We will prove that solutions of the Allen–Cahn equations that satisfy the equipartition of the energy can be transformed into solutions of the Euler equations with constant pressure. As a consequence, we obtain De Giorgi type results, that is, the level sets of entire solutions are hyperplanes. Also, we will determine the structure of solutions of the Allen–Cahn system in two dimensions that satisfy the equipartition. In addition, we apply the Leray projection on the Allen–Cahn system and provide some explicit entire solutions. Finally, we obtain some examples of smooth entire solutions of the Euler equations. For specific type of initial conditions, some of these solutions can be extended to the Navier–Stokes equations. The motivation of this paper is to find a transformation that relates the solutions of the Allen–Cahn equations to solutions of the minimal surface equation of one dimension less. We prove this result for equipartitioned solutions in dimension three.

We establish a priori L^infty -estimates for non-negative solutions of a semilinear nonlocal Neumann problem. As a consequence of these estimates, we get non-existence of non-constant solutions under suitable assumptions on the diffusion coefficient and on the nonlinearity. Moreover, we prove an existence result for radial, radially non-decreasing solutions in the case of a possible supercritical nonlinearity, extending to the case 0<sle 1/2 the analysis started in [7].

In this paper, we study the Cauchy–Dirichlet problem ∂tu-divDξf(t,Du)=0inΩT,u=uoon∂PΩT,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{ll} \\partial _t u - {\ ext {div}} \\left( D_\\xi f(t, Du)\\right) = 0 &{} \\quad \\hbox {in} \\ \\Omega _T, \\\\ u = u_o &{} \\quad \\hbox { on} \\ \\partial _{\\mathcal {P}} \\Omega _T,\\\\ \\end{array} \\right. \\end{aligned}$$\\end{document}where Omega subset mathbb {R}^n is a convex and bounded domain, f:[0,T]times {mathbb {R}}^n rightarrow {mathbb {R}} is L^1-integrable in time and convex in the second variable. Assuming that the initial and boundary datum u_o:{overline{Omega }}rightarrow {mathbb {R}} satisfies the bounded slope condition, we prove the existence of a unique variational solution that is Lipschitz continuous in the space variable.

We consider a system of forward backward stochastic differential equations (FBSDEs) with a time-delayed generator driven by Lévy-type noise. We establish a non-linear Feynman–Kac representation formula associating the solution given by the FBSDEs system to the solution of a path dependent nonlinear Kolmogorov equation with both delay and jumps. Obtained results are then applied to study a generalization of the so-called large investor problem, where the stock price evolves according to a jump-diffusion dynamic.

Here, we consider periodic homogenization for time-fractional Hamilton–Jacobi equations. By using the perturbed test function method, we establish the convergence, and give estimates on a rate of convergence. A main difficulty is the incompatibility between the function used in the doubling variable method, and the non-locality of the Caputo derivative. Our approach is to provide a lemma to prove the rate of convergence without the doubling variable method with respect to the time variable, which is a key ingredient.