterize the full branch of positive solutions on the two-dimensional ball.
• To identify a sharp threshold for the absence of free boundaries inside Ω.
• To extend uniqueness results beyond the special cases of p=1 or radial symmetry.

Proposed method

• Formulate the problem via a dual variational problem (P)λ involving a constrained minimization over L∞ functions.
• Use the duality between (F)I and (P)λ to translate solution properties between the two formulations.
• Apply spectral theory and bifurcation analysis to study the solution curve λ ↦ (αλ, ψλ) in the parameter λ.
• Employ contraction mapping and a priori bounds to prove local uniqueness and regularity for small λ.
• Use monotonicity arguments and eigenvalue estimates to show dαλ/dλ < 0 and dEλ/dλ > 0 for λ in the interval [0, 1/p Λ(Ω,2p)).
• Leverage the sharp threshold 1/p Λ(Ω,2p) to characterize the maximal existence interval and boundary behavior.

Experimental results

Research questions

• RQ1 Is there a universal threshold beyond which no free boundary exists inside Ω for general domains and p > 1?
• RQ2 Can the solution curve (αλ, ψλ) be uniquely parameterized and shown to be real analytic and monotonic?
• RQ3 Does the uniqueness of positive solutions hold for p > 1 on non-radial domains?
• RQ4 What is the precise relationship between the solution curve and the best constant in the Sobolev embedding H¹₀(Ω) ↪ L²ᵖ(Ω)?
• RQ5 How does the boundary density and energy evolve along the solution branch, and is this behavior universal across domains?

Key findings

• For any smooth, bounded domain Ω⊂ℝᴺ and p ∈[1, pN), the solution (αλ, ψλ) of (P)λ is unique for all λ ∈[0, 1/p Λ(Ω,2p)).
• The solution curve is a real analytic simple curve with dαλ/dλ < 0 and dEλ/dλ > 0, indicating monotonic decrease in αλ and increase in energy Eλ.
• The threshold λ = 1/p Λ(Ω,2p) is sharp for p=1, where it coincides with the first Dirichlet eigenvalue λ(1)(Ω), and marks the point where αλ → 0.
• On the two-dimensional ball D₂, the full branch of positive solutions is described: αλ decreases from 1 to 0 and ψλ increases as λ increases to 1/p Λ(D₂,2p).
• The energy Eλ satisfies Eλ = E₀(Ω) + O(λ) as λ → 0+, with E₀(Ω) = ½ ∫∫ GΩ(x,y) dxdy ≤ |B₁|⁻²/ᴺ 4(N+2).
• For p > 1, the result resolves the long-standing open problem of uniqueness of variational solutions, showing that such solutions exist and are unique precisely for λ < λ∗∗(Ω,p) = (I∗∗(Ω,p))¹/𝑞.