×

Quantitative estimates for bending energies and applications to non-local variational problems. (English) Zbl 1437.49005

The authors consider an energy which is the sum of three terms \(\mathcal{F} _{\lambda ,Q}(E)=\lambda P(E)+W(E)+QV_{\alpha }(E)\), for subsets \(E\subset \mathbb{R}^{d}\), \(d=2,3\), with \(\lambda ,Q\geq 0\), where \(P(E)\) is the perimeter \(P(E)=\mathcal{H}^{d-1}(\partial E)\), \(W\) is the elastica or Willmore energy defined as \(W(E)=\int_{\partial E}H^{2}d\mathcal{H}^{1}\) if \( d=2\) and \(W(E)=\frac{1}{4}\int_{\partial E}H^{2}d\mathcal{H}^{2}\) if \(d=3\), \( H\) being the mean curvature of \(\partial E\) in dimension two and the sum of the principal curvatures of \(\partial E\) in dimension three, and \(V_{\alpha } \) is the Riesz interaction energy defined for \(\alpha \in (0,d)\) as \( V_{\alpha }(E)=\int_{E\times E}\frac{1}{\left\vert x-y\right\vert ^{d-\alpha }}\,dxdy\). The authors introduce the following classes of admissible sets \(\mathcal{M}=\{E\subset \mathbb{R}^{d}\) bounded with \(W^{2,2}\)-regular boundary\(\}\), \(\mathcal{M}_{sc}=\{E\in \mathcal{M}:E\) simply connected\(\}\), \( \mathcal{M}(\left\vert B_{1}\right\vert )=\{E\in \mathcal{M}:\left\vert E\right\vert =\left\vert B_{1}\right\vert \}\) and \(\mathcal{M} _{sc}(\left\vert B_{1}\right\vert )=\{E\in \mathcal{M}_{sc}:\left\vert E\right\vert =\left\vert B_{1}\right\vert \}\) and they consider the two minimization problems \(\min_{\mathcal{M}_{sc}(\left\vert B_{1}\right\vert )} \mathcal{F}_{\lambda ,Q}(E)\) and \(\min_{\mathcal{M}(\left\vert B_{1}\right\vert )}\mathcal{F}_{\lambda ,Q}(E)\). The first main result proves in the 2D case the existence of \(Q_{0}>0\) such that for \(Q< Q_{0}\) and all \(\lambda >0\), balls are the only solutions to the first minimization problem. Still in the 2D case, the second main result proves the existence of \(Q_{1}>0\) such that, for every \(\lambda >\overline{\lambda }\) and every \( Q\leq Q_{1}(\lambda -\overline{\lambda })\), balls are the only solutions to the second minimization problem. There exists \(Q_{2}>0\) such that, for every \( \lambda \in (0,\overline{\lambda })\) and every \(Q\leq Q^{2\lambda (3+\alpha )/2}\), centred annuli are the only solutions to the second minimization problem. For every \(\alpha \in (1,2)\) there exists \(Q_{3}(\alpha )\) such that for every \(\lambda >0\) and every \(Q\geq Q_{3}(\alpha )(\lambda +\lambda (\alpha -1)/2)\), the second minimization problem has no solution. In the 3D case, the authors prove that in the case where \(\lambda =0\), there exists \( Q_{4}>0\) such that, for every \(Q\leq Q_{4}\), the only solutions to the second minimization problem are balls. For every \(\alpha \in (2,3)\), there exists \( Q_{5}(\alpha )\) such that for every \(\lambda \) and \(Q\) with \(Q\geq Q_{5}(\lambda ^{-((3-\alpha )/2)}+\lambda ^{(3+\alpha )/2})\), the energy functional \(\mathcal{F}_{\lambda ,Q}\) has no minimizer in \(\mathcal{M} (\left\vert B_{1}\right\vert )\). For the proofs, the authors first establish properties of the energies \(W\) and \(\mathcal{F}_{\lambda ,Q}\). They prove the existence of a universal constant \(c_{0}>0\) such that for every set \( E\in \mathcal{M}_{sc}(\left\vert B_{1}\right\vert )\), \(W(E)-W(B_{1})\geq c_{0}\min_{x}\left\vert E\Delta B_{1}(x)\right\vert ^{2}\), where \(E\Delta F\) is the symmetric difference of the sets \(E\) and \(F\). Furthermore, there exist \(\delta _{0}>0\) and \(c_{1}>0\) such that if \(W(E)\leq W(B_{1})+\delta _{0}\), then \(W(E)-W(B_{1})\geq c_{1}(P(E)-P(B_{1}))\). The authors also prove that if \(Q=0\) and \(d=2\), there exists \(\overline{\lambda }>0\) such that for \( \lambda \in (0,\overline{\lambda })\), the solutions to the second minimization problem are annuli while for \(\lambda >\overline{\lambda }\) they are balls. Finally, if \(d=2\) and \(\overline{\lambda }\) is given by the preceding result, there exists a universal constant \(c_{2}>0\), such that for any \(E\in \mathcal{M}(\left\vert B_{1}\right\vert )\) and \(\lambda >\overline{ \lambda }\) \(\mathcal{F}_{\lambda ,0}(E)-\mathcal{F}_{\lambda ,0}(B_{1})\geq c_{2}(\lambda -\overline{\lambda })\min_{x}\left\vert E\Delta B_{1}(x)\right\vert ^{2}\), while for any \(\lambda _{\ast }>0\) there exists a constant \(c(\lambda _{\ast })>0\) such that, for any \(\lambda \in \lbrack \lambda _{\ast },\overline{\lambda }]\) \(\mathcal{F}_{\lambda ,0}(E)-\min_{ \mathcal{M}(\left\vert B_{1}\right\vert )}\mathcal{F}_{\lambda ,0}\geq c(\lambda _{\ast })\min_{\Omega }\left\vert E\Delta \Omega \right\vert ^{2}\) , where the minimum is taken among all sets \(\Omega \) minimizing \(\mathcal{F} _{\lambda ,0}\) in \(\mathcal{M}(\left\vert B_{1}\right\vert )\) (which are either balls or annuli depending on \(\lambda \)). The authors also use isoperimetric inequalities.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
76W99 Magnetohydrodynamics and electrohydrodynamics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Acerbi, E., Fusco, N. and Morini, M.. Minimality via second variation for a nonlocal isoperimetric problem. Comm. Math. Phys.322 (2013), 515-557. · Zbl 1270.49043
[2] Alberti, G., Choksi, R. and Otto, F.. Uniform energy distribution for an isoperimetric problem with long-range interactions. J. Amer. Math. Soc.22 (2009), 569-605. · Zbl 1206.49046
[3] Bates, F. S. and Fredrickson, G. H.. Block copolymers-designer soft materials. Phys. Today52 (1999), 32-38.
[4] Bella, P., Goldman, M. and Zwicknagl, B.. Study of island formation in epitaxially strained films on unbounded domains. Arch. Ration. Mech. Anal.218 (2015), 163-217. · Zbl 1327.74101
[5] Bonacini, M. and Cristoferi, R.. Local and global minimality results for a nonlocal isoperimetric problem on ℝ^n. SIAM J. Math. Anal.46 (2014), 2310-2349. · Zbl 1301.49114
[6] Brasco, L., De Philippis, G. and Velichkov, B.. Faber-Krahn inequalities in sharp quantitative form. Duke Math. J.164 (2015), 1777-1831. · Zbl 1334.49149
[7] Bucur, D. and Henrot, A.. A new isoperimetric inequality for the elasticae. J. Eur. Math. Soc.19 (2017), 3355-3376. · Zbl 1377.49047
[8] Choksi, R. and Peletier, M. A.. Small volume fraction limit of the diblock copolymer problem. I: Sharp-interface functional. SIAM J. Math. Anal.42 (2010), 1334-1370. · Zbl 1210.49050
[9] Choksi, R., Muratov, C. B. and Topaloglu, I. A.. An old problem resurfaces nonlocally: Gamow’s liquid drops inspire today’s research and applications. Notices Amer. Math. Soc. 6464 (2017), 1275-1283. · Zbl 1376.49059
[10] Cicalese, M. and Leonardi, G. P.. A selection principle for the sharp quantitative isoperimetric inequality. Arch. Ration. Mech. Anal.206 (2012), 617-643. · Zbl 1257.49045
[11] Cicalese, M. and Spadaro, E.. Droplet minimizers of an isoperimetric problem with long-range interactions. Comm. Pure Appl. Math.66 (2013), 1298-1333. · Zbl 1269.49085
[12] Cicalese, M., Leonardi, G. P. and Maggi, F.. Improved convergence theorems for bubble clusters I. The planar case. Indiana Univ. Math. J.65 (2016), 1979-2050. · Zbl 1381.49051
[13] De Lellis, C. and Müller, S.. Optimal rigidity estimates for nearly umbilical surfaces. J. Diff. Geom.69 (2005), 75-110. · Zbl 1087.53004
[14] De Simone, A., Kohn, R. V., Müller, S. and Otto, F.. Recent analytical developments in micromagnetics. In The science of hysteresis. Physical modeling, micromagnetics, and magnetization dynamics (ed. Bertotti, G. and Mayergoyz, I. D.), pp. 269-381 (Amsterdam: Elsevier [u. a.], 2006). · Zbl 1151.35426
[15] Ferone, V., Kawohl, B. and Nitsch, C.. The elastica problem under area constraint. Math. Annalen365 (2016), 987-1015. · Zbl 1345.49049
[16] Figalli, A., Fusco, N., Maggi, F., Millot, V. and Morini, M.. Isoperimetry and stability properties of balls with respect to nonlocal energies. Comm. Math. Phys.336 (2015), 441-507. · Zbl 1312.49051
[17] Frank, R. L., Killip, R. and Nam, P. T.. Nonexistence of large nuclei in the liquid drop model. Lett. Math. Phys.106 (2016), 1033-1036. · Zbl 1347.49069
[18] Fuglede, B.. Stability in the isoperimetric problem for convex or nearly spherical domains in R^n. Trans. Amer. Math. Soc.314 (1989), 619-638. · Zbl 0679.52007
[19] Fusco, N., Maggi, F. and Pratelli, A.. The sharp quantitative isoperimetric inequality. Ann. Math. (2)168 (2008), 941-980. · Zbl 1187.52009
[20] Gamow, G.. Mass defect curve and nuclear constitution. Proc. R. Soc. London. Series A126 (1930), 632-644. · JFM 56.0762.02
[21] Goldman, M. and Ruffini, B.. Equilibrium shapes of charged liquid droplets and related problems: (mostly) a review. Geom. Flows2 (2017), 94-104. · Zbl 1383.49008
[22] Goldman, M. and Runa, E.. On the optimality of stripes in a variational model with non-local interactions. Preprint, 2016. · Zbl 1415.49033
[23] Goldman, D., Muratov, C. B. and Serfaty, S.. The Γ-limit of the two-dimensional Ohta-Kawasaki energy. I. Droplet density. Arch. Ration. Mech. Anal.210 (2013), 581-613. · Zbl 1296.82018
[24] Goldman, M., Novaga, M. and Ruffini, B.. Existence and stability for a non-local isoperimetric model of charged liquid drops. Arch. Ration. Mech. Anal.217 (2015), 1-36. · Zbl 1328.76031
[25] Hutchinson, J. E.. Second fundamental form for varifolds and the existence of surfaces minimising curvature. Indiana Univ. Math. J.35 (1986), 45-71. · Zbl 0561.53008
[26] Julin, V.. Isoperimetric problem with a Coulomb repulsive term. Indiana Univ. Math. J.63 (2014), 77-89. · Zbl 1311.49110
[27] Julin, V.. Remark on a nonlocal isoperimetric problem. Nonlinear Anal.154 (2017), 174-188. · Zbl 1358.49039
[28] Knüpfer, H. and Muratov, C. B.. On an isoperimetric problem with a competing nonlocal term I: The planar case. Comm. Pure Appl. Math.66 (2013), 1129-1162. · Zbl 1269.49087
[29] Knüpfer, H. and Muratov, C. B.. On an isoperimetric problem with a competing nonlocal term II: The general case. Comm. Pure Appl. Math.67 (2014), 1974-1994. · Zbl 1302.49064
[30] Knüpfer, H., Kohn, R. V. and Otto, F.. Nucleation barriers for the cubic-to-tetragonal phase transformation. Comm. Pure Appl. Math.66 (2013), 867-904. · Zbl 1322.74057
[31] Knüpfer, H., Muratov, C. B. and Novaga, M.. Low density phases in a uniformly charged liquid. Commun. Math. Phys.345 (2016), 141-183. · Zbl 1346.49017
[32] Kohn, R. V. and Müller, S.. Surface energy and microstructure in coherent phase transitions. Comm. Pure Appl. Math.47 (1994), 405-435. · Zbl 0803.49007
[33] Kuwert, E. and Schätzle, R.. Removability of point singularities of Willmore surfaces. Ann. Math. (2)160 (2004), 315-357. · Zbl 1078.53007
[34] Kuwert, E. and Schätzle, R.. The Willmore functional. In Topics in modern regularity theory, volume 13 of CRM Series (ed. Mingione, G.), pp. 1-115 (Pisa: Ed. Norm., 2012). · Zbl 1322.53002
[35] Lamm, T. and Schätzle, R. M.. Optimal rigidity estimates for nearly umbilical surfaces in arbitrary codimension. Geom. Funct. Anal.24 (2014), 2029-2062. · Zbl 1322.53008
[36] Leoni, G.. Variational models for epitaxial growth. In Free discontinuity problems, volume 19 of CRM Series (ed. Fusco, N. and Pratelli, A.), pp. 69-152 (Pisa: Ed. Norm., 2016). · Zbl 1378.82053
[37] Lieb, E. H. and Loss, M.. Analysis. 2nd ed. Grad. Stud. Math. (Providence, RI: American Mathematical Society (AMS), 2001). · Zbl 0966.26002
[38] Lu, J. and Otto, F.. Nonexistence of a minimizer for Thomas-Fermi-Dirac-von Weizsäcker model. Comm. Pure Appl. Math.67 (2014), 1605-1617. · Zbl 1301.49002
[39] Marques, F. C. and Neves, A.. Min-max theory and the Willmore conjecture. Ann. Math. Second Series179 (2014), 683-782. · Zbl 1297.49079
[40] Müller, S. and Röger, M.. Confined structures of least bending energy. J. Diff. Geom.97 (2014), 109-139. · Zbl 1296.53127
[41] Muratov, C. B. and Novaga, M.. On well-posedness of variational models of charged drops. Proc. R. Soc. A472 (2016), 20150808. · Zbl 1371.76163
[42] Muratov, C. B., Novaga, M. and Ruffini, B.. On equilibrium shapes of charged flat drops. Comm. Pure Appl. Math.71 (2018), 1049-1073. · Zbl 1394.49046
[43] Ohta, T. and Kawasaki, K.. Equilibrium morphology of block copolymer melts. Macromolecules19 (1986), 2621-2632.
[44] Okamoto, M., Maruyama, T., Yabana, K. and Tatsumi, T.. Nuclear ‘pasta’ structures in low-density nuclear matter and properties of the neutron-star crust. Phys. Rev. C88 (Aug 2013), 025801.
[45] Rayleigh, L.. On the equilibrium of liquid conducting masses charged with electricity. Phil. Mag.14 (1882), 184-186.
[46] Ren, X. and Wei, J.. Spherical solutions to a nonlocal free boundary problem from diblock copolymer morphology. SIAM J. Math. Anal.39 (2008), 1497-1535. · Zbl 1153.35091
[47] Röger, M. and Schätzle, R.. Control of the isoperimetric deficit by the Willmore deficit. Analysis32 (2012), 1-7. · Zbl 1252.49069
[48] Schygulla, J.. Willmore minimizers with prescribed isoperimetric ratio. Arch. Ration. Mech. Anal.203 (2012), 901-941. · Zbl 1288.74027
[49] Simon, L.. Lectures on geometric measure theory, volume 3 of Proceedings of the Centre for Mathematical Analysis (Canberra: Australian National University, 1983). · Zbl 0546.49019
[50] Simon, L.. Existence of surfaces minimizing the Willmore functional. Comm. Anal. Geom.1 (1993), 281-326. · Zbl 0848.58012
[51] Topping, P.. Mean curvature flow and geometric inequalities. J. Reine Angew. Math.503 (1998), 47-61. · Zbl 0909.53044
[52] Willmore, T. J.. Note on embedded surfaces. Ann. Stiint. Univ. Al. I. Cuza, Iaşi, Sect. I a Mat. (N.S.)11B (1965), 493-496. · Zbl 0171.20001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.