Conformal Geometry and The Composite Membrane Problem

We consider smooth bounded surfaces with a smooth boundary and a prescribed background metric g_0. We now consider all metrics g conformal to g_0 which have a prescribed volume M. We now minimize the first eigenvalue of the Laplace operator of g over the metrics conformal to g_0 and having the prescribed volume. We show that this problem is equivalent to the study of the Composite Membrane Problem, a free boundary problem studied earlier by the author and his collaborators in all dimensions. Thus complete answers, existence of the limit metric, regularity of the minimizing eigenfunction and various qualitative properties of the metric are easily obtained from the solution of the Composite Membrane problem. The problem of minimizing eigenvalues over conformal classes has a higher dimensional analog for the critical GJMS operator and leads to new classes and questions for higher order unstable free boundary problems. In particular in dimension 4 we are lead to free boundary problems involving the Paneitz operator and in odd dimensions to fractional free boundary problems of unstable type.

Proof. The proof follows by simply noting that the numerator in the Rayleigh quotient in (0.2) is a conformal invariant as we are in two dimensions. That is Set ρ = e 2u , and so the problem (0.2) by virtue of the constraints can be re-written as: For In the case our background metric g 0 = dx 2 +dy 2 is the flat metric, then this last minimization problem (0.3) is exactly the one treated in Theorem 13, [3], i.e. the Composite Membrane Problem.
The existence and regularity of the minimization problem associated with the composite membrane problem is the subject of many articles, [3], [4], [10], [1], [9], [5] and [6] among others. Thus these articles now give complete information about the minimization of eigenvalues in conformal classes, the existence and regularity of the limit metric and the associated eigenfunction and more crucially the optimal C 1,1 regularity of the minimizing eigenfunction.
The limit metric and thus the associated eigenfunction need not be unique due to a symmetry breaking phenomena [3]. We point out that unlike a traditional minimization problem for eigenfunctions, we have to find a minimizing pair (u ∞ , φ ∞ ). We remark that our regularity results rely on blow-up analysis and so curvature has no role to play in regularity issues. We may summarize the results in [3], [4], [5] and [6] as applied to eigenvalue minimization (0. 2) in the form of a theorem. The proofs essentially follow by using ρ = e 2u . We also restrict to the case g 0 is flat.
Theorem 0.2. There exists a limit metric ρ ∞ g 0 = e 2u∞ g 0 and an associated limit eigenfunc- where D ⊆ Ω and D c denotes the complement of D in Ω.
(4) D c has finitely many components, and the free boundary ∂D c consists of finitely many, simple, closed real-analytic curves.
(5) Due to symmetry breaking the function u ∞ associated to the limiting metric and the eigenfunction φ ∞ is not necessarily unique. If Ω is a disk, then u ∞ and the eigenfunction is unique. Additional hypotheses on convex Ω does guarantee uniqueness of (u ∞ , φ ∞ ), see [5]. (6) If Ω is simply-connected, then D is connected.
We now pass to a higher dimensional analog of the problem stated above. This concerns the critical GJMS operator and its conformal invariance properties. The GJMS hierarchy of conformally invariant operators was constructed in [7] and include the Paneitz operator and the Yamabe operator.
Specifically we consider (Ω n , g 0 ) and the associated critical GJMS operator P g 0 n/2 . For us the results proved in [7], [8] prove crucial. The operator P g 0 n/2 has the property that if one considers the metric g = e 2u g 0 , then the GJMS operator in the new metric P g n/2 satisfies the relation, (see [7] [8]) (0.4) P g n/2 (φ) = e −nu P g 0 n/2 (φ).
The operator P g n/2 is an elliptic, self-adjoint operator with leading term (−∆ g ) n/2 . So in particular it is fourth order in dimension 4. This fourth order operator in dimension 4 is the Paneitz operator. The operators in odd dimensions are non-local pseudo-differential operators. One may consult [2] for the role in Conformal Geometry of the fractional operators that arise.
We have the following proposition whose proof follows the same scheme as the two dimensional case where instead we use (0.4). The proposition leads to a higher order free boundary problem involving now the critical GJMS operator. In dimension 4 the operator that arises is the Paneitz operator and for odd n the operator is a fractional operator leading to fractional free boundary problems which are unstable. with given M > 0, λ = e −nA > 0, Λ = e nA < ∞ and 0 < λ ≤ ρ ≤ Λ < ∞ and where ρ = e nu .