Nodal sets for solutions of elliptic equations

Hardt, Robert; Simon, Leon

doi:10.4310/jdg/1214443599

Cited by 199 publications

(217 citation statements)

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“…In fact, since u is harmonic, the level sets of u have locally finite H n−1 -measure (see [13] and [17]). Moreover, by the properness of u, they are compact and thus their hypersurface area is finite.…”

Section: Statements Of the Main Resultsmentioning

confidence: 99%

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On the Geometry of the Level Sets of Bounded Static Potentials

Agostiniani

Mazzieri

2017

Commun. Math. Phys.

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Abstract. In this paper we present a new approach to the study of asymptotically flat static metrics arising in general relativity. In the case where the static potential is bounded, we introduce new quantities which are proven to be monotone along the level set flow of the potential function. We then show how to use these properties to detect the rotational symmetry of the static solutions, deriving a number of sharp inequalities. As a consequence of our analysis, a simple proof of the classical 3-dimensional Black Hole Uniqueness Theorem is recovered and some geometric conditions are discussed under which the same statement holds in higher dimensions.MSC (2010): 35B06, 53C21, 83C57, 35N25.

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Section: Statements Of the Main Resultsmentioning

confidence: 99%

“…Concerning the nonregular level sets of ϕ, we first observe that by the results in [13] and [17], one has that the (n − 1)-dimensional Hausdorff measure of the level sets of ϕ is locally finite. Hence, the properness of ϕ forces the level sets to have finite (n−1)-dimensional Hausdorff measure.…”

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confidence: 99%

On the Geometry of the Level Sets of Bounded Static Potentials

Agostiniani

Mazzieri

2017

Commun. Math. Phys.

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“…Related results for the 3D NavierStokes equations (with general forcing) can be found in [4]. The upper bounds on the Hausdorff measures of the level sets associated with solutions of some other partial differential equations were obtained in [5,9,10,15,17].…”

Section: Abdelhafid Younsi Nodeamentioning

confidence: 95%

Effect of hyperviscosity on the geometry of the vorticity

Younsi

2012

Nonlinear Differ. Equ. Appl.

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Abstract.We consider the two-dimensional modified Navier-Stokes equations by adding a p-Laplacian. We provide upper bounds on the Hausdorff length of the level sets of the vorticity. We express them in terms of the Landau-Lifschitz estimates of the number of degrees of freedom in turbulent flow. The proof, based on very standard methods, relies on a straightforward relation between the size of the nodal set of the vorticity and the growth of hyperdissipation.Mathematics Subject Classification. 35K55, 35Q30, 76D05, 76F20, 76F70.

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“…Recently this bound was refined to Cλ 3/4−ε in [12]. In higher dimensions the estimate H n−1 ({ϕ λ = 0}) ≤ Cλ C √ λ by Hardt and Simon ( [9]) was the only known upper bound till now. We prove that…”

Section: Preliminariesmentioning

confidence: 98%