{"id":190,"date":"2010-06-22T07:45:40","date_gmt":"2010-06-22T11:45:40","guid":{"rendered":"http:\/\/blogs.williams.edu\/Morgan\/?p=190"},"modified":"2019-05-09T05:04:09","modified_gmt":"2019-05-09T10:04:09","slug":"variation-formulae-for-perimeter-and-volume-densities","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2010\/06\/22\/variation-formulae-for-perimeter-and-volume-densities\/","title":{"rendered":"Variation Formulae for Perimeter and Volume Densities"},"content":{"rendered":"

For R<\/strong>n+1<\/sup> with volume density f<\/em> and perimeter density g<\/em>, for a normal variation u<\/em> of a surface with classical mean curvature H<\/em>, the first variation of volume and perimeter are given by:<\/p>\n

\\delta ^1V=-\\int uf,<\/p>\n

\\delta ^1P=-\\int [(g\/f)nH - (1\/f)(\\partial g\/\\partial n)] uf.<\/p>\n

For a volume-preserving variation, the second variation of perimeter is given by:
\n\\delta ^2P=\\int g|\\nabla u|^2-g|\\sigma|^2u^2-f\\frac{\\partial (g\/f)}{\\partial n}u^2nH+u^2\\frac{\\partial ^2g}{\\partial n^2}-\\frac 1fu^2\\frac{\\partial f}{\\partial n}\\frac{\\partial g}{\\partial n},<\/p>\n

where \\sigma is the second fundamental form, so that \\sigma^2 is the sum of the squares of the principal curvatures.<\/p>\n

The proof is the same as for the case of simple density, f=g; see\u00a0Rosales\u00a0et al. [R]<\/a>, Section 3.<\/p>\n

It follows that balls about the origin are stable for a radial density if and only if:<\/p>\n

a simple density (f(r)<\/em>=g(r)<\/em>) is log convex;<\/p>\n

a perimeter density g<\/em>(r<\/em>)\u00a0(f<\/em>=1) satisfies (n<\/em>\/r<\/em>)g<\/em>‘ + g<\/em>” \u2265 0;<\/p>\n

a volume density f(r)<\/em> (g<\/em>=1) is nonincreasing;<\/p>\n

a conformal change of metric f(r)<\/em> = \u03bbn+1<\/sup>, g(r)<\/em> = \u03bbn<\/sup> satisfies \u2212\u03bb\u03bb’\/r + \u03bb\u03bb” \u2212 2\u03bb’2<\/sup> \u2265 0.<\/p>\n

The conformal case can be checked by using the standard second variation formula for a Riemannian metric:<\/p>\n

\\delta ^2P=\\int |\\nabla u|^2-|\\sigma|^2u^2-Ric(u,u).<\/p>\n

In the planar case (n=1), the curvature \u03ba is given by<\/p>\n

\\kappa = -\\frac{dP}{dA} = -\\frac{\\lambda + r\\lambda^\\prime}{\\lambda ^2r}<\/p>\n

and the Gauss curvature is given by the formula<\/a><\/p>\n

G=-\\lambda^{-2}\\Delta log\\lambda
\n= \u03bb-4<\/sup>(\u03bb\u03bb’\/r + \u03bb\u03bb” \u22122\u03bb’2<\/sup>).<\/p>\n

Therefore |\u03c3|2<\/sup> = n<\/em>\u03ba2<\/sup> and Ric(u,u<\/em>) = nGu<\/em>2<\/sup>. Also, the gradient of u<\/em> is larger by a factor of \u03bb in the background metric. Now the second variation formula yields the same condition for stability of spheres. (For the lowest eigenvalue, the gradient term and the 1\/r<\/em>2<\/sup> term cancel out.)<\/p>\n

Remark<\/em> February 14, 2011. In the plane with density r<\/em>p<\/em><\/sup> (p<\/em>>0), isoperimetric regions are balls with the origin on the boundary [Dahlberg et al.<\/a>]. For unequal radial volume and perimeter densities, such balls are not even in equilibrium.<\/p>\n

Corollary\u00a0<\/em>June 25, 2016. In a cone over a round n<\/em>-sphere of mean curvature 1\/a<\/em>, volume density\u00a0f(r)<\/em>, and perimeter density\u00a0g(r)<\/em>,\u00a0\u00a0a geodesic sphere about the origin of radius\u00a0r <\/em>has nonnegative second variation\u00a0if and only if<\/p>\n

(1) \u00a0n(a^2 -1) + nr(f\/g)(g\/f)' + r^2 g"\/g - r^2 (1\/fg)(f'g') \\ge 0. <\/p>\n

For a single density g = f, (1) becomes<\/p>\n

(2)\u00a0n(a^2 - 1)+r^2 (log f)" \\ge 0. <\/p>\n

which for a=1\u00a0reduces to the standard log convexity condition.\u00a0For volume density f = r^m and perimeter density g = r^k, (1) and (2) become<\/p>\n

(1a) m \\le k-1+na^2 \/(k+n),<\/p>\n

in agreement with Alvino et al. [A, (5.53)] when a=1;<\/p>\n

(2a) n(a^2-1) \\ge m.<\/p>\n

In the conformal case m=k(n+1)\/n, (1a) becomes k \\le n(a-1). For n=1, the cone is equivalent to the \\theta = \\pi \/a sector, and (2a) becomes\u00a0\\theta \\le \\pi \/\\sqrt{m+1}\u00a0of Diaz et al. [D, Prop. 4.16].<\/p>\n

Proof.\u00a0<\/em>The lowest mode on S^n comes from translation in R^{n+1}, i.e. u = x_1, with |\\nabla u|^2 on the average n<\/em>\u00a0times as large as u^2 when a=1\u00a0and n a^2\u00a0times as large in general. |\\sigma|^2 is n\/r^2.<\/p>\n

Theorem<\/strong> [E, 6.3].\u00a0Let S<\/em> be a smooth Riemannian disk, sphere, or annulus of revolution with metric ds^2 = dr^2 + \\phi(r)^2 d\\theta^2 and density f(r)<\/em>. Then the circle of revolution at distance r<\/em> has nonnegative second variation\u00a0if and only if<\/p>\n

Q(r) = \\phi'(r)^2 - \\phi(r)\\phi"(r) - \\phi(r)^2 (log f)" \\le 1.<\/p>\n

Proposition<\/strong> (May 7, 2019). Similarly, in an (n+1)D ambient of revolution, the sphere at distance r has nonnegative second variation if and only if<\/em><\/p>\n

Q(r) = \\phi'(r)^2 - \\phi(r)\\phi"(r) - \\phi(r)^2 (log f)"\/n \\le 1.<\/p>\n

Proof<\/em>. The Gauss curvature of a radial section is -\\phi"\/\\phi [E, 6.3], so the outward Ricci curvature is -n\\phi"\/h. Each principal curvature equals the mean curvature<\/p>\n

dA\/ndV = (log A)'\/n = log(\\phi^n)'\/n = \\phi'\/\\phi,<\/p>\n

so the second fundamental form II satisfies II^2 = nh'^2\/h^2. By Rosales et al. [R, Rmk. 3.7], the second variation of a sphere S about the origin for a normal (i.e. radial) variation u<\/em> that preserves volume to first order (\u222bu<\/em> = 0) satisfies<\/p>\n

\\delta^2(u) = \\int |\\nabla u|^2 - u^2(Ric(N,N) - (log f)" + II^2).<\/p>\n

By the generalized Wirtinger Inequality [O, (3.1)] (Poincar\u00e9 Inequality), the second variation is nonnegative if and only if<\/p>\n

n\/h^2 \\le Ric(N,N) - (log f)" + II^2 = -n\\phi"\/\\phi - (log f)" + n\\phi'^2\/\\phi^2,<\/p>\n

i.e.,<\/p>\n

1 \\ge \\phi'^2 - \\phi\\phi" - \\phi^2(log f)"\/n. QED.<\/p>\n

Corollary.<\/strong> In an (n+1)D ambient of revolution, for decreasing ambient curvature and convex density, spheres about the origin are stable for fixed volume, while for increasing curvature and concave density, they are unstable. <\/em><\/p>\n

Proof.<\/em>\u00a0For infinitesimal spheres, the ambient is Euclidean and Q<\/em> = 0.\u00a0For constant density f<\/em>, Q<\/em> and the Gauss curvature K<\/em> of a radial section satisfy Q' = \\phi^2 K' (trivial one-line computation as in [R, Lemma 1.6]), [E, Sect. 6], [M]). In particular, Q<\/em> and K<\/em> increase or decrease together. The convexity or concavity of the density only causes Q<\/em> to be still smaller or larger.<\/p>\n

[A] A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo, M.R. Posteraro, Some isoperimetric inequalities on R^N with respect to weights |x|^\\alpha, arXiv.org (2016). .<\/p>\n

[D] Alexander D\u00edaz, Nate Harman, Sean Howe, David Thompson. Isoperimetric problems in sectors with density, Adv. Geom. 12 (2012), 589\u2013619; arXiv.org (2010).<\/p>\n

[E]\u00a0Max Engelstein, Anthony Marcuccio, Quinn Maurmann, and Taryn Pritchard, Isoperimetric problems on the sphere and on surfaces with density, New York J. Math. 15 (2009), 97\u2013123.<\/p>\n

[M] Frank Morgan,\u00a0 Isoperimetric symmetry breaking: a counterexample to a generalized form of the log-convex density conjecture, Anal. Geom. Metr. Spaces 4 (2016), 314-316.<\/p>\n

[O] Robert Osserman, The Isoperimetric Inequality<\/a>, Bull. Amer. Math. Soc. 84 (1978), 1182-1238.<\/p>\n

[R]\u00a0C\u00e9sar Rosales, Antonio Ca\u00f1ete, Vincent Bayle, and Frank Morgan. On the isoperimetric problem in Euclidean space with density. Calc. Var. PDE 31 (2008), 27-46; arXiv.org<\/a> (2006).<\/p>\n","protected":false},"excerpt":{"rendered":"

For Rn+1 with volume density f and perimeter density g, for a normal variation u of a surface with classical mean curvature H, the first variation of volume and perimeter are given by: , . For a volume-preserving variation, the second variation of perimeter is given by: , where is the second fundamental form, so […]<\/p>\n","protected":false},"author":269,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[14043],"tags":[],"class_list":["post-190","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/190","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/users\/269"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/comments?post=190"}],"version-history":[{"count":121,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/190\/revisions"}],"predecessor-version":[{"id":2887,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/190\/revisions\/2887"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=190"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=190"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=190"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}