Abstract. An entire solution of the Allen-Cahn equation ∆u = f (u), where f has exactly three zeros at ±1 and 0, is balanced and odd, e.g. f (u) = u(u 2 − 1), is called a 2k-ended solution if its nodal set is asymptotic to 2k half lines, and if along each of these half lines the function u looks like the one dimensional, heteroclinic solution. In this paper we consider the family of four ended solutions whose ends are almost parallel at ∞. We show that this family can be parametrized by the family of solutions of the two component Toda system. As a result we obtain the uniqueness of four ended solutions with almost parallel ends. Combining this result with the classification of connected components in the moduli space of the four ended solutions we can classify all such solutions. Thus we show that four end solutions form, up to rigid motions, a one parameter family. This family contains the saddle solution, for which the angle between the nodal lines is π 2 as well as solutions for which the angle between the asymptotic half lines of the nodal set is arbitrary small (almost parallel nodal sets).

Abstract. In this paper, we construct a wealth of bounded, entire solutions of the Allen-Cahn equation in the plane. The asymptotic behavior at infinity of these solutions is determined by 2L half affine lines, in the sense that, along each of these half affine lines, the solution is close to a suitable translated and rotated copy of a one dimensional heteroclinic solution. The solutions we construct belong to a smooth 2L-dimensional family of bounded, entire solutions of the Allen-Cahn equation, in agreement with the result of [3] and, in some sense, they provide a description of a collar neighborhood of part of the compactification of the moduli space of 2L-ended solutions for the Allen-Cahn equation. Our construction is inspired by a construction of minimal surfaces by M. Traizet [12].

Abstract. The comparison principle (uniqueness) for the Hamilton-Jacobi equation is usually established through arguments involving a distance function. In this article we illustrate the subtle nature of choosing such a distance function, using a special example of one dimensional Hamiltonian with coefficient singularly (non-Lipschitz) depending upon the state variable. The standard method of using Euclidean distance as a test function fails in such situation. Once the comparison is established, we apply it to obtain a new result on small noise Freidlin-Wentzell type probabilistic large deviation theorem for certain singular diffusion processes.This

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In this paper a novel variational method is developed to construct fourend solutions in R 2 for the Allen-Cahn equation. Four-end solutions have been constructed by Del Pino, Kolwazyck, Pacard and Wei when the angle θ of the ends is close to π/2 or 0 using the Lyapunov-Schmid reduction method, and later by Kolwazyck, Liu and Pacard using a continuation method for general θ ∈ (0, pi/2). By a special mountain pass argument in a restricted space, namely a set of monotone paths of monotone functions, a family of solutions in bounded domain is constructed with very good control of their nodal sets, which then leads to a four-end solution after sending the domains to R 2. In this way, not only a family of four end solutions is constructed for any angle θ ∈ (0π/2). The Morse index of such a four-end solution is also shown to be one.

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In mixed‐powder‐lubricated interfaces, the lubrication performance is strongly affected by evolution of the boundary layer features. Therefore, the current investigation uses a laser microscopy system to observe the behaviours of boundary layer formed on the brass specimens during the process of graphite‐lubricated 0.45 C steel (HRC56) sliding upon brass alloy (HV160). The evolution process of mixed powder lubrication can be divided into 4 typical stages: running‐in, stable, deterioration, and damage. Multiple micro phenomena are observed in the boundary layers at different stages. These include extent of layer coverage, flaking, local adhesion, lamellar shedding, blisters, pits, and abrasion damage. These phenomena indicate that the boundary layer deteriorates from full coverage to a damaged powdery film, and eventually the serious wear on metal substrate. Moreover, an interface model of mixed powder lubrication is established to explain the changes in boundary layers. The model displays that the typically micro events above are related to the quantitative degree of overlapping of the asperities between opposing surfaces. At first, many particles bear the load collectively and adapt to velocity differences through inner shearing in powder layer. Later, plastic deformation and abrasion between asperities play a major role in surface velocity accommodation, and the metal substrates bear the load more directly.