(IMTech–UPC and ICREA). Received 24 Feb, 2021.
The paper [1] solves a semilinear version of Hilbert’s XIXth problem that remained open since the 1970s. In dimensions n ≤ 9, it establishes the smoothness of local minimizers of the functional
where Ω is an open set of R^{n}, u : Ω ⊂ R^{n} → R, and F : R → R is a given function. Making a first variation u + εv, integrating by parts, and defining f = F ′, one easily sees that critical points of the functional satisfy the reaction-diffusion equation or semilinear elliptic equation
The setting is relevant since, in numerous physical phenomena and geometric problems, observable states try to minimize a certain functional. In classical mechanics, and also in (1), the functional is called action and it is the integral of a Lagrangian. In geometry, two important examples are geodesics in a Riemannian manifold (where the functional is their length) and minimal surfaces in Euclidean space (where the functional is area). In all these problems, one considers minimizers —or more generally, critical points— of the functional among functions, or surfaces, with prescribed boundary values (for instance, the end points of the geodesic).
In Hilbert’s XIXth problem the functional is given by A(u) = 
for some convex function L : R^{n} → R and, therefore, A is convex. As a consequence, if it admits a critical point, this will be an absolute minimizer which accordingly fulfils the principle of least action in mechanics. However, for many potentials F in (1), as well as for minimal surfaces, the functional is not convex and the states that one observes in nature are only local minimizers (minimizers among small perturbations). Such critical points are stable, in the sense that the second variation of the functional at such state is nonnegative definite.
A situation in which this occurs concerns catenoids: the soap film or minimal surface formed between two coaxial parallel circular rings. In [2], catenoids are experimentally produced in a lab and photographed while the distance between the two circular wires increases. For a range of distances, the observed catenoids are minimal surfaces which are stable but not absolute minimizers, since the two flat disks spanned by the wires have less area.
In the 1960s the Italian school proved that the Simons’ cone 
is an absolute minimizer of area in R^{2m} if n = 2m ≥ 8. At the same time, a sequence of outstanding contributions (J. Simons’ being a prominent one) established the smoothness, when n ≤ 7, of every minimal hypersurfaces of R^{n} that is an aboslute minimizer. It is a long-standing open problem to extend this regularity result to the larger class of stable minimal surfaces. It is only known to hold for surfaces in R^{3}. See the survey [3] for more details.
The paper [1] takes on the analogue question, the regularity of stable solutions, for the reaction-diffusion equation (2). The problem was motivated by a combustion model for the thermal self-ignition of a chemically active mixture of gases in a container Ω. Here, u = u(x) is the temperature of the point x ∈ Ω. For a large class of nonlinearities, which include the
so called Gelfand problem −∆u = λe^{u} arising from Arrhenius law in chemical kinetics, the situation is similar to the one of catenoids. Indeed, for a certain range λ ∈ (0, λ∗) of parameters, there exists a stable solution (that is, a solution at which the action has a nonnegative definite second variation) which is not an absolute minimizer (since the action functional is unbounded from below). In terms of the associated nonlinear heat equation v_{t} − ∆v = λe^{v} , it corresponds to the stationary temperature of the container observed for very large times when the initial temperaure is constant v(·, 0) = 0 and ignition fails. When λ ≥ λ∗ (successful ignition), no solution exists —in the same way that catenoids did not exist for large distances between the wires.
When n ≥ 3, Ω = B1 is the unit ball,
a simple computation shows that we are in the presence of a singular solution of (2) vanishing on ∂B1. Similarly to the Simons’ cone in minimal surfaces theory, this solution turns out to be stable in high dimensions, precisely when n ≥ 10. On the other hand, in the 1970s Crandall and Rabinowitz [4] established that, for f (u) = e^{u} or f (u) = (1 + u)^{p} with p > 1, stable solutions in any smooth bounded domain Ω are bounded (and hence smooth, by classical elliptic regularity theory) when n ≤ 9. This result was the main reason for Haïm Brézis to raise the following question in the 1990s (which we cite almost literally from a later reference):
Brezis [5, Open problem 1]: Is there something “sacred” about dimension 10? More precisely, is it possible in “low” dimensions to construct some f (and some Ω) for which a singular stable solution exists? Alternatively, can one prove in “low” dimensions that every stable solution is smooth for every f and every Ω?
The last twenty five years have produced a large literature on the topic; see the monograph [6]. The main developments proving that stable solutions to (2) are smooth (no matter what the nonlinearity f is) were made
- by Nedev [7] in 2000, when n ≤ 3 (and f is convex);
- by the author and Capella [8] in 2006, when Ω = B1 (u is radially symmetric) and n ≤ 9;
- by the author [9] in 2010, when n ≤ 4 (and Ω is convex).
Note that the 2006 result in the radially symmetric case, [8], accomplished the optimal range n ≤ 9 for every nonlinearity f. This gave hope for the result to be true also in the general non-radial case, though no certainty was assured. The work [1] finally solves Brezis’ open problem:
Theorem In dimensions n ≤ 9, under the only requirement for the nonlinearity f to be nonnegative, every (energy) stable solution of (2) is smooth in the open set Ω.
Furthermore, adding the vanishing boundary condition u = 0 on ∂Ω, the article proves regularity up to the boundary when Ω is of class C3 and n ≤ 9, assuming now f to be nonnegative, nondecreasing, and convex. Both results come along with new universal Hölder-continuity estimates:
where α ∈ (0, 1) and C are dimensional constants, while C_{Ω} depends only on Ω. These estimates are rather surprising for a nonlinear problem, since they make no reference to the reaction nonlinearity f . The stability of the solution is crucial for their validity. For the expert reader, [1] also establishes another open problem, now from [10]: an apriori H1 = W 1,2 estimate for stable solutions in all dimensions n.
If the nonnegativeness of f is a needed requirement in the theorem remains as an open question. It is only known to be unnecessary for n ≤ 4, as well as for n ≤ 9 in the radial case.
An analogue result for equations involving the p-Laplacian has been proved in 2020 by the author, Miraglio, and Sanchón, [11]. In terms of dimensions it is optimal for p > 2, but not for p < 2. On the other hand, for the recently very active area of fractional Laplacians, an optimal result is largely open —even in the radial case. The optimal dimensions for regularity have only been accomplished in a 2014 work of Ros-Oton [12] for the Gelfand nonlinearity f (u) = λeu in domains which are symmetric and convex with respect to all coordinate directions.
References
[1] Cabré, X.; Figalli, A.; Ros-Oton, X.; Serra, J. Stable solutions to semilinear elliptic equations are smooth up to dimension 9. Acta Math. 224 (2020), 187-252.
[2] Ito, M.; Sato, T. In situ observation of a soap-film catenoid – a simple educational physics experiment. Eur. J. Phys. 31 (2010), 357-365.
[3] Colding, T. H.; Minicozzi, W. P. In search of stable geometric structures. Notices Amer. Math. Soc. 66 (2019), 1785-1791.
[4] Crandall, M. G.; Rabinowitz, P. H. Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems. Arch. Ration. Mech. Anal. 58 (1975), 207-218.
[5] Brezis, H. Is there failure of the inverse function theorem? Morse theory, minimax theory and their applications to nonlinear differential equations, 23-33, New Stud. Adv. Math., 1, Int. Press, Somerville, MA, 2003.
[6] Dupaigne, L. Stable Solutions of Elliptic Partial Differential Equations. Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 143, CRC Press, Boca Raton, 2011.
[7] Nedev, G. Regularity of the extremal solution of semilinear elliptic equations. C. R. Acad. Sci. Paris 330 (2000), 997-1002.
[8] Cabré, X.; Capella, A. Regularity of radial minimizers and extremal solutions of semilinear elliptic equations. J. Funct. Anal. 238 (2006), 709-733.
[9] Cabré, X. Regularity of minimizers of semilinear elliptic problems up to dimension 4. Comm. Pure Appl. Math. 63 (2010), 1362-1380.
[10] Brezis, H; Vázquez. J. L. Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complut. Madrid 10 (1997), 443-469.
[11] Cabré, X.; Miraglio, P.; Sanchón, M. Optimal regularity of stable solutions to nonlinear equations involving the p-Laplacian. Adv. Calc. Var. 2020, https://doi.org/10.1515/acv-2020-0055.
[12] Ros-Oton, X. Regularity for the fractional Gelfand problem up to dimension 7. J. Math. Anal. Appl. 419 (2014), 10-19.