Geometric aspects of $p$-capacitary potentials

Fogagnolo, Mattia; Mazzieri, Lorenzo; Pinamonti, Andrea

doi:10.1016/j.anihpc.2018.11.005

Geometric aspects of

p

-capacitary potentials

Fogagnolo, Mattia ; Mazzieri, Lorenzo ; Pinamonti, Andrea

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 4, pp. 1151-1179.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez l'article sur le site de la revue.

Résumé

We provide monotonicity formulas for solutions to the p-Laplace equation defined in the exterior of a convex domain. A number of analytic and geometric consequences are derived, including the classical Minkowski inequality as well as new characterizations of rotationally symmetric solutions and domains. The proofs rely on the conformal splitting technique introduced by the second author in collaboration with V. Agostiniani.

DOI : 10.1016/j.anihpc.2018.11.005

Mots-clés : p-Laplacian, Monotonicity formulas, Minkowski inequality

@article{AIHPC_2019__36_4_1151_0,
     author = {Fogagnolo, Mattia and Mazzieri, Lorenzo and Pinamonti, Andrea},
     title = {Geometric aspects of $p$-capacitary potentials},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1151--1179},
     publisher = {Elsevier},
     volume = {36},
     number = {4},
     year = {2019},
     doi = {10.1016/j.anihpc.2018.11.005},
     mrnumber = {3955113},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2018.11.005/}
}

TY  - JOUR
AU  - Fogagnolo, Mattia
AU  - Mazzieri, Lorenzo
AU  - Pinamonti, Andrea
TI  - Geometric aspects of $p$-capacitary potentials
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2019
SP  - 1151
EP  - 1179
VL  - 36
IS  - 4
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2018.11.005/
DO  - 10.1016/j.anihpc.2018.11.005
LA  - en
ID  - AIHPC_2019__36_4_1151_0
ER  -

%0 Journal Article
%A Fogagnolo, Mattia
%A Mazzieri, Lorenzo
%A Pinamonti, Andrea
%T Geometric aspects of $p$-capacitary potentials
%J Annales de l'I.H.P. Analyse non linéaire
%D 2019
%P 1151-1179
%V 36
%N 4
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2018.11.005/
%R 10.1016/j.anihpc.2018.11.005
%G en
%F AIHPC_2019__36_4_1151_0

Fogagnolo, Mattia; Mazzieri, Lorenzo; Pinamonti, Andrea. Geometric aspects of $p$-capacitary potentials. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 4, pp. 1151-1179. doi : 10.1016/j.anihpc.2018.11.005. http://www.numdam.org/articles/10.1016/j.anihpc.2018.11.005/

Bibliographie
Cité par

[1] Alexandrov, A.D. Zur Theorie der gemischten Volumina von konvexen Körpern, II. Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen, Mat. Sb. (N.S.), Volume 2 (1937), pp. 1205–1238 (in Russian) | JFM

[2] Alexandrov, A.D. Zur Theorie der gemischten Volumina von konvexen Körpern, III. Die Erweiterung zweeier Lehrsatze Minkowskis über die konvexen Polyeder auf beliebige konvexe Flachen, Mat. Sb. (N.S.), Volume 3 (1938), pp. 27–46 (in Russian) | Zbl

[3] Agostiniani, V.; Mazzieri, L. Riemannian aspects of potential theory, J. Math. Pures Appl. (9), Volume 104 (2015) no. 3, pp. 561–586 | DOI | MR

[4] Agostiniani, V.; Mazzieri, L. Monotonicity formulas in potential theory (ArXiv Preprint Server) | arXiv | MR

[5] Agostiniani, V.; Mazzieri, L. On the geometry of the level sets of bounded static potentials, Commun. Math. Phys., Volume 355 (2017) no. 1, pp. 261–301 | DOI | MR

[6] Agostiniani, V.; Mazzieri, L. Comparing monotonicity formulas for electrostatic potentials and static metrics, Rend. Lincei Mat. Appl., Volume 28 (2017), pp. 7–20 | MR

[7] Akman, M.; Gong, J.; Hineman, J.; Lewis, J.; Vogel, A. The Brunn–Minkowski inequality and a Minkowski problem for nonlinear capacity | arXiv

[8] Bianchini, C.; Ciraolo, G. Wulff shape characterizations in overdetermined anisotropic elliptic problems | arXiv

[9] Borghini, S.; Mascellani, G.; Mazzieri, L. Some sphere theorems in linear potential theory, Trans. Am. Math. Soc. (2018) (in press) | DOI | MR

[10] Borghini, S.; Mazzieri, L. On the mass of static metrics with positive cosmological constant – I, Class. Quantum Gravity, Volume 35 (2018) | DOI | MR

[11] Borghini, S.; Mazzieri, L. On the mass of static metrics with positive cosmological constant – II (ArXiv Preprint Server) | arXiv | MR

[12] Bour, V.; Carron, G. Optimal integral pinching results, Ann. Sci. Éc. Norm. Supér., Volume 48 (2015) no. 1, pp. 41–70 | MR

[13] Chang, S.C.; Chen, J.T.; Wei, S.W. Liouville properties of $p$ -harmonic maps with finite $q$ -energy, Trans. Am. Math. Soc., Volume 368 (2016) no. 2, pp. 787–825 | DOI | MR

[14] Chang, S.-Y.A.; Wang, Y. On Aleksandrov–Fenchel inequalities for $k$ -convex domains, Milan J. Math., Volume 79 (2011) no. 1, pp. 13–38 | MR | Zbl

[15] Cheeger, J.; Naber, A.; Valtorta, D. Critical set of elliptic equations, Commun. Pure Appl. Math., Volume 68 (2015), pp. 173–209 | DOI | MR | Zbl

[16] Colding, T.H. New monotonicity formulas for Ricci curvature and applications. I, Acta Math., Volume 209 (2012) no. 2, pp. 229–263 | DOI | MR | Zbl

[17] Colding, T.H.; Minicozzi, W.P. Ricci curvature and monotonicity for harmonic functions, Calc. Var. Partial Differ. Equ., Volume 49 (2014) no. 3, pp. 1045–1059 | MR | Zbl

[18] Colesanti, A.; Nyström, K.; Salani, P.; Xiao, J.; Yang, D.; Zhang, G. The Hadamard variational formula and the Minkowsky problem for $p$ -capacity, Adv. Math., Volume 285 (2015), pp. 1511–1588 | DOI | MR

[19] Colesanti, A.; Salani, P. On the Brunn–Minkowski inequality for the $p$ -capacity of convex bodies, Math. Ann., Volume 327 (2003), pp. 459–479 | DOI | MR | Zbl

[20] DiBenedetto, E. $C^{1, α}$ local regularity of weak solutions of degenerate elliptic equation, Nonlinear Anal., Volume 7 (1983), pp. 827–850 | DOI | MR | Zbl

[21] Dipierro, S.; Pinamonti, A.; Valdinoci, E. Rigidity results for elliptic boundary value problems: stable solutions for quasilinear equations with Neumann or Robin boundary conditions, Int. Math. Res. Not. IMRN (2018) (in press) | DOI | MR

[22] Dipierro, S.; Pinamonti, A.; Valdinoci, E. Rigidity results in diffusion Markov triples, J. Funct. Anal. (2018) (in press) | DOI | MR

[23] Dipierro, S.; Pinamonti, A.; Valdinoci, E. Classification of stable solutions for boundary value problems with nonlinear boundary conditions on Riemannian manifolds with nonnegative Ricci curvature, Adv. Nonlinear Anal. (2018) (in press) | DOI | MR

[24] Enciso, A.; Peralta-Salas, D. Critical points and level sets in exterior boundary problems, Indiana Univ. Math. J., Volume 58 (2009) no. 4, pp. 1947–1969 | DOI | MR | Zbl

[25] Evans, L.C. A new proof of local $C^{1, α}$ regularity for solutions of certain degenerate P.D.E., J. Differ. Equ., Volume 45 (1982), pp. 356–373 | DOI | MR | Zbl

[26] Hirsch, M.W. Differential Topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York–Heidelberg, 1976 (MR 0448362) | DOI | MR | Zbl

[27] Kichenassamy, S.; Veron, L. Singular solutions of the $p$ -Laplace equation, Math. Ann., Volume 275 (1986), pp. 599–615 | DOI | MR | Zbl

[28] Farina, A.; Mari, L.; Valdinoci, E. Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds, Commun. Partial Differ. Equ., Volume 38 (2013) no. 10, pp. 1818–1862 | DOI | MR | Zbl

[29] Farina, A.; Sciunzi, B.; Valdinoci, E. Bernstein and De Giorgi type problems: new results via a geometric approach, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5), Volume 7 (2008) no. 4, pp. 741–791 | Numdam | MR | Zbl

[30] Farina, A.; Sire, Y.; Valdinoci, E. Stable solutions of elliptic equations on Riemannian manifolds, J. Geom. Anal., Volume 23 (2013) no. 3, pp. 1158–1172 | DOI | MR | Zbl

[31] Fenchel, W. Über Krümmung und Windung geschlossener Raumkurven, Math. Ann., Volume 101 (1929), pp. 238–252 | DOI | JFM | MR

[32] Freire, A.; Schwartz, F. Mass-capacity inequalities for conformally flat manifolds with boundary, Commun. Partial Differ. Equ., Volume 39 (2014), pp. 98–119 | DOI | MR | Zbl

[33] Garofalo, N.; Lewis, J.L. A symmetry result related to some overdetermined boundary value problems, Am. J. Math., Volume 111 (1989) no. 1, pp. 9–33 | DOI | MR | Zbl

[34] Garofalo, N.; Sartori, E. Symmetry in exterior boundary value problems for quasilinear elliptic equations via blow-up and a priori estimates, Adv. Differ. Equ., Volume 4 (1999) no. 2, pp. 137–161 | MR | Zbl

[35] Gerhardt, C. Flow of nonconvex hypersurfaces into spheres, J. Differ. Geom., Volume 32 (1990), pp. 299–314 | DOI | MR | Zbl

[36] Gilbarg, D.; Trudinger, N.S. Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, Heidelberg, 2001 | DOI | MR | Zbl

[37] Hawking, S.W.; Ellis, G.F.R. The Large Scale Structure of Space–Time, Cambridge Monographs on Mathematical Physics, vol. 1, Cambridge University Press, London, 1973 | DOI | MR | Zbl

[38] Guan, P.; Li, J. The quermassintegral inequalities for $k$ -convex starshaped domains, Adv. Math., Volume 221 (2009), pp. 1725–1732 | DOI | MR | Zbl

[39] Ladyzhenskaya, O.; Ural'tseva, N. Linear and Quasilinear Elliptic Equations, Academic Press, New York, London, 1968 | MR | Zbl

[40] Huisken, G.; Ilmanen, T. The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differ. Geom., Volume 59 (2001), pp. 353–437 | DOI | MR | Zbl

[41] Huisken, G. An isoperimetric concept for the mass in general relativity, March 2009 http://video.ias.edu/node/234 (video available at)

[42] Lewis, J.L. Capacitary functions in convex rings, Arch. Ration. Mech. Anal., Volume 66 (1977), pp. 201–224 | DOI | MR | Zbl

[43] Lewis, J.L. Regularity of the derivative of solutions of certain degenerate elliptic equations, Indiana Univ. Math. J., Volume 32 (1983), pp. 849–858 | DOI | MR | Zbl

[44] Magnanini, R.; Poggesi, G. On the stability for Alexandrov's soap bubble theorem, J. Anal. Math. (2018) (in press, ArXiv Preprint Server, 2016) | arXiv | MR

[45] Magnanini, R.; Poggesi, G. Serrin's problem and Alexandrov's soap bubble theorem: enhanced stability via integral identities, Indiana Univ. Math. J. (2018) (in press, ArXiv Preprint Server, 2017) | arXiv | MR

[46] Maz'ya, V. Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, 2011 | DOI | MR | Zbl

[47] Meyers, N.G. A theory of capacities for potential of functions in Lebesgue classes, Math. Scand., Volume 26 (1970), pp. 255–292 | DOI | MR | Zbl

[48] Moser, R. The inverse mean curvature flow and $p$ -harmonic functions, J. Eur. Math. Soc., Volume 9 (2007), pp. 77–83 | DOI | MR | Zbl

[49] Poggesi, G. Radial symmetry for p-harmonic functions in exterior and punctured domains, Appl. Anal. (2018) (in press) | DOI | MR

[50] Reichel, W. Radial symmetry for elliptic boundary-value problems on exterior domains, Arch. Ration. Mech. Anal., Volume 137 (1997), pp. 81–394 | DOI | MR | Zbl

[51] Schoen, R.; Simon, L.; Yau, S.T. Curvature estimates for minimal hypersurfaces, Acta Math., Volume 134 (1975), pp. 275–288 | DOI | MR | Zbl

[52] Sternberg, P.; Zumbrun, K. Connectivity of phase boundaries in strictly convex domains, Arch. Ration. Mech. Anal., Volume 141 (1998) no. 4, pp. 375–400 | DOI | MR | Zbl

[53] Uraltseva, N. Degenerate quasilinear elliptic systems, Zap. Naučn. Semin. Leningrad. Otel. Mat. Inst. Steklov, Volume 7 (1968), pp. 184–222 | MR | Zbl

[54] Uhlenbeck, K. Regularity for a class of nonlinear elliptic system, Acta Math., Volume 138 (1977), pp. 219–240 | DOI | MR | Zbl

[55] Urbas, J. On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures, Math. Z., Volume 205 (1990), pp. 355–372 | DOI | MR | Zbl

[56] Valtorta, D. On the p-Laplace operator on Riemannian manifolds (PhD thesis) | arXiv

[57] Willmore, T.J. Mean curvature of immersed surfaces, An. Şti. Univ. “All. I. Cuza” Iaşi Secţ. I a Mat. (N.S.), Volume 14 (1968), pp. 99–103 | MR | Zbl

[58] Xiao, J. $P$ -Capacity vs surface-area, Adv. Math. (2017), pp. 1318–1336 | MR | Zbl

Cité par Sources :