The main topic of my thesis lies on the convergence of semilinear elliptic partial differential equations, mainly the Allen-Cahn equation (ACE) and its applications to the mean curvature flow. Firstly, we are going to classify the solutions of semilinear elliptic equations (not only the ACE) under some necessary and technical conditions such as monotonicity and boundedness by the moving plane method established by Berestycki, Cafarelli and Nirenberg in 1990s (in lower dimensions); and secondly, we will present the modified convergence method established by Cabré and Savin in 2000s to show certain rigidity of ACE (in higher dimensions), which is related to the De Giorgi conjecture (We refer readers to Professor Wei's survey [Chan18] to see more details and recent works on De Giorgi's conjecture); Finally, we will study the convergence of Allen-Cahn equation (and vector-valued version) under the mean curvature flow with further methods with the harmonic map.
To be simple, the solution u is always assumed to be monotone along xn-axis and be bounded in [-1,1].
The semilinear elliptic equation Δu-F'(u)=0 may be regarded as a critical point of the following functional E with kinetic energy |∇u|2 and potential energy F(u)
where u is a function defined on Ω ⊂ ℝn and F is a nonnegative function in C2. Particularly, in our work, we focus on the potential F with double-well property, called the double-well potential, denoted by W, namely
A well-known model is W(s) = ¼(1-s2)2, which is related to the so-called Allen-Cahn equation, i.e.,
In this case, the functional above, called Ginzburg-Landau energy functional, denoted by J, is related to the phase transition theory [Braides02], in particular, the van der Walls-Cahn-Hilliard theory [Cahn58][Allen79]. Intuitionally, one may imagine that there is a cup of ice-water mixture, where the functional J can be regarded as the energy of the mixture and u is the density of ice (or water) with respect to the mixture. Thanks to the Hamilton's principle of least action, the ice-water system tends to stay at the least level of such energy; and then we obtain the critical equation written above.
We now turn to the dynamical model from the static case by adding the assumption of thermal energy, which accelerates the ice to melt. On the other hand, we obtain the dynamical equation, introduced by Allen and Cahn [Allen79], i.e.,
which is in fact the gradient flow of Ginzburg-Landau functional J, when W=¼(1-s2)2, namely
It is evident that the Allen-Cahn equation is the steady equation of the dynamical version. Hence, one may be interested in the limitation of such flow, which will be illustrated with more details in the next part after we introduce the blow-down. In fact, geometrically, through the ε↘0+ blow-down, the asymptotic behavior converges like the mean curvature flow.
Furthermore, the vector-value Allen-Cahn equation
where W(u):ℝm→ℝ is a nonnegative radial function that vanishes precisely on two concentric spheres, which extends the definition of double-well potential to higher dimensions, was introduced, such as Bronsard and Stoth [Bronsard98], who established the case for m=2. On the other hand, it also can be regarded as a typical example of so-called reaction-diffusion system introduced by Rubinstein, Sternberg and Keller [Rubinstein89][Rubinstein89b].
In our paper, inspired many previous seminal works in previous about six decades, we are interested the symmetry (depend on merely on variable [Berestycki91]) of critical points of such functional J - or solutions of equations, equivalently - and furthermore, the convergence (in Γ-convergence [Braides02] and mean curvature flow sense) of equations.
In 1978, De Giorgi [DeGiorgi79] guessed that the monotone and bounded entire solution u to the Allen-Cahn equation
is dependent of merely ONE variable as n ≤ 8. Noting that all level sets of u, say {u=s}1≤s≤1 are graphs evidently, since u is monotone, we will illustrate the reason that De Giorgi believe the level sets are flat, which is equivalent to depending on one variable up to n ≤ 8 later. Firstly, we state the conjecture formally.
To understand this conjecture, we now present a more intuitional and geometric problem, which was completely solved in the previous century, called the Bernstein problem.
The problem was named and solved as n ≤ 3 by Bernstein [Bernstein27]. Indeed, the surface Σ is the graph of entire solution v to following minimal surfaces equation (MSE),
where derives from the variational of the area functional, i.e.,
However, unfortunately, one may encounter some difficulties when apply Bernstein's cutoff method in higher dimensions. After about fifty years, Fleming [Fleming62] established a ground-breaking method called blowing-down and minimal cone to solve this problem in the same dimension, nevertheless, which not only motivated the following works in 1960s to solve this problem completely but also created a seminal field, namely so-called geometric measure theory. Subsequently, De Giorgi [DeGiorgi65] discovered a "one-dimensional gap" of minimal cone (If there is no non-flat minimal cone in ℝN-1 then the analogue of Bernstein's theorem is true for graphs in ℝN), which implies the Bernstein problem in ℝ4. Then, from 1966 to 1968, Almgren [Almgren66] and Simons [Simons68] proved it as N=5 and N≤7 respectively. Finally, in 1969, Bombieri, De Giorgi and Simons [Bombieri69] showed there exists non-flat minimal cones in ℝN, as N≥9 by so-called Simons cone. Hence, the Bernstein problem was solved perfectly. We refer the readers to [Giusti84] for more detailed historical viewpoints and discussions.
Back to De Giorgi's conjecture, we rewrite the Ginzburg-Landau energy
where W is a double-well potential. We say u is a minimizer of J(·,Ω), if
particularly, saying a global minimizer, if Ω=ℝn.
To simplify, let Ω=B1={x∈ℝn:|x|<1}. Introduce the rescaling of u, say uε(x):=u(x/ε), x∈B1,ε>0. If u minimizes J(·,Bε-1), then uε minimizes the rescaling functional Jε in B1, namely
Intuitionally, fixing v, the main contribution in Jε comes from the potential W(v) as ε↘0+, which assumes the minimum when v equals either 1 or -1. Hence, v may jump from the region of v=1 to v=-1 through the interface dramatically.
Let Σs:={v=s}, by Cauchy-Schwarz inequality and co-area formula, we obtain
where Hn-1(Σs) is n-1 dimensional Hausdorff measure of the level set {v=s}=Σs. We hope that Jε is minimized, which requires every level sets of v are minimal surfaces that minimizes the term Hn-1(Σs). Therefore, it is reasonable to guess that all level sets of u are flat at least n ≤ 8.
Furthermore, we have the equality from the inequality above, i.e., ε|∇v|=√(2W(v)). Let
where dΣ0:B1→ℝ≥0 is the signed distance function from x to the level set Σ0. Clearly
where g0 is actually the unique minimizer of Jε in 1 dimension and is increasing. It is also useful to note that g0 is bounded since W is double-well. Particularly, if W=¼(1-s2)2,ε=1, then g0=tanh(x/√2), which is exactly the particular solution of Allen-Cahn equation in 1 dimension, namely
Thus, one may rewrite the conjecture more precisely.
Now, we present several milestone works on the De Giorgi conjecture. Firstly, inspired by the methods established by Berestycki, Caffarelli and Nirenberg in [Berestycki97], Ghoussoub and Gui [Ghoussoub98] proved it for general W in n=2. Subsequently, by observing "one dimensional gap" of energy estimate in previous work, Ambrosio and Cabré [Ambrosio00] proved it in n=3; and furthermore, Alberti, Ambrosio and Cabré [Alberti01] proved it for general W in the same dimension by the argument of calibration. Motivated by the argument of asymptotic flatness in [Alberti01], Savin [Savin09] solved it as 4≤n≤8 under a technical assumption,
by modifying a flatness convergence theorem from De Giorgi (see more details in [Giusti84]), derived from Modica [Modica78]. Finally, del Pino, Kowalczyk and Wei [DelPino11] established a counterexample as n≥9 which was long believed to exist, based on Bombieri, De Giorgi and Simons' work ([Bombieri69]). Up to this day, the conjecture is still OPEN as 4≤n≤8 for dropping the technical condition above.
However, one may ask a weaker question by strengthening the technical assumption to uniform convergence, which is called the Gibbons conjecture that was solved completely by Farina [Farina99] and Barlow, Bass and Gui [Barlow00] and Berestycki, Hamel, and Monneau [Berestycki00], independently.
Thanks to the rescaling of u by uε and the blow-down as ε↘0+, we obtain the dynamical equation in scaling version, i.e., ∂tu=εΔu-1/εW'(u)=εΔu+1/ε(u-u3), which is in fact the typical Reaction-diffusion equation [Rubinstein89][Rubinstein89b]. By an evident scaling, we have
We are interested in the asymptotic behavior of u by the flow above as ε↘0+, which has been studied by Modica [Modica77][Modica78][Modica87] in steady case, namely Δu-1/ε2W'(u)=0. However, thanks to the argument of De Giorgi's conjecture, we hope that the asymptotic behavior of u by the flow performs as the same profile as the static case, namely
where Σt means the interface Σ0 at time t and dΣt(t) is the signed distance from x to the surface Σt. By calculation, one may obtain the following mean curvature flow with respect to dΣt(x), i.e.,
Geometrically, the velocity of the surface Σt is exactly equal to the mean curvature (vector). There are plenty of seminal works in such dynamical case, such as [Bronsard91][Chen92][Evans92][Ilmanen93].
Finally, we extend the Allen-Cahn equation to vector-value case, which was first introduced in [Bronsard98]. Generally, let u:Ω ⊂ ℝn→ℝm be a vector-value function and W:ℝm→ℝ≥0 be a double-well potential in ℝm, which vanishes on two disjoint connected submanifolds, say M1,M2, namely
Hence, we obtain the phase transition model in vector-valued version, i.e.,
The interface (still denoted by Σt) flows by the mean curvature flow, while u performs as a harmonic map heat flow away from the interface to the submanifolds (phases) either M1 or M2 by formal calculation in [Rubinstein89][Rubinstein89b]. However, this result lacks a rigorous proof, hence, the problem is still OPEN, known as so-called Keller-Rubinstein-Sternberg problem, see [Fei23].
There are some milestone results on this problem. For example, firstly, Bronsard and Stoth [Bronsard98] considered the W as follows,
They rigorously proved the conjecture in this special case. Subsequently, Lin, Pan and Wang [Lin12] studied the case for W attaining the global minimum at two disjoint compact connected submanifolds N± in ℝm and showed that the energy functional
has the first term approximation as ε↘0+ like
where c represents the energy of the minimal connected orbit between N+ and N-; and D is the energy of the minimal harmonic mapping to N±. After that, there are many seminal works for the special W, such that [Fei18][Lin19][Laux21][Lin23][Fei23]. For the latest results, we refer the readers to Dong and Wang's paper [Dong25].