Some basic facts on the system $\Delta u - W_{u} (u) = 0$

Alikakos, Nicholas D.

doi:10.1090/s0002-9939-2010-10453-7

Cited by 39 publications

(68 citation statements)

References 28 publications

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“…[12]) to the vector case. The Modica estimate states that for a non-negative potential W ∈ C 2 (R, R), and for every bounded entire solution u ∈ C 3 (R n , R) of the equation (1) ∆u = W ′ (u), then (2) 1 2 |∇u(x)| 2 ≤ W (u(x)), ∀x ∈ R n .…”

Section: Introductionmentioning

confidence: 99%

Gradient estimates for semilinear elliptic systems and other related results

Smyrnelis¹

2015

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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Abstract. A periodic connection is constructed for a double well potential defined in the plane. This solution violates Modica's estimate as well as the corresponding Liouville Theorem for general phase transition potentials. Gradient estimates are also established for several kinds of elliptic systems. They allow us to prove in some particular cases the Liouville Theorem. Finally, we give an alternative form of the stress-energy tensor for solutions defined in planar domains. As an application, we deduce a (strong) monotonicity formula.

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Section: Introductionmentioning

confidence: 99%

Gradient estimates for semilinear elliptic systems and other related results

Smyrnelis¹

2015

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…If F (u(x 0 )) = 0 for some x 0 ∈ R n , then u must be constant. (3) Again, making use of the Modica-type estimate, the following monotonicity formula has been established in [2]. Let u : R n → R be an entire, bounded solution of the nonlinear Poisson equation.…”

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confidence: 99%

Some remarks on energy inequalities for harmonic maps with potential

Branding

2017

Arch. Math.

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Abstract. In this note we discuss how several results characterizing the qualitative behavior of solutions to the nonlinear Poisson equation can be generalized to harmonic maps with potential between complete Riemannian manifolds. This includes gradient estimates, monotonicity formulas, and Liouville theorems under curvature and energy assumptions.Mathematics Subject Classification. 58E20, 53C43, 35J61.

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“…From the monotonicity formula (see (1.4) in [2]), which holds for general Lipschitz W ≥ 0, it follows that any nonconstant solution to ∆u − W u (u) = 0 satisfies the lower bound…”

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confidence: 99%

Density estimates for vector minimizers and applications

Alikakos¹,

Fusco²

2015

DCDS-A

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We extend the Caffarelli-Córdoba estimates to the vector case (L. Caffarelli and A. Córdoba, Uniform Convergence of a singular perturbation problem, Comm. Pure Appl. Math. 48 (1995)). In particular, we establish lower codimension density estimates. These are useful for studying the hierarchical structure of minimal solutions. We also give applications.2010 Mathematics Subject Classification. 35J20, 35J47, 35J50.(c) the phase separation system ∆u i − j =i u i u j = 0, i = 1, . . . , m (Caffarelli and Lin [15]) and its variants. All these systems are included in the general framework of Rubinstein, Sternberg and Keller [27], [28].We assumewhere · denotes the Euclidean inner product in R m and | · | the associated norm.In the present paper we limit ourselves to uniformly bounded solutions to (1.1) that are minimal with respect to their boundary conditions (called minimizers by some authors),2 0 (Ω; R m ) (1.6) for every open bounded set Ω ⊂ D. Moreover we assume |u − a| < M, |∇u| < M, on R n .(1.7)The basic estimate for minimal solutions (cr. Lemma 2.1) isWe generally omit the center and write it B R . In this paper, we give two different extensions of the Caffarelli-Cordoba density estimates [14] to the vector case, together with a few applications. In the scalar case, among other things, these estimates refine the linking of the phase transition model to minimal surfaces and have played a major role in the resolution of the De Giorgi conjecture in higher dimensions (Savin [29]). Other extensions of the density estimates in different contexts have been provided by Farina and Valdinoci [20], Savin and Valdinoci [30],[31], Sire and Valdinoci [32], and very recently by Cesaroni, Muratov and Novaga [16].We denote L k the k-dimensional Lebesgue measure and let

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Some basic facts on the system $\Delta u - W_{u} (u) = 0$

Cited by 39 publications

References 28 publications

Gradient estimates for semilinear elliptic systems and other related results

Gradient estimates for semilinear elliptic systems and other related results

Some remarks on energy inequalities for harmonic maps with potential

Density estimates for vector minimizers and applications

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