Minimizers for boundary reactions: renormalized energy, location of singularities, and applications

Abstract: The Casten-Holland and Matano theorem for interior reactions states that no nonconstant stable solutions exist in convex domains $Ω$ of $\mathbb{R}<sup>n$ under zero Neumann boundary conditions. In this paper we establish that the analogous statement fails for boundary reactions when $n=2$ (that is, for harmonic functions in $Ω$ with a Neumann reaction term on its boundary $\partialΩ$). For instance, nonconstant stable solutions exist when $Ω$ is a square, or a smooth strictly convex approximation of it. In regular polygons of many sides, which approach the circle, we can prove the existence of as many nonconstant stable solutions as wished. Instead, in the circle such stable solutions do not exist. More importantly, we can predict the existence or not of nonconstant stable solutions, as well as the location of its boundary "vortices" $(p,q)$, through the properties of a real function defined on $\partialΩ\times\partialΩ$ (the renormalized energy) which depends only on the conformal structure of the domain $Ω$. This requires the development of a new Ginzburg-Landau theory for real-valued functions and the analysis of the half-Laplacian on the real line.

• The paper proves that nonconstant stable solutions exist in rectangles and smooth strictly convex domains, overturning the classical rigidity expected from interior-reaction problems.
• The authors show that isolated local minima of the conformally defined renormalized energy determine the locations of two boundary transition points and generate stable solutions for sufficiently small ε.
• A new half-plane classification rules out homoclinic layers, while sharp logarithmic energy bounds connect blow-up analysis, Γ-convergence, and the construction of arbitrarily many stable patterns near disks.

Background and problem

The paper by Cabré, Cónsul, and Kurzke studies stable solutions of boundary reaction problems in planar domains. For a bounded smooth domain $\Omega\subset\mathbb{R}^2$ and a bistable nonlinearity $f=-G'$, they consider

$$\begin{cases} \Delta u = 0 & \text{in } \Omega,\ \partial_\nu u = \varepsilon^{-1} f(u) & \text{on } \partial\Omega, \end{cases}$$

which is the Euler–Lagrange equation of the functional $$E_\varepsilon(u)=\frac12\int_\Omega|\nabla u|^2+\frac1\varepsilon\int_{\partial\Omega}G(u)$$, with prototypical potential $G(u)=\frac14(1-u^2)^2$. Stability means nonnegativity of the second variation on $\partial\Omega$.

The motivation is the classical Casten–Holland–Matano theorem: for interior reactions with zero Neumann data, convex domains admit no nonconstant stable solutions. The authors show that this statement fails for boundary reactions: nonconstant stable solutions exist in squares and smooth strictly convex perturbations thereof, resolving a question that had been open since the 1990s (two earlier announcements of the convex-domain analogue were incorrect). The numerical evidence came from Cónsul–Jorba's 2005 finite-element computations, which found stability of two-vortex solutions in the unit square for $\varepsilon<0.352$.

Main results

Three results structure the paper:

• Existence in rectangles: for $\Omega=(0,L)\times(0,H)$ with $L\le H$ (or smooth strictly convex domains close to it), there exist nonconstant stable solutions for small $\varepsilon$, converging on the boundary to the characteristic function jumping at the midpoints $f=-G'$0 and $f=-G'$1.
• Arbitrarily many stable solutions near a disk: for every $f=-G'$2 there is a smooth strictly convex domain — arbitrarily close to a ball as $f=-G'$3, or a regular polygon with sufficiently many vertices — admitting at least $f=-G'$4 distinct nonconstant stable solutions. This contrasts sharply with Cónsul's 1996 result that the ball admits none; stability is thus not continuous under domain convergence to a disk.
• General criterion via renormalized energy: if $f=-G'$5 has an isolated local minimizer $f=-G'$6, then for all small $f=-G'$7 there is a nonconstant stable solution converging in $f=-G'$8 to the characteristic function $f=-G'$9 jumping at $$\begin{cases} \Delta u = 0 & \text{in } \Omega,\ \partial_\nu u = \varepsilon^{-1} f(u) & \text{on } \partial\Omega, \end{cases}$$0 and $$\begin{cases} \Delta u = 0 & \text{in } \Omega,\ \partial_\nu u = \varepsilon^{-1} f(u) & \text{on } \partial\Omega, \end{cases}$$1.

The criterion holds for general bistable potentials satisfying $$\begin{cases} \Delta u = 0 & \text{in } \Omega,\ \partial_\nu u = \varepsilon^{-1} f(u) & \text{on } \partial\Omega, \end{cases}$$2, vanishing only at $$\begin{cases} \Delta u = 0 & \text{in } \Omega,\ \partial_\nu u = \varepsilon^{-1} f(u) & \text{on } \partial\Omega, \end{cases}$$3, with $$\begin{cases} \Delta u = 0 & \text{in } \Omega,\ \partial_\nu u = \varepsilon^{-1} f(u) & \text{on } \partial\Omega, \end{cases}$$4; non-even potentials are allowed.

The renormalized energy

Since the limiting boundary datum $$\begin{cases} \Delta u = 0 & \text{in } \Omega,\ \partial_\nu u = \varepsilon^{-1} f(u) & \text{on } \partial\Omega, \end{cases}$$5, its harmonic extension has infinite Dirichlet energy. The authors nevertheless extract a finite renormalized contribution: removing $$\begin{cases} \Delta u = 0 & \text{in } \Omega,\ \partial_\nu u = \varepsilon^{-1} f(u) & \text{on } \partial\Omega, \end{cases}$$6-balls around the jump points,

$$\begin{cases} \Delta u = 0 & \text{in } \Omega,\ \partial_\nu u = \varepsilon^{-1} f(u) & \text{on } \partial\Omega, \end{cases}$$7

The function $$\begin{cases} \Delta u = 0 & \text{in } \Omega,\ \partial_\nu u = \varepsilon^{-1} f(u) & \text{on } \partial\Omega, \end{cases}$$8 depends only on the conformal structure of $$\begin{cases} \Delta u = 0 & \text{in } \Omega,\ \partial_\nu u = \varepsilon^{-1} f(u) & \text{on } \partial\Omega, \end{cases}$$9:

$$E_\varepsilon(u)=\frac12\int_\Omega|\nabla u|^2+\frac1\varepsilon\int_{\partial\Omega}G(u)$$0

for any conformal map $$E_\varepsilon(u)=\frac12\int_\Omega|\nabla u|^2+\frac1\varepsilon\int_{\partial\Omega}G(u)$$1 (well defined by Möbius invariance of the cross-ratio), and admits equivalent expressions through Dirichlet and Neumann Green's functions, e.g. $$E_\varepsilon(u)=\frac12\int_\Omega|\nabla u|^2+\frac1\varepsilon\int_{\partial\Omega}G(u)$$2. For the disk, $$E_\varepsilon(u)=\frac12\int_\Omega|\nabla u|^2+\frac1\varepsilon\int_{\partial\Omega}G(u)$$3, which is maximal at antipodal points and has no other critical points — consistent with nonexistence of stable solutions there. The construction parallels Bethuel–Brezis–Hélein theory but for real-valued functions taking values in $$E_\varepsilon(u)=\frac12\int_\Omega|\nabla u|^2+\frac1\varepsilon\int_{\partial\Omega}G(u)$$4 rather than $$E_\varepsilon(u)=\frac12\int_\Omega|\nabla u|^2+\frac1\varepsilon\int_{\partial\Omega}G(u)$$5.

A $$E_\varepsilon(u)=\frac12\int_\Omega|\nabla u|^2+\frac1\varepsilon\int_{\partial\Omega}G(u)$$6-convergence-type result connects the energies: any sequence with traces converging to $$E_\varepsilon(u)=\frac12\int_\Omega|\nabla u|^2+\frac1\varepsilon\int_{\partial\Omega}G(u)$$7 satisfies

$$E_\varepsilon(u)=\frac12\int_\Omega|\nabla u|^2+\frac1\varepsilon\int_{\partial\Omega}G(u)$$8

and comparison functions attaining the matching upper bound exist. Here $$E_\varepsilon(u)=\frac12\int_\Omega|\nabla u|^2+\frac1\varepsilon\int_{\partial\Omega}G(u)$$9 depends only on the nonlinearity; it is computed explicitly in the appendix for the Peierls–Nabarro nonlinearity $G(u)=\frac14(1-u^2)^2$0, giving $G(u)=\frac14(1-u^2)^2$1.

Note that the leading-order $G(u)=\frac14(1-u^2)^2$2-limit of Alberti–Bouchitté–Seppecher, which counts transition points, carries no information about local minima; the renormalized energy at the next order is what determines both existence and vortex location.

Classification in the half-plane

The sharp lower bound requires understanding the blow-up problem in the upper half-plane, equivalent to $G(u)=\frac14(1-u^2)^2$3 on $G(u)=\frac14(1-u^2)^2$4:

$G(u)=\frac14(1-u^2)^2$5

The paper proves a new classification theorem: every bounded solution with $G(u)=\frac14(1-u^2)^2$6 as $G(u)=\frac14(1-u^2)^2$7 is either constant or a translate/reflection of the unique layer solution. In particular, no homoclinic solutions exist — a result new even for $G(u)=\frac14(1-u^2)^2$8; previously it was known only for the integrable Peierls–Nabarro case via Toland's work. The proof combines maximum-principle arguments (using harmonic comparison functions built from derivatives of $G(u)=\frac14(1-u^2)^2$9) to obtain decay $\partial\Omega$0, then a Pohožaev identity forcing $\partial\Omega$1. They also establish the layer-solution energy expansion $\partial\Omega$2.

Proof strategy for the main theorem

Minimizing $\partial\Omega$3 over a closed $\partial\Omega$4-neighborhood $\partial\Omega$5 of a smoothed $\partial\Omega$6 yields solutions of a penalized problem with Lagrange multiplier bounded by $\partial\Omega$7 — small enough to vanish under $\partial\Omega$8-scale blow-up. Pohožaev identities on star-shaped subdomains produce covering results: the approximate transition set can be covered by boundedly many disjoint balls whose mutual distances are large compared to $\partial\Omega$9. Blow-up on each ball yields either constants or layers; a Struwe-style merging argument then shows exactly two heteroclinic balls suffice for constrained minimizers. Combining the sharp local lower bounds near the two vortices with weak lower semicontinuity of the Dirichlet energy away from them gives

$\varepsilon<0.352$0

with $\varepsilon<0.352$1. Comparing against the upper bound forces $\varepsilon<0.352$2; isolated minimality then gives $\varepsilon<0.352$3, $\varepsilon<0.352$4, so the constraint becomes inactive, the multiplier vanishes, and $\varepsilon<0.352$5 solves the true boundary reaction problem stably.

Domains with nontrivial local minimizers

For the rectangle, an explicit series computation of the Dirichlet Green's function shows that $\varepsilon<0.352$6 is an isolated local minimizer of $\varepsilon<0.352$7 whenever $\varepsilon<0.352$8; the proof reduces to positivity of a series $\varepsilon<0.352$9, guaranteed when $\Omega=(0,L)\times(0,H)$0 exceeds the root $\Omega=(0,L)\times(0,H)$1 of $\Omega=(0,L)\times(0,H)$2.

For arbitrary numbers of minimizers, the authors use Schwarz–Christoffel maps onto polygons, smoothed via Study's theorem (images of shifted half-planes remain convex). Since $\Omega=(0,L)\times(0,H)$3 near prevertices, jumps near corners are energetically costly; choosing the smoothing parameter $\Omega=(0,L)\times(0,H)$4 makes $\Omega=(0,L)\times(0,H)$5 smaller at cell centers than on cell boundaries, producing $\Omega=(0,L)\times(0,H)$6 isolated local minimizers. A regular-polygon variant yields domains with inradius and circumradius arbitrarily close — i.e., arbitrarily close to a disk — yet supporting many stable solutions.

Limitations and open questions

Several restrictions are explicit. The existence criterion requires an isolated local minimizer of $\Omega=(0,L)\times(0,H)$7; the method does not apply to degenerate critical points, and the general case of more than two jumps is only sketched ("can be treated with some modifications"). The analysis is confined to simply connected planar domains, exploiting conformal maps; higher-dimensional analogues are untouched. The result guarantees stability for each sufficiently small $\Omega=(0,L)\times(0,H)$8 but does not determine the threshold $\Omega=(0,L)\times(0,H)$9 (numerics suggest $L\le H$0 for the square). Independently, Sonego has announced existence of a nonconstant stable solution in every sufficiently elongated convex domain for some $L\le H$1, by a different method that does not identify vortex locations. Whether the Casten–Holland–Matano dichotomy admits a complete conformally invariant characterization — i.e., whether absence of nontrivial local minimizers of $L\le H$2 characterizes domains without stable patterns — remains open, as does the classification of all half-plane solutions beyond the homoclinic-free class treated here.

Conclusion

The paper resolves the long-standing question of whether boundary reactions obey the convex-domain rigidity known for interior reactions: they do not. The mechanism is fully characterized by a renormalized energy $L\le H$3 computable from Green's functions or conformal maps, whose isolated local minima both predict the existence of nonconstant stable solutions and locate their boundary vortices. Technically, the work develops a Ginzburg–Landau theory for $L\le H$4-valued boundary data, including a sharp classification of half-plane solutions and precise energy asymptotics, providing a template for variational analyses of singularly perturbed boundary value problems.