De Giorgi's Conjecture for the Allen-Cahn Equation and Related Problems for Classical and Fractional Laplacians

We present results related to a famous conjecture of Enrico De Giorgi for a special class of bounded monotone solutions, called layer solutions, to nonlinear equations of the form −Δu = f(u) in ℝⁿ and (−Δ)^su = f(u) in ℝⁿ, where (−Δ)^s denotes the fractional Laplace operator with fractional exponent s ∈ (0,1). Here, we assume that f : ℝ → ℝ is at least of class C¹. We begin by defining the fractional Laplace operator, and prove many of its fundamental properties. We then present the extension problem in ℝ₊ⁿ⁺¹ := {(x,y) ∈ ℝⁿ⁺¹ : x ∈ ℝⁿ, y > 0} for the operator (−Δ)^s introduced by Caffarelli and Silvestre (2007) and develop a fundamental solution and Poisson kernel for (−Δ)^s in ℝ₊ⁿ⁺¹. Subsequently, we prove De Giorgi's conjecture for layer solutions to the first equation above in dimensions n ≤ 3. Precisely, we show that layer solutions are necessarily one-dimensional. We then turn our attention to the fractional De Giorgi conjecture, and present the fractional versions of the results obtained in the classical case (i.e. similar results for the second equation above). To supplement, we discuss some Pohozaev-type monotonicity formulae for the operators Δ and (−Δ)^s, along with some closely related problems. We close with a brief discussion of some open problems and topics of further interest to the author.