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On De Giorgi's conjecture in dimension N>9
A celebrated conjecture due to De Giorgi states that any bounded solution of the equation u + (1 u 2 )u = 0 in R N with @yNu > 0 must be such that its level setsfu = g are all hyperplanes, at leastExpand
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Concentration on curves for nonlinear Schrödinger Equations
We consider the problem where p > 1, e > 0 is a small parameter, and V is a uniformly positive, smooth potential. Let Γ be a closed curve, nondegenerate geodesic relative to theExpand
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Variational reduction for Ginzburg–Landau vortices
Let Ω be a bounded domain with smooth boundary in R2. We construct non-constant solutions to the complex-valued Ginzburg–Landau equation e2Δu+(1−|u|2)u=0 in Ω, as e→0, both under zero Neumann andExpand
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MULTIPLE-END SOLUTIONS TO THE ALLEN-CAHN EQUATION IN R2
We construct a new class of entire solutions for the Allen–Cahn equation Δu+(1−u2)u=0, in R2(∼C). Given k⩾1, we find a family of solutions whose zero level sets are, away from a compact set,Expand
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Moduli space theory for the allen-cahn equation in the plane
In this paper we study entire solutions of the Allen-Cahn equation \Delta u - F'(u) = 0, where F is an even, bistable function. We are particularly interested in the description of the moduli spaceExpand
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Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature
Let $${(\mathcal {M},\tilde{g})}$$ be an N-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen–Cahn equation$$\varepsilon ^2 \Delta _{\tilde g} u \, + \, (1 -Expand
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Traveling Waves with Multiple and Nonconvex Fronts for a Bistable Semilinear Parabolic Equation
We construct new examples of traveling wave solutions to the bistable and balanced semilinear parabolic equation in \input amssym ${\Bbb R}^N+1$, . Our first example is that of a traveling waveExpand
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Theory of light-matter interaction in nematic liquid crystals and the second Painlevé equation
We study global minimizers of an energy functional arising as a thin sample limit in the theory of light-matter interaction in nematic liquid crystals. We show that depending on the parametersExpand
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A counterexample to a conjecture by De Giorgi in large dimensions
We consider the Allen–Cahn equation Δu+u(1−u2)=0in RN. A celebrated conjecture by E. De Giorgi (1978) states that if u is a bounded solution to this problem such that ∂xNu>0, then the level setsExpand
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Nondegeneracy of the saddle solution of the Allen-Cahn equation
A solution of the Allen-Cahn equation in the plane is called a saddle solution if its nodal set coincides with the coordinate axes. Such solutions are known to exist for a large class ofExpand
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