We study perturbations of the Allen-Cahn equation and prove the convergence to forced mean curvature flow in the sharp interface limit. We allow for perturbations that are square-integrable with respect to the diffuse surface area measure. We give a suitable generalized formulation for forced mean curvature flow and apply previous results for the Allen-Cahn action functional. Finally we discuss some applications.
In this paper we consider the Allen-Cahn equation with constraint. In 1994, Chen and Elliott studied the asymptotic behavior of the solution of the Allen-Cahn equation with constraint. They proved that the zero level set of the solution converges to the classical solution of the mean curvature flow under the suitable conditions on initial data. In 1993, Ilmanen proved the existence of the mean curvature flow via the Allen-Cahn equation without constraint in the sense of Brakke. We proved the same conclusion for the Allen-Cahn equation with constraint.
We show that strictly stable components of Allen-Cahn minimal hypersurfaces always occur with multiplicity one. We also establish the uniqueness of solutions converging to nondegenerate hypersurfaces with multiplicity one. Our results work in all dimensions and without variational assumptions on the Allen-Cahn solutions.
In this paper we find solutions $u_ε$ to a certain class of vector-valued parabolic Allen-Cahn equation that as $ε\to 0$ develops as interface a given triod evolving under curve shortening flow.
In this paper we prove an analogue of the Brakke's $\varepsilon$-regularity theorem for the parabolic Allen-Cahn equation. In particular, we show uniform $C^{2,α}$ regularity for the transition layers converging to smooth mean curvature flows as $\varepsilon\rightarrow0$. The proof utilises Allen-Cahn versions of the monotonicity formula, parabolic Lipschitz approximation and blowups. A corresponding gap theorem for entire eternal solutions of the parabolic Allen-Cahn is also obtained. As an application of the regularity theorem, we give an affirmative answer to a question of Ilmanen that ther…
In this paper, we consider bounded positive solutions to the Allen-Cahn equation on complete noncompact Riemannian manifolds without boundary. We derive gradient estimates for those solutions. As an application, we get a Liouville type theorem on manifolds with nonnegative Ricci curvature.
This paper presents a conditional convergence result of solutions to the Allen--Cahn equation with arbitrary potentials to a De Giorgi type $ \mathrm{BV} $-solution to multiphase mean curvature flow. Moreover we show that De Giorgi type $\mathrm{BV} $-solutions are De Giorgi type varifold solutions, and thus our solution is unique in a weak-strong sense.
This paper presents a new partial differential equation to build Brakke's motion, which is a weak notion of mean curvature flow. We call the equation a modified Allen-Cahn equation abbreviated to MAC. After that we introduce its benefit: An improved estimate on the discrepant energy that means the equipartition of the first energy. This equation is obtained by rendering the parameter in Allen-Cahn equation time-dependent and adding a term like Z-transform. Furthermore, we mention the existence of a Brakke's motion through our MAC.
We define a (mean curvature flow) entropy for Radon measures in $\mathbb{R}^n$ or in a compact manifold. Moreover, we prove a monotonicity formula of the entropy of the measures associated with the parabolic Allen-Cahn equations. If the ambient manifold is a compact manifold with non-negative sectional curvature and parallel Ricci curvature, this is a consequence of a new monotonicity formula for the parabolic Allen-Cahn equation. As an application, we show that when the entropy of the initial data is small enough (less than twice of the energy of the one-dimensional standing wave), the limit …
We study a singular limit problem of the Allen-Cahn equation with Neumann boundary conditions and general initial data of uniformly bounded energy. We prove that the time-parametrized family of limit energy measures is Brakke's mean curvature flow with a generalized right angle condition on the boundary.
We make use of the flexibility of infinite-index solutions to the Allen-Cahn equation to show that, given any compact hypersurface $Σ$ of R^d, with $d\geq 4$, there is a bounded entire solution of the Allen-Cahn equation on R^d whose zero level set has a connected component diffeomorphic (and arbitrarily close) to a rescaling of $Σ$. More generally, we prove the existence of solutions with a finite number of compact connected components of prescribed topology in their zero level sets.
We use a min-max procedure on the Allen-Cahn energy functional to construct geodesics on closed, 2-dimensional Riemannian manifolds, as motivated by the work of Guaraco. Borrowing classical blowup and curvature estimates from geometric analysis, as well as novel Allen-Cahn curvature estimates due to Wang-Wei, we manage to study the fine structure of potential singular points at the diffuse level, and show that the problem reduces to that of understanding "entire" singularity models constructed by del Pino-Kowalczyk-Pacard with Morse index 1. The argument is completed by a Morse index estimate …
We prove the existence of global minimizers of Allen-Cahn equation in dimensions $8$ and above. More precisely, given any strictly area-minimizing Lawson's cones, there are global minimizers whose nodal sets are asymptotic to the cones. As a consequence of Jerison-Monneau's program we establish the existence of many counter-examples to the De Giorgi's conjecture different from the Bombierie-De Giorgi-Giusti minimal graph.
Let $M^n$ be a connected closed smooth manifold, where $n\geq 2$. In this article, we adapt the techniques in Micheletti and Pistoia (2009) and Ghimenti and Micheletti (2011) to prove the generic nondegeneracy for solutions of the Van der Waals-Allen-Cahn-Hilliard equation under a volume constraint in $M$.
We prove that given a minimal hypersurface $Γ$ in a compact Riemannian manifold $M$ without boundary, if all the Jacobi fields of $Γ$ are generated by ambient isometries, then we can find solutions of the Allen-Cahn equation $-\varepsilon^2Δu +W'(u)=0$ on $M$, for sufficiently small $\varepsilon>0$, whose nodal sets converge to $Γ$. This extends the results of Pacard-Ritoré (in the case of closed manifolds and zero mean curvature).
We construct new families of two-ended $O(m)\times O(n)$-invariant solutions to the Allen- Cahn equation Δu+u-u3=0 in $\mathbb{R}^{N+1}$, with $N\ge 7$, whose zero level sets diverge logarithmically from the Lawson cone at infinity. The construction is based on a careful study of the Jacobi-Toda system on a given $O(m)\times O(n)$-invariant manifold, which is asymptotic to the Lawson cone at infinity.
We prove a rigidity theorem in the style of Urbano for the Allen-Cahn equation on the three-sphere: the critical points with Morse index five are symmetric functions that vanish on a Clifford torus. Moreover they realise the fifth width of the min-max spectrum for the Allen-Cahn functional. We approach this problem by analysing the nullity and symmetries of these critical points. We then prove a suitable Frankel-type theorem for their nodal sets, generally valid in manifolds with positive Ricci curvature. This plays a key role in establishing the conclusion, and further allows us to derive anc…
In this paper we consider the Allen-Cahn equation $$ -Δu = u-u^3 \ \mbox{in} \ {\mathbb R}^3 $$ We prove that for each $k\in\left( \sqrt{2},+\infty\right),$ there exists a solution to the equation which has growth rate $k$, i.e. $$ \| u-H(\cdot -k \ln r + c_k) \|_{L^\infty} \to 0$$ The main ingredients of our proof consist: (1) compactness of solutions with growth $k$, (2) moduli space theory of analytical variety of formal dimension one.
In this article we study eternal solutions to the Allen-Cahn equation in the 3-sphere, in view of the connection between the gradient flow of the associated energy functional, and the mean curvature flow. We construct eternal integral Brakke flows that connect Clifford tori to equatorial spheres, and study a family of such flows, in particular their symmetry properties. Our approach is based on the realization of Brakke's motion by mean curvature as a singular limit of Allen-Cahn gradient flows, as studied by Ilmanen, and Tonegawa, and it uses the classification of ancient gradient flows in sp…
The Allen-Cahn equation is a semilinear PDE which is deeply linked to the theory of minimal hypersurfaces via a singular limit. We prove curvature estimates and strong sheet separation estimates for stable solutions (building on recent work of Wang-Wei) of the Allen-Cahn equation on a 3-manifold. Using these, we are able to show for generic metrics on a 3-manifold, minimal surfaces arising from Allen-Cahn solutions with bounded energy and bounded Morse index are two-sided and occur with multiplicity one and the expected Morse index. This confirms, in the Allen-Cahn setting, a strong form of th…
In this paper, we prove even symmetry of monotone traveling wave solutions to the balanced Allen-Cahn equation in the entire plane. Related results for the unbalanced Allen-Cahn equation are also discussed.
In this paper we present a new family of solutions to the singularly perturbed Allen-Cahn equation $α^2 Δu + u(1-u^2)=0, \quad \hbox{in }Ω\subset \R^N $ where $N=3$, $Ω$ is a smooth bounded domain and $\A>0$ is a small parameter. We provide asymptotic behavior which shows that, as $α\to 0$, the level sets of the solutions collapse onto a bounded portion of a complete embedded minimal surface with finite total curvature that intersects orthogonally $\partial Ω$ of the domain and that is non-degenerate respect to $Ω$. We provide explicit examples of surfaces to which our result applies.
In this paper, we are interested in bounded, entire, solutions of the Allen-Cahn equation which are defined in Euclidean space and whose zero set is asymptotic to a given minimal cone. In particular, in dimension larger than or equal to 8, we prove the existence of stable solutions of the Allen-Cahn equation whose zero sets are not hyperplanes.
For the Allen-Cahn equation, it is well known that there is a monotone standing wave joining with the balanced wells of the potential. In this paper we study the existence of traveling wave solutions for the Allen-Cahn equation on an infinite channel. Such traveling wave solutions possess a large number of oscillation and they are obtained with the aid of variational arguments.
We study entire viscosity solutions of the Allen-Cahn type equation for the truncated Laplacian that are either one dimensional or radial, in order to shed some light on a possible extension of the Gibbons conjecture in this degenerate elliptic setting.
In this paper, we study the relation between the second inner variations of the Allen-Cahn functional and its Gamma-limit, the area functional. Our result implies that the Allen-Cahn functional only approximates well the area functional up to the first order. However, as an application of our result, we prove, assuming the single-multiplicity property of the limiting energy, that the Morse indices of critical points of the Allen-Cahn functional are bounded from below by the Morse index of the limiting minimal hypersurface.
In this set of notes, we present some recent developments on the fractional Allen-Cahn equation $$ (-Δ)^s u = u-u^3,$$ with special attention to $Γ$-convergence results, energy and density estimates, convergence of level sets, Hamiltonian estimates, rigidity and symmetry results.
We establish existence and non-existence results for entire solutions to the fractional Allen-Cahn equation in $\mathbb R^3$, which vanish on helicoids and are invariant under screw-motion. In addition, we prove that helicoids are surfaces with vanishing nonlocal mean curvature.
We consider a Dirichlet problem for the Allen-Cahn equation in a smooth, bounded or unbounded, domain $Ω\subset {\bf R}^n.$ Under suitable assumptions, we prove an existence result and a uniform exponential estimate for symmetric solutions. In dimension n=2 an additional asymptotic result is obtained. These results are based on a pointwise estimate obtained for local minimizers of the Allen-Cahn energy.
We prove optimal lower bounds for the growth of the energy over balls of minimizers to the vectorial Allen-Cahn energy in two spatial dimensions, as the radius tends to infinity. In the case of radially symmetric solutions, we can prove a stronger result in all dimensions.
In this paper we study the fractional analogue of the Allen-Cahn energy in bounded domains, and we show that it admits a number of critical points which goes to infinity as the perturbation parameter tends to zero.
For a solution of the Allen-Cahn equation in $\mathbb{R}^2$, under the natural linear growth energy bound, we show that the blowing down limit is unique. Furthermore, if the solution has finite Morse index, the blowing down limit satisfies the multiplicity one property.
This article presents some qualitative results for entire solutions of the fully nonlinear elliptic equations of Allen Cahn type . Precisely under some additional assumptions on the forcing term, if the solution is bounded and converges uniformly at infinity in a fixed direction to its extrema, then the solution depends only on one variable.
This paper studies two well-known models for two-phase fluid flow at constant temperature, the isothermal Navier-Stokes-Allen-Cahn and the isothermal Navier-Stokes-Cahn-Hilliard equations, both of which consist of equations for the (total) fluid density rho, the (mass-averaged)velocity u and the concentration (of one of the phases,) c. Assuming in either case that both phases are incompressible with different densities, each of the models is shown to reduce to a system of evolution equations in rho and u alone. In the case of the Navier-Stokes-Allen-Cahn model, this reduced system is the class…
We consider solutions of the Allen-Cahn equation in the whole Grushin plane and we show that if they are monotone in the vertical direction, then they are stable and they satisfy a good energy estimate. However, they are not necessarily one-dimensional, as a counter-example shows.
In this paper we establish an improvement of tilt-excess decay estimate for the Allen-Cahn equation, and use this to give a new proof of Savin's theorem on the uniform $C^{1,α}$ regularity of flat level sets, which then implies the one dimensional symmetry of minimizers in $\mathbb{R}^n$ for $n\leq 7$. This generalizes Allard's $\varepsilon$-regularity theorem for stationary varifolds to the setting of Allen-Cahn equations.
In this paper, we prove even symmetry and monotonicity of certain solutions of Allen-Cahn equation in a half plane. We also show that entire solutions with {\it finite Morse index} and {\it four ends} must be evenly symmetric with respect to two orthogonal axes. A classification scheme of general entire solutions with {\it finite Morse index} is also presented using energy quantization.
Formal asymptotic expansions have long been used to study the singularly perturbed Allen-Cahn type equations and reaction-diffusion systems, including in particular the FitzHugh-Nagumo system. Despite their successful role, it has been largely unclear whether or not such expansions really represent the actual profile of solutions with rather general initial data. By combining our earlier result and known properties of eternal solutions of the Allen-Cahn equation, we prove validity of the principal term of the formal expansions for a large class of solutions.
In this paper we study axially symmetric solutions of Allen-Cahn equation with finite Morse index. It is shown that there does not exist such a solution in dimensions between $4$ and $10$. In dimension $3$, we prove that these solutions have finitely many ends. Furthermore, the solution has exactly two ends if its Morse index equals $1$.
We investigate the behavior, as a small parameter tends to zero, of a nonlocal Allen-Cahn equation. Given a rather general initial data, we perform a rigorous analysis of both the generation and the motion of interface, and obtain a new estimate for its thickness. The limit problem involves motion by mean curvature together with a nonlocal effect (which allows the possibility of nontrivial stationary state).
We give a short and self-contained proof for rates of convergence of the Allen-Cahn equation towards mean curvature flow, assuming that a classical (smooth) solution to the latter exists and starting from well-prepared initial data. Our approach is based on a relative entropy technique. In particular, it does not require a stability analysis for the linearized Allen-Cahn operator. As our analysis also does not rely on the comparison principle, we expect it to be applicable to more complex equations and systems.
We are concerned with solutions to the parabolic Allen-Cahn equation in Riemannian manifolds. For a general class of initial condition we show non positivity of the limiting energy discrepancy. This in turn allows to prove almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) which gives a local uniform control of the energy densities at small scales. Such results will be used in [40] to extend previous important results from [31] in Euclidean space, showing convergence of solutions to the parabolic Allen-Cahn equations to Brakke's motion by mean curvature in space…
By using the unique continuation principle for linear elliptic systems, we can simplify the proof of a recent variational maximum principle due to Alikakos and Fusco. At the same time, this approach allows us to relax an assumption from the latter reference.
