1989
DOI: 10.1007/bf00945002
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On the convexity of geometric functional of level for solutions of certain elliptic partial differential equations

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Cited by 22 publications
(36 citation statements)
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“…In order to derive the temperature in the microcanonical ensemble, according to the definition T (E) = (∂S(E)/∂E) −1 , we shall use the following generalization [18] of the Federer-Laurence derivation formula [9][10][11][12][13]. The flux Φ ξ with a non-vanishing component in the direction orthogonal to the constant energy hypersurfaces, but tangent to the level hyper-surfaces of V , can be defined by the vector ξ = n H − (n H · n V )n V , where n H = ▽H/ ▽ H and n V = ▽V / ▽ V .…”
mentioning
confidence: 99%
“…In order to derive the temperature in the microcanonical ensemble, according to the definition T (E) = (∂S(E)/∂E) −1 , we shall use the following generalization [18] of the Federer-Laurence derivation formula [9][10][11][12][13]. The flux Φ ξ with a non-vanishing component in the direction orthogonal to the constant energy hypersurfaces, but tangent to the level hyper-surfaces of V , can be defined by the vector ξ = n H − (n H · n V )n V , where n H = ▽H/ ▽ H and n V = ▽V / ▽ V .…”
mentioning
confidence: 99%
“…In [7], among other results, in the case of the Laplace equation on a plane convex annulus, certain isoperimetric inequalities, which imply that I(a) is log-convex, are proved. These inequalities were successively generalized and extended in [5] to the star-shaped case, and in [3] for equation (4.1) and a general annular domain. Here, we shall generalize these results to general dimension, that is, we shall give what we think is their natural extension for equation (4.1).…”
Section: R Magnaninimentioning
confidence: 99%
“…All arguments in this proof are to be intended almost everywhere with respect to ae(a 1 5 a 2 ). Set A # 1, and let us recall two formulae expressing the derivatives of £(oc) as surface integrals (see [5]):…”
Section: (4>9)mentioning
confidence: 99%
“…[36]. By using by the Federer-Laurence derivation formula [39,40,42,43], in the case of the proposed entropy we get…”
mentioning
confidence: 97%