Decoupling of DeGiorgi-Type Systems via Multi-Marginal Optimal Transport

Abstract

We exhibit a surprising relationship between elliptic gradient systems of PDEs, multi-marginal MongeKantorovich optimal transport problem, and multivariable Hardy-Littlewood inequalities. We show that the notion of an orientable elliptic system, conjectured in [7] to imply that (in low dimensions) solutions with certain monotonicity properties are essentially 1-dimensional, is equivalent to the definition of a compatible cost function, known to imply uniqueness and structural results for optimal measures to certain Monge-Kantorovich problems [13]. Orientable nonlinearities and compatible cost functions turned out to be also related to submodular functions, which appear in rearrangement inequalities of HardyLittlewood type. We use this equivalence to establish a decoupling result for certain solutions to elliptic PDEs and show that under the orientability condition, the decoupling has additional properties, due to the connection to optimal transport.

Optimal potential energy growth lower bounds for minimizing solutions to the vectorial Allen–Cahn equation in two space dimensions

“…In fact, as was observed in [12] ( [14]), if a solution u satisfies the vectorial analog of Modica's gradient bound (3), it follows that its energy satisfies the strong monotonicity property (4). Armed with (7), and doing some more work in the case n D 2 ([12]), it is easy to show that, if x 0 2 R n , the energy of each nonconstant solution to the system (1) satisfies:…”

“…In the vectorial case, that is when m ≥2, in the absence of the maximum principle, it is not known in general whether the analog of the gradient bound holds; see and for examples and counterexamples respectively of its validity in the case of the phase transition systems that we describe following , and for counterexamples in the case of the Ginzburg‐Landau system . Actually, for potentials W ≥0 that vanish on a codimension one manifold , easy counterexamples are provided by the one‐dimensional periodic solutions in which shadow closed, non‐degenerate geodesics of (whenever the latter exist).…”

Multi-marginal optimal transport on Riemannian manifolds

“…Note that the c-concave function u c i is superdifferentiable everywhere (since the relevant domains, i.e. support of measures, etc, are compact), and that at any point x i where u i (x i ) + u c i (y) = Furthermore, as for y ∈ȳ(S) we have v(y) = u c i (y), we have equality in (16) for each such x i .…”

“…In recent years, the multi-marginal case, m ≥ 3 has attracted increasing attention, due to emerging applications in areas such as economics [5] [7], physics [9] [3], cyclical monotonicity [10,14,15], and systems of elliptic equations [16]. In contrast to the two marginal case, the structure of solutions for general cost functions is not well understood.…”

Counterexamples to Modica's gradient estimate for systems arising in multi-phase transitions

“…In this regard, it is of interest to know whether the analog of Modica's gradient estimate (2) holds for bounded, entire solutions to this class of systems (see the related comments in [3] and Open Problem 1 in [9]). It is worth noting that a class of such systems which satisfy this property has been provided in [10]. On the other hand, a counterexample to this property, for such systems, was provided very recently by [14].…”