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These surfaces are solutions of a variational problem that describes the shape of a drop of incompressible fluid in equilibrium by the action of surface tension when it rotates about L with constant angular velocity. The effect of gravity is neglected. In this paper we study the geometric configurations of such surfaces, focusing the relationship between the geometry of the surface and the one of its boundary. As special cases, we will consider two families of such surfaces: axisymmetric surfaces and embedded surfaces with planar boundary.",{"EN":209},"Stationary rotating surfaces in Euclidean space",{"VOID":211},"[\"4228482698975726486\"]",{"VOID":213},"Aguiar D.: Stability of rotating liquid films. Q. J. Mech. Appl. Math. 55, 327–343 (2002)\nAlbano S., Gonzalez E.H.A.: Rotating drops. Indiana Univ. Math. J. 32, 687–702 (1983)\nAlexandrov A.D.: Uniqueness theorems for surfaces in the large V. Vestnik Leningrad Univ. Math. 13, 5–8 (1958) (English translation: AMS Transl. 21, 412–416 (1962))\nAthanassenas M.: Rotating drops trapped between parallel planes. Ann. Sci. Norm. Super Pisa Cl. Sci. 26, 749–762 (1998)\nAuchmuty J.E.G.: Existence of axisymmetric equilibrium figures. Arch. Ration. Mech. Anal. 65, 249–261 (1977)\nAussillous P., Queré D.: Shapes of rolling liquid drops. J. Fluid Mech. 512, 133–151 (2004)\nBeer A.: Einleitung in der mathematische Theorie der Elasticität und Capillarität, part 2. A. Gissen Verlag, Leipzig (1869)\nBrito F., Sa Earp R., Meeks W., Rosenberg H.: Structure theorems for constant mean curvature surfaces bounded by a planar curve. Indiana Univ. Math. J. 40, 333–343 (1991)\nBrown R.A., Scriven L.E.: The shape and stability of rotating liquid drops. Proc. R. Soc. Lond. A 371, 331–357 (1980)\nBrown R.A., Scriven L.E.: New class of asymmetric shapes of rotating liquid drops. Phys. Rev. Lett. 45, 180–183 (1980)\nBrulois, F.: The limit of stability of axisymmetric rotating drops. In: Variational methods for free surface interfaces, pp. 145–153 (Menlo Park, Ca. 1985). Springer, New York, (1987)\nCaffarelli L.A., Friedman A.: The shape of axisymmetric rotating fluid. J. Funct. Anal. 35, 109–142 (1980)\nCardoso V., Gualtieri L.: Equilibrium configurations of fluids and their stability in higher dimensions. Class. Quant. Grav. 23, 7151–7198 (2006)\nChandrasekhar, S.: Ellipsoidal figures of equilibrium. Yale Univ. Press, New Haven, Conn. (1962)\nChandrasekhar S.: The stability of a rotating liquid drop. Proc. R. Soc. Lond. A 286, 1–26 (1965)\nCongedo G.: Rotating drops in a vessel. Existence of local minima. Rend. Sem. Mat. Univ. Padova 72, 135–156 (1984)\nCongedo G., Emmer M., Gonzalez E.H.A.: Rotating drops in a vessel. Rend. Sem. Math. Univ. Padova 70, 167–186 (1983)\nFinn R.: Equilibrium capillary surfaces, Grundlehren der Mathematischen Wissenschaften, vol. 284. Springer, New York (1986)\nGulliver R.: Tori of prescribed mean curvature and the rotating drop. Soc Math. de France, Astérisque 118, 167–179 (1984)\nHeine C.J.: Computations of form and stability of rotating drops with finite elements. IMA J. Num. Anal. Adv. 26, 723–751 (2006)\nHeinz H.: On the nonexistence of a surface of constant mean curvature with finite area and prescribed rectifiable boundary. Arch. Rat. Mech. Anal. 35, 249–252 (1969)\nHopf H.: Differential geometry in the large. Lecture notes in mathematics 1000. Springer, Berlin (1983)\nHynd R., McCuan J.: On toroidal rotating drops. Pac. J. Math. 224, 279–289 (2006)\nKapouleas N.: Slowly rotating drops. Comm. Math. Phys. 129, 139–159 (1990)\nKoiso M.: Symmetry of hypersurfaces of constant mean curvature with symmetric boundary. Math. Z. 191, 567–574 (1986)\nKoiso M., Palmer B.: Geometry and stability of bubbles with gravity. Indiana Univ. Math. J. 54, 65–98 (2005)\nKopal Z.: Figures of equilibrium in celestial bodies. Univ. of Wisconsin Press, Madikson (1960)\nLangbein D.W.: Capillary surfaces: shape—stability—dynamics, in particular under weightlessness. Springer-Verlag, Berlin (2002)\nLee C.P., Anilkumar A.V., Hmelo A.B., Wang T.G.: Equilibrium of liquid drops under the effects of rotation and acoustic flattening: results from USML-2 experiments in Space. J. Fluid Mech. 354, 43–67 (1998)\nLópez R.: A criterion on instability of rotating cylindrical surfaces. Arch. Math. 94, 91–99 (2010)\nMcCuan, J.: Retardation of plateau-rayleigh instability: a distinguishing characteristic among perfectly wetting fluids. MSRI Preprint # 1997-011. arXiv:math\u002F9701214v1 (1997)\nPlateau, J.A.F.: Experimental and theoretical researches on the figures of equilibrium of a liquid mass withdrawn from the action of gravity. Annu. Rep. Board Regents Smithson. Inst., pp. 207–285 (1863)\nPoincaré H.: Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation. Acta Math. 7, 259–380 (1885)\nRoss D.K.: The shape and energy of a revolving liquid mass held together by surface tension. Aust. J. Phys. 21, 823–835 (1968)\nRoss, J., Brulois, F.: The stability of axisymmetric rotating drops. Variational methods for equilibrium problems of fluids. Meet. Trento\u002FItaly 1983, Astérisque, vol. 118, pp. 219–226. (1984)\nSerrin J.: On surfaces of constant mean curvature which span a given space curve. Math. Z. 112, 77–88 (1969)\nSmith D.R., Ross J.E.: Universal shapes and bifurcation for rotating incompressible fluid drops. Methods Appl. Anal. 1, 210–228 (1994)\nWang T.G., Trinh E.H., Croonquist A.P., Elleman D.D.: Shapes of rotating free drops: Spacelab experimental results. Phys. Rev. Lett. 56, 452–455 (1986)\nWavre R.: Figures Planétaires et Géodésie. Gauthier-Villars, Paris (1932)\nWente, H.C.: Existence theorems for surfaces of constant mean curvature and perturbations of a liquid globule in equilibrium. Ph.D. thesis, Harvard University, Cambridge, MA, (1966)\nWente H.C.: The symmetry of sessile and pendant drops. Pac. J. Math. 88, 387–397 (1980)\nWente H.C.: The symmetry of rotating fluid bodies. Manuscr. math. 39, 287–296 (1982)",{"VOID":215},"10.1007\u002Fs00526-010-0312-8","PUBLICATION","VERIFIED","2024-05-17T00:58:41.496+00:00","Auto Verify","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs00526-010-0312-8",[222],{"id":223,"sortIndex":21,"researcher":20,"roles":224,"affiliations":226,"properties":235,"displayName":237,"givenName":20,"familyName":20},"0dce2101-e613-4b90-a025-81f545b5a7b5",[225],"AUTHOR",[227],{"id":228,"sortIndex":21,"affiliation":229,"properties":20},"ad5f4d86-aebf-4c6d-9b9e-ef1d54d3536c",{"id":228,"createTime":20,"updateTime":20,"relativeEntities":230,"slug":20,"properties":231,"entityType":20,"verifyStatus":20,"verifyTime":20,"verifyNote":20,"languages":20,"translateLanguages":20,"viewCount":20,"url":20,"parentIds":234,"statistic":20},[],{"title":232},{"VI":233},"Departamento de Geometría y Topología, Universidad de Granada, Granada, Spain",[],{"title":236,"gsAuthor":238},{"VI":237},"Rafael 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We study a perturbed semilinear problem with Neumann boundary condition \n                \n                  \n \\[ \\cases{ -\\varepsilon^2\\Delta u+u=u^p & {\\rm in} \\Omega \\cr &\\cr u>0 & {\\rm in} \\Omega\\cr &\\cr {{\\partial u}\\over{\\partial\\nu}}=0& {\\rm in} \\partial\\Omega,\\cr} \\] \n                \n               where \n                  \n$\\Omega$\n                 is a bounded smooth domain of \n                  \n${mathbb{R}}^N$\n                , \n                  \n$N\\ge2$\n                , \n                  \n$\\varepsilon>0$\n                , \n                  \n$1 \u003C p \u003C {{N+2}\\over{N-2}}$\n                 if \n                  \n$N\\ge3$\n                 or \n                  \n$p>1$\n                 if \n                  \n$N=2$\n                 and \n                  \n$\\nu$\n                 is the unit outward normal at the boundary of \n                  \n$\\Omega$\n                . We show that for any fixed positive integer K any “suitable” critical point \n                  \n$(x_0^1,\\dots,x_0^K)$\n                 of the function \n                \n                  \n\\begin{eqnarray*} \\lefteqn{\\varphi_K(x^1,\\dots,x^K)} &=& \\min\\left\\{{\\rm dist}(x^i,{\\partial\\Omega}),{|x^j-x^l|\\over2} \\mid i,j,l=1.\\dots,K, j\\ne l\\right\\} \\end{eqnarray*} \n                \n               generates a family of multiple interior spike solutions, whose local maximum points \n                  \n$x_\\varepsilon^1,\\dots,x_\\varepsilon^K$\n                 tend to \n                  \n$x_0^1,\\dots,x_0^K$\n                 as \n                  \n$\\varepsilon$\n                 tends to zero.",{"EN":310},"Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point 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Based in this result and using blow up analysis we establish a sharp form of Trudinger-Moser type inequality for this class of weighted Sobolev spaces. 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Equ. 16, 1223–1253 (1991)",{},{"id":20,"text":561,"url":20,"identifiers":562},"Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext, Springer, New York (2011)",{},{"id":20,"text":564,"url":20,"identifiers":565},"Carleson, L., Chang, S.Y.A.: On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math. 110, 113–127 (1986)",{},{"id":567,"text":568,"url":569,"identifiers":570},"0ee9cb53-fd89-4355-8a84-7f67477077ee","Černý, R., Cianchi, A., Hencl, S.: Concentration-compactness principles for Moser-Trudinger inequalities: new results and proofs. Ann. Mat. Pura Appl. 192, 225–243 (2013)","http:\u002F\u002Flink.springer.com\u002F10.1007\u002Fs10231-011-0220-3",{"doi":571},"10.1007\u002Fs10231-011-0220-3",{"id":549,"text":573,"url":551,"identifiers":574},"Chen, W., Li, C.: Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63, 615–622 (1991)",{"doi":553},{"id":549,"text":576,"url":551,"identifiers":577},"Clément, P., de Figueiredo, D.G., Mitidieri, E.: Quasilinear elliptic equations with critical exponents. Topol. Methods Nonlinear Anal. 7, 133–170 (1996)",{"doi":553},{"id":549,"text":579,"url":551,"identifiers":580},"de Figueiredo, D.G., Gonçalves, J.V., Miyagaki, O.H.: On a class of quasilinear elliptic problems involving critical exponents. Commun. Contemp. Math. 2, 47–59 (2000)",{"doi":553},{"id":549,"text":582,"url":551,"identifiers":583},"de Figueiredo, D.G., do Ó, J.M., Ruf, B.: On an inequality by N. Trudinger and J. Moser and related elliptic equations. Comm. Pure Appl. Math. 55, 135–152 (2002)",{"doi":553},{"id":549,"text":585,"url":551,"identifiers":586},"de Figueiredo, D.G., do Ó, J.M., Ruf, B.: Elliptic equations and systems with critical Trudinger-Moser nonlinearities. Discrete Contin. Dyn. Syst. 30, 455–476 (2011)",{"doi":553},{"id":549,"text":588,"url":551,"identifiers":589},"de Oliveira, J.F., do Ó, J. M.: Trudinger-Moser type inequalities for weighted Sobolev spaces involving fractional dimensions. Proc. Am. Math. Soc. (to appear)",{"doi":553},{"id":549,"text":591,"url":551,"identifiers":592},"do Ó, J.M.: Semilinear Dirichlet problems for the \\(N\\)-Laplacian in \\(\\mathbb{R}^N\\) with nonlinearities in the critical growth range. Differ. Integr. Equ. 9, 967–979 (1996)",{"doi":553},{"id":594,"text":595,"url":596,"identifiers":597},"d8f17450-5527-4bf0-9742-3960394a4827","Flucher, M.: Extremal functions for the Trudinger-Moser inequality in 2 dimensions. Comment. Math. Helv. 67, 471–497 (1992)","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF02566514",{"doi":598},"10.1007\u002FBF02566514",{"id":549,"text":600,"url":551,"identifiers":601},"Hudson, S., Leckband, M.: Extremals for a Moser-Jodeit exponential inequality. Pacific J. Math. 206, 113–128 (2002)",{"doi":553},{"id":603,"text":604,"url":605,"identifiers":606},"5d05fbca-c7f6-40e7-a257-bd2e5f7dbfae","Hudson, S., Leckband, M.: Extremals for Moser inequalities. Arch. Ration. Mech. Anal. 171, 43–54 (2004)","http:\u002F\u002Flink.springer.com\u002F10.1007\u002Fs00205-003-0280-7",{"doi":607},"10.1007\u002Fs00205-003-0280-7",{"id":20,"text":609,"url":20,"identifiers":610},"Jacobsen, J.: Radial Solutions of Quasilinear Elliptic Differential Equations. Handbook of Differential Equations, pp. 359–435. Elsevier\u002FNorth-Holland, Amsterdam (2004)",{},{"id":549,"text":612,"url":551,"identifiers":613},"Jacobsen, J., Schmitt, K.: The Liouville-Bratu-Gelfand problem for radial operators. J. Differ. Equ. 184, 283–298 (2002)",{"doi":553},{"id":549,"text":615,"url":551,"identifiers":616},"Kufner, A., Opic, B.: Hardy-type Inequalities, Pitman Res. Notes in Math., vol 219. Longman Scientific and Technical, Harlow (1990)",{"doi":553},{"id":20,"text":618,"url":20,"identifiers":619},"Leckband, M.: Moser’s inequality on the ball \\(B^n\\) for functions with mean value zero. Comm. Pure Appl. Math. 58, 789–798 (2005)",{},{"id":549,"text":621,"url":551,"identifiers":622},"Lin, K.C.: Extremal functions for Moser’s inequality. Trans. Amer. Math. Soc. 348, 2663–2671 (1996)",{"doi":553},{"id":624,"text":625,"url":626,"identifiers":627},"74f5e9e8-5e92-4817-8e6c-785809de2ebc","Li, Y.: Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds. Sci. China Ser. A 48(5), 618–648 (2005)","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1360\u002F04ys0050",{"doi":628},"10.1360\u002F04ys0050",{"id":549,"text":630,"url":551,"identifiers":631},"Li, Y., Liu, P., Yang, Y.: Moser-Trudinger inequalities of vector bundle over a compact Riemannian manifold of dimension \\(2\\). Calc. Var. Partial Differ. Equ. 28, 59–83 (2007)",{"doi":553},{"id":549,"text":633,"url":551,"identifiers":634},"Lions, P.L.: The concentration-compactness principle in the calculus of variations. The limit case, Part 1. Rev. Mat. Iberoamericana 1, 145–201 (1985)",{"doi":553},{"id":549,"text":636,"url":551,"identifiers":637},"Lu, G., Yang, Y.: Sharp constant and extremal function for the improved Moser-Trudinger inequality involving \\(L^p\\) norm in two dimension. Discrete Contin. Dyn. Syst. 25, 963–979 (2009)",{"doi":553},{"id":20,"text":639,"url":20,"identifiers":640},"McLeod, J.B., Peletier, L.A.: Observations on Moser’s inequality. Arch. Ration. Mech. Anal. 106, 261–285 (1989)",{},{"id":20,"text":642,"url":20,"identifiers":643},"Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970\u002F71)",{},{"id":20,"text":645,"url":20,"identifiers":646},"Pohozaev, S.I.: The Sobolev embedding in the case \\(pl = n\\). In: Proceedings of the Technical Scientific Conference on Advances of Scientific Research 1964–1965. Mathematics Section, Moscov. Energet. Inst., pp. 158–170 (1965)",{},{"id":549,"text":648,"url":551,"identifiers":649},"Struwe, M.: Critical points of embeddings of \\(H^{1, n}_{0}\\) into Orlicz spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire 5, 425–464 (1988)",{"doi":553},{"id":651,"text":652,"url":653,"identifiers":654},"ce815e5d-472f-41ec-984e-af0b75833c3a","Struwe, M.: Positive solutions of critical semilinear elliptic equations on non-contractible planar domains. J. Eur. Math. Soc. 2, 329–388 (2000)","https:\u002F\u002Fems.press\u002Fjournals\u002Fjems\u002Farticles\u002F118",{"doi":655},"10.1007\u002Fs100970000023",{"id":549,"text":657,"url":551,"identifiers":658},"Tian, G.J., Wang, X.J.: Moser-Trudinger type inequalities for the Hessian equation. J. Funct. Anal. 259, 1974–2002 (2010)",{"doi":553},{"id":549,"text":660,"url":551,"identifiers":661},"Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51, 126–150 (1984)",{"doi":553},{"id":549,"text":663,"url":551,"identifiers":664},"Trudinger, N.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)",{"doi":553},{"id":549,"text":666,"url":551,"identifiers":667},"Yang, Y.: A sharp form of Moser-Trudinger inequality in high dimension. J. Func. Anal. 239, 100–126 (2006)",{"doi":553},{"id":20,"text":669,"url":20,"identifiers":670},"Yudovich, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations. Dok. Akad. Nauk. SSSR 138, 804–808 (1961). [English translation in Soviet Math. Doklady 2 (1961), 746–749]",{},{"id":672,"createTime":673,"updateTime":674,"relativeEntities":675,"slug":676,"properties":677,"entityType":216,"verifyStatus":217,"verifyTime":688,"verifyNote":219,"languages":20,"translateLanguages":20,"viewCount":21,"primaryUrl":689,"fullTextUrl":20,"authors":690,"publicationType":240,"publisherRelationship":742,"citationCount":21,"citationInfo":794,"publishDate":797,"publishYear":795,"citationAnalyzeStatus":19,"lastCitationAnalyze":674,"indexDatabases":798,"openAccess":20,"references":20,"isForceReanalyzing":299},"c69a3c7b-ce54-4998-813a-a846ad00587c","2023-12-23T17:29:10.271+00:00","2026-07-28T07:09:42.766+00:00",[],"Regularity-of-radial-minimizers-of-reaction-equations-involving-the-p-Laplacian",{"abstract":678,"title":680,"gsPaper":682,"references":684,"doi":686},{"EN":679},"We consider semi-stable, radially symmetric, and decreasing solutions of  − Δ\n                  p\n                \n                        u = g(u) in the unit ball of \n                  \n                    \n                  \n                  $${\\mathbb{R}^n}$$\n                 , where p > 1, Δ\n                  p\n                 is the p-Laplace operator, and g is a locally Lipschitz function. For this class of radial solutions, which includes local minimizers, we establish pointwise, L\n                        \n                  q\n                , and W\n                        1,q\n                         estimates which are optimal and do not depend on the specific nonlinearity g. Among other results, we prove that every radially decreasing and semi-stable solution u belonging to W\n                        1,p\n                        (B\n                        1) is bounded whenever n \u003C p + 4p\u002F(p − 1). Under standard assumptions on the nonlinearity g(u) = λf (u), where λ > 0 is a parameter, it is proved that the corresponding extremal solution u\n                        * is semi-stable, and hence, it enjoys the regularity stated in our main result.",{"EN":681},"Regularity of radial minimizers of reaction equations involving the p-Laplacian",{"VOID":683},"[\"3781228618362144845\"]",{"VOID":685},"Brézis H., Vázquez J.L.: Blow-up solutions of some nonlinear elliptic problems. Rev. Math. Univ. Compl. Madr. 10, 443–469 (1997)\nCabré, X.: Boundedness of minimizers of semilinear elliptic problems up to dimension four (in preparation)\nCabré, X.: Extremal solutions and instantaneous complete blow-up for elliptic and parabolic problems. In: Perspectives in Nonlinear Partial Differential Equations: in honor of Haïm Brezis. Contemp. Math., vol. 446, pp. 159–174. American Mathematical Society (2007)\nCabré X., Capella A.: On the stability of radial solutions of semilinear elliptic equations in all of \\({\\mathbb{R}^n}\\) . C. R. Math. Acad. Sci. Paris 338, 769–774 (2004)\nCabré X., Capella A.: Regularity of radial minimizers and extremal solutions of semilinear elliptic equations. J. Funct. Anal. 238, 709–733 (2006)\nCabré X., Capella A.: Regularity of minimizers for three elliptic problems: minimal cones, harmonic maps, and semilinear equations. Pure Appl. Math. Q. 3, 801–825 (2007)\nCabré X., Sanchón M.: Semi-stable and extremal solutions of reaction equations involving the p-Laplacian. Comm. Pure Appl. Anal. 6, 43–67 (2007)\nCrandall M.G., Rabinowitz P.H.: Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems. Arch. Ration. Mech. Anal. 58, 207–218 (1975)\nDamascelli L., Sciunzi B.: Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations. J. Differ. Equ. 206, 483–515 (2004)\nDávila J.: Singular solutions of semi-linear elliptic problems, preprint\nDávila J., Dupaigne L.: Perturbing singular solutions of the Gelfand problem. Commun. Contemp. Math. 9, 639–680 (2007)\nDávila, J., Dupaigne, L., Guerra, I., Montenegro, M.: Stable solutions for the bilaplacian with exponential nonlinearity, SIAM J. Math. Anal. 39 (2007), 565–592 (electronic)\nDávila, J., Dupaigne L., Montenegro, M.: The extremal solution of a boundary reaction problem, preprint\nEidelman S., Eidelman Y.: On regularity of the extremal solution of the Dirichlet problem for some semilinear elliptic equations of the second order. Houst. J. Math. 31, 957–960 (2005)\nEsposito P.: Compactness of a nonlinear eigenvalue problem with singular nonlinearity. Commun. Contemp. Math. 10, 17–45 (2008)\nFerrero A.: On the solutions of quasilinear elliptic equations with a poly- nomial–type reaction term. Adv. Differ. Equ. 9, 1201–1234 (2004)\nGarcía-Azorero J., Peral I.: On an Emden–Fowler type equation. Nonlinear Anal. 18, 1085–1097 (1992)\nGarcía-Azorero J., Peral I., Puel J.P.: Quasilinear problems with exponential growth in the reaction term. Nonlinear Anal. 22, 481–498 (1994)\nGhoussoub N., Guo Y.: On the partial differential equations of electrostatic MEMS devices: stationary case. SIAM J. Math. Anal. 38, 1423–1449 (2007)\nJoseph D.D., Lundgren T.S.: Quasilinear Dirichlet problems driven by positive sources. Arch. Ration. Mech. Anal. 49, 241–269 (1973)\nLieberman G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 11, 1203–1219 (1988)\nMignot F., Puel J.P.: Sur une classe de problèmes non linéaires avec nonlinéarité positive, croissante, convexe. Comm. Partial Differ. Equ. 5, 791–836 (1980)\nNedev G.: Regularity of the extremal solution of semilinear elliptic equations. C. R. Acad. Sci. Paris Sér. I Math. 330, 997–1002 (2000)\nPeral, I.: Multiplicity of solutions for the p-Laplacian. International Center for Theoretical Physics Lecture Notes, Trieste (1997)\nSakaguchi S.: Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 14, 403–421 (1987)\nSanchón M.: Boundedness of the extremal solution of some p-Laplacian problems. Nonlinear Anal. 67, 281–294 (2007)\nSanchón M.: Existence and regularity of the extremal solution of some nonlinear elliptic problems related to the p-Laplacian. Potential Anal. 27, 217–224 (2007)\nVillegas S.: Asymptotic behavior of stable radial solutions of semilinear elliptic equations in \\({\\mathbb{R}^n}\\) . J. Math. Pures et Appl. 88, 241–250 (2007)\nVillegas, S.: Sharp estimates for semi-stable radial solutions of semilinear elliptic equations, preprint",{"VOID":687},"10.1007\u002Fs00526-008-0192-3","2024-08-31T00:32:12.391+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs00526-008-0192-3",[691,708,725],{"id":692,"sortIndex":21,"researcher":20,"roles":693,"affiliations":694,"properties":703,"displayName":705,"givenName":20,"familyName":20},"ce4b38bd-4930-4af6-a652-ab9611587996",[225],[695],{"id":696,"sortIndex":21,"affiliation":697,"properties":20},"349a7515-c423-41fb-bf0f-d66898533cca",{"id":696,"createTime":20,"updateTime":20,"relativeEntities":698,"slug":20,"properties":699,"entityType":20,"verifyStatus":20,"verifyTime":20,"verifyNote":20,"languages":20,"translateLanguages":20,"viewCount":20,"url":20,"parentIds":702,"statistic":20},[],{"title":700},{"VI":701},"Departament de Matemàtica Aplicada I, ICREA and Universitat Politècnica de Catalunya, Barcelona, 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the spatial generalized N-centre problem \n                  \n                    \n                  \n                  $$\\begin{aligned} \\ddot{x} = -\\sum _{i=1}^{N} \\frac{m_i (x - c_i)}{\\vert x - c_i \\vert ^{\\alpha +2}},\\quad x \\in \\mathbb {R}^3 {\\setminus } \\{c_1,\\ldots ,c_N \\}, \\end{aligned}$$\n                  \n                    \n                  \n                where \n                  \n                    \n                  \n                  $$m_i > 0$$\n                  \n                    \n                  \n                 and \n                  \n                    \n                  \n                  $$\\alpha \\in [1,2)$$\n                  \n                    \n                  \n                , we prove the existence of positive energy entire solutions with prescribed scattering angle. The proof relies on variational arguments, within an approximation procedure via (free-time) boundary value problems. A self-contained appendix describing a general strategy to rule out the occurrence of collisions is also included.",{"EN":809},"The spatial N-centre problem: scattering at positive energies",{"VOID":811},"[\"15133220590061960063\"]",{"EN":813},"",{"VOID":815},"Ambrosetti, A., Coti Zelati, V.: Periodic Solutions of Singular Lagrangian Systems, Progress in Nonlinear Differential Equations and their Applications, vol. 10. Birkhäuser Boston, Inc., Boston (1993)\nBahri, A., Rabinowitz, P.H.: A minimax method for a class of Hamiltonian systems with singular potentials. J. Funct. Anal. 82, 412–428 (1989)\nBolotin, S.V.: Nonintegrability of the problem of \\(n\\) centers for \\(n>2\\) (Russian). Vestnik Moskov Univ. Ser. I Mat. Mekh. 3, 65–68 (1984)\nBolotin, S.V., Kozlov, V.V.: Topological approach to the generalized n-centre problem (Russian). Uspekhi Mat. Nauk 72, 65–96 (2017). (English version on arXiv:1705.04671)\nBolotin, S.V., Negrini, P.: Chaotic behavior in the 3-center problem. J. Differ. Equ. 190, 539–558 (2003)\nBolotin, S.V., Negrini, P.: Regularization and topological entropy for the spatial \\(n\\)-center problem. Ergod. Theory Dyn. Syst. 21, 383–399 (2001)\nBoscaggin, A., Dambrosio, W., Papini, D.: Parabolic solutions for the planar \\(N\\)-centre problem: multiplicity and scattering. Ann. Mat. Pura Appl. 197(3), 869–882 (2018)\nBoscaggin, A., Dambrosio, W., Terracini, S.: Scattering parabolic solutions for the spatial \\(N\\)-centre problem. Arch. Ration. Mech. Anal. 223, 1269–1306 (2017)\nBoscaggin, A., Ortega, R., Zhao, L.: Periodic solutions and regularization of a Kepler problem with time-dependent perturbation. Trans. Am. Math. Soc. (to appear). Preprint available online at http:\u002F\u002Fwww.ugr.es\u002F~ecuadif\u002Ffiles\u002FBosOrtZha.pdf\nDimare, L.: Chaotic quasi-collision trajectories in the 3-centre problem. Celest. Mech. Dyn. Astron. 107, 427–449 (2010)\nFelmer, P., Tanaka, K.: Hyperbolic-like solutions for singular Hamiltonian systems. NoDEA Nonlinear Differ. Equ. Appl. 7, 43–65 (2000)\nGranas, A., Dugundji, J.: Fixed Point Theory. Springer Monographs in Mathematics. Springer, New York (2003)\nKlein, M., Knauf, A.: Classical Planar Scattering by Coulombic Potentials. Springer, Berlin (1992)\nKnauf, A.: The \\(n\\)-centre problem of celestial mechanics for large energies. J. Eur. Math. Soc. (JEMS) 4, 1–114 (2002)\nPinzari, G.: An analysis of the Sun–Earth–Asteroid systems based on the two-centre problem. preprint (2017). arXiv:1702.03680.pdf\nSoave, N., Terracini, S.: Symbolic dynamics for the \\(N\\)-centre problem at negative energies. Discrete Contin. Dyn. Syst. 32, 3245–3301 (2012)\nSoave, N., Terracini, S.: Avoiding collisions under topological constraints in variational problems coming from celestial mechanics. J. Fixed Point Theory Appl. 14, 457–501 (2013)\nSperling, H.J.: The collision singularity in a perturbed two-body problem. Celest. Mech. 1, 213–221 (1969\u002F1970)\nTanaka, K.: A note on generalized solutions of singular Hamiltonian systems. Proc. Am. Math. Soc. 122, 275–284 (1994)\nTanaka, K.: A prescribed energy problem for a singular Hamiltonian system with a weak force. J. Funct. Anal. 113, 351–390 (1993)\nTanaka, K.: Noncollision solutions for a second order singular Hamiltonian system with weak force. Ann. Inst. H. Poincaré Anal. Non Linéaire 10, 215–238 (1993)\nWaldvogel, J.: Quaternions for regularizing celestial mechanics: the right way. Celest. Mech. Dyn. Astron. 102, 149–162 (2008)\nWhittaker, E.T.: A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: With an Introduction to the Problem of Three Bodies, 4th edn. Cambridge University Press, New York (1959)\nYu, G.: Periodic solutions of the planar \\(N\\)-center problem with topological constraints. Discrete Contin. Dyn. 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We consider the problem \n                \n                  \n \\[ \\left \\{ \\begin{array}{rcl} \\varepsilon^2 \\Delta u - u + f(u) = 0 & \\mbox{ in }& \\Omega u > 0 \\mbox{ in} \\Omega, u = 0 & \\mbox{ on }& \\partial\\Omega, \\end{array} \\right.\\nonumber \\] \n                \n               where \n                  \n$\\Omega$\n                 is a smooth domain in \n                  \n$R^N$\n                , not necessarily bounded, \n                  \n$\\varepsilon >0$\n                 is a small parameter and f is a superlinear, subcritical nonlinearity. It is known that this equation possesses a solution that concentrates, as \n                  \n$\\varepsilon$\n                 approaches zero, at a maximum of the function \n                  \n$d(x)=d(\\cdot,\\partial\\Omega)$\n                , the distance to the boundary. We obtain multi-peak solutions of the equation given above when the domain \n                  \n$\\Omega$\n                 presents a distance function to its boundary d with multiple local maxima. We find solutions exhibiting concentration at any prescribed finite set of local maxima, possibly degenerate, of d. The proof relies on variational arguments, where a penalization-type method is used together with sharp estimates of the critical values of the appropriate functional. Our main theorem extends earlier results, including the single peak case. We allow a degenerate distance function and a more general nonlinearity.",{"EN":928},"Multi-peak solutions for some singular perturbation problems",{"VOID":930},"[\"4713741020370640407\"]",{"VOID":932},"10.1007\u002Fs005260050147","2024-06-25T04:00:53.856+00:00","http:\u002F\u002Flink.springer.com\u002F10.1007\u002Fs005260050147",[936,953,968],{"id":937,"sortIndex":21,"researcher":20,"roles":938,"affiliations":939,"properties":948,"displayName":950,"givenName":20,"familyName":20},"6843b07f-3ae3-4f99-9f73-ec83715be2b5",[225],[940],{"id":941,"sortIndex":21,"affiliation":942,"properties":20},"17b382c7-7363-422b-9560-f75291e1293e",{"id":941,"createTime":20,"updateTime":20,"relativeEntities":943,"slug":20,"properties":944,"entityType":20,"verifyStatus":20,"verifyTime":20,"verifyNote":20,"languages":20,"translateLanguages":20,"viewCount":20,"url":20,"parentIds":947,"statistic":20},[],{"title":945},{"VI":946},"Departamento de Ingeniería Matemática F.C.F.M., Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile (e-mail: delpino@dim.uchile.cl \u002F pfelmer@dim.uchile.cl)\n, , CL",[],{"title":949,"gsAuthor":951},{"VI":950},"Manuel del Pino",{"VOID":952},"[\"uUUiCyEAAAAJ\"]",{"id":954,"sortIndex":161,"researcher":20,"roles":955,"affiliations":956,"properties":963,"displayName":965,"givenName":20,"familyName":20},"e330ec4d-a82a-44c0-bf5b-a372d811c12f",[225],[957],{"id":941,"sortIndex":21,"affiliation":958,"properties":20},{"id":941,"createTime":20,"updateTime":20,"relativeEntities":959,"slug":20,"properties":960,"entityType":20,"verifyStatus":20,"verifyTime":20,"verifyNote":20,"languages":20,"translateLanguages":20,"viewCount":20,"url":20,"parentIds":962,"statistic":20},[],{"title":961},{"VI":946},[],{"title":964,"gsAuthor":966},{"VI":965},"Patricio L. 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establish a Rademacher type theorem involving Hamiltonians H(x, p) under very weak conditions in both of Euclidean and Carnot-Carathéodory spaces. In particular, H(x, p) is assumed to be only measurable in the variable x, and to be quasiconvex and lower-semicontinuous in the variable p. Without the lower-semicontinuity in the variable p, we provide a counter example showing the failure of such a Rademacher type theorem. Moreover, by applying such a Rademacher type theorem we build up an existence result of absolute minimizers for the corresponding \n                \n                  \n                \n                $$L^\\infty $$\n                \n              -functional. These improve or extend several known results in the literature.",{"EN":1049},"A Rademacher type theorem for Hamiltonians H(x, p) and an application to absolute minimizers",{"VOID":1051},"[\"8696738879797810669\"]",{"VOID":1053},"Aronsson, G.: Minimization problems for the functional \\(\\sup _x F(x, f(x), f^{\\prime } (x))\\). Ark. Mat. 6, 33–53 (1965)\nAronsson, G.: Minimization problems for the functional \\(\\sup _x F(x, f(x), f^{\\prime } (x))\\). II. Ark. Mat. 6, 409–431 (1966)\nAronsson, G.: Extension of functions satisfying Lipschitz conditions. Ark. Mat. 6, 551–561 (1967)\nG. Aronsson, On the partial differential equation \\(u^2_x u_{xx} + 2u_x u_y u_{xy} + u^2_y u_{yy} = 0.\\) Ark. Mat. 7, (1968), 395-425\nAronsson, G.: Minimization problems for the functional \\(\\sup _x F(x, f(x), f^{\\prime } (x))\\). III. Ark. Mat. 7, 509–512 (1969)\nArmstrong, S.N., Crandall, M.G., Julin, V., Smart, C.K.: Convexity criteria and uniqueness of absolutely minimizing functions. Arch. Ration. Mech. Anal. 200(2), 405–443 (2011)\nAronsson, G., Crandall, M.G., Juutinen, P.: A tour of the theory of absolutely minimizing functions. Bull. Am. Math. Soc. (N.S.) 41, 439–505 (2004)\nN. Barron, Viscosity solutions and analysis in \\(L^\\infty \\). In: Nonlinear Analysis, Differential Equations and Control (Montreal, QC, 1998). NATO Sci. Ser. C Math. Phys. Sci. 528. Dordrecht: Kluwer Acad. Publ., pp. 1-60.(1999)\nBarron, E.N., Jensen, R.R., Wang, C.Y.: The Euler equation and absolute minimizers of \\(L^\\infty \\) functionals. Arch. Ration. Mech. Anal. 157, 255–283 (2001)\nBieske, T.: On \\(\\infty \\)-harmonic functions on the Heisenberg group. Comm. Partial Diff. Eq. 27(3–4), 727–761 (2002)\nBoutet de Monvel, A., Lenz, D., Stollmann, P.: Schnol’s theorem for strongly local forms. Israel J. Math. 173, 189–211 (2009)\nChampion, T., De Pascale, L., Prinari, F.: \\(\\Gamma \\)-Convergence and absolute minimizers for supremal functionals. ESAIM Control Optim. Calc. Var. 10, 14–27 (2004)\nV. M. Chernikov, S. K. Vodop’yanov, Sobolev Spaces and hypoelliptic equations I,II. Siberian Advances in Mathematics. 6 (1996) no. 3, 27-67, no. 4, 64-96. Translation from: Trudy In-ta matematiki RAN. Sib. otd-nie. 29 (1995), 7-62\nChampion, T., De Pascale, L.: Principles of comparison with distance functions for absolute minimizers. J. Convex Anal. 14, 515–541 (2007)\nCrandall, M.: An efficient derivation of the Aronsson equation. Arch. Ration. Mech. Anal. 167, 271–279 (2003)\nDavini, A.: Smooth approximation of weak Finsler metrics. Diff. Integr. Eq. 18(5), 509–530 (2005)\nDragoni, F., Manfredi, J.J., Vittone, D.: Weak Fubini property and infinity harmonic functions in Riemannian and sub-Riemannian manifolds. Trans. Am. Math. Soc. 365(2), 837–859 (2013)\nFriedrichs, K.O.: The identity of weak and strong extensions of differential operators. Trans. Am. Math. Soc. 55, 132–151 (1944)\nFranchi, B., Hajłasz, P., Koskela, P.: Definitions of Sobolev classes on metric spaces. Annales de l’Institut Fourier 49(6), 1903–1924 (1999)\nFranchi, B., Serapioni, R., Serra Cassano, F.: Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields. Houston J. Math. 22, 859–890 (1996)\nFranchi, B., Serapioni, R., Serra Cassano, F.: Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields. Boll. Un. Mat. Ital. 7, 83–117 (1997)\nFrank, R.L., Lenz, D., Wingert, D.: Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory. J. Funct. Anal. 266(8), 4765–4808 (2014)\nGarofalo, N., Nhieu, D.M.: Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces. Commun. Pure Appl. Math. 49, 1081–1144 (1996)\nGarofalo, N., Nhieu, D.: Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces. J. Anal. Math. 74, 67–97 (1998)\nGariepy, R., Wang, C.Y., Yu, Y.: Generalized cone comparison principle for viscosity solutions of the Aronsson equation and absolute minimizers. Commun. Partial Diff. Eq. 31, 1027–1046 (2006)\nGuo, C.Y., Xiang, C., Yang, D.: \\(L^\\infty \\)-variational problems associated to measurable Finsler structures. Nonlinear Anal. 132, 126–140 (2015)\nHajlasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145, 688 (2000)\nJ. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson, Sobolev spaces on metric measure spaces: an approach based on upper gradients, Cambridge Studies in Advanced Mathematics Series, Cambridge University Press, 2015\nJensen, R.: Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch. Ration. Mech. Anal. 123, 51–74 (1993)\nJensen, R., Wang, C.Y., Yu, Y.: Uniqueness and nonuniqueness of viscosity solutions to Aronsson’s equation. Arch. Ration. Mech. Anal. 190(2), 347–370 (2008)\nJerison, D.: The Poincaré inequality for vector fields satisfying Hörmander’s condition. Duke Math. J. 53, 503–523 (1986)\nD. Jerison, A. Sanchez-Calle, Subelliptic, second order differential operators. In: Complex analysis, III (College Park, Md., 1985-86). pp. 46-77, Lecture Notes in Math., 1277, Springer, 1987\nJuutinen, P.: Minimization problems for Lipschitz functions via viscosity solutions. Ann. Acad. Sci. Fenn. Math. Diss. No. 115, 53 (1998)\nJuutinen, P.: Absolutely minimizing Lipschitz extensions on a metric space. Ann. Acad. Sci. Fenn. Math. 27(1), 57–67 (2002)\nJuutinen, P., Shanmugalingam, N.: Equivalence of AMLE, strong AMLE, and comparison with cones in metric measure space. Math. Nachr. 279, 1083–1098 (2006)\nKoskela, P., Shanmugalingam, N., Zhou, Y.: \\(L^\\infty \\)-Variational problem associated to Dirichlet forms. Math. Res. Lett. 19, 1263–1275 (2012)\nKoskela, P., Zhou, Y.: Geometry and analysis of Dirichlet forms. Adv. Math. 231, 2755–2801 (2012)\nKoskela, P., Shanmugalingam, N., Zhou, Y.: Intrinsic geometry and analysis of diffusion process and \\(L^\\infty \\)-variational problem. Arch. Rational Mech. Anal. 214(1), 99–142 (2014)\nLe Donne, E., Speight, G.: Lusin approximation for horizontal curves in step 2 carnot groups. Calculus Var. Partial Diff. Eq. 55, 1–22 (2016)\nMonti, R., Cassano, F.S.: Surface measures in Carnot- Caratheodory spaces. Calc. Var. Partial Diff. Eq. 13, 339–376 (2001)\nNagel, A., Stein, E.M., S, Wainger,: Balls andmetrics defined by vectorfields L basic properties. Acta Math. 155, 103–147 (1985)\nPansu, P.: Metriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Annals Math. 129, 1–60 (1989)\nSpeight Lusin, G.: Approximation and Horizontal Curves in Carnot Groups. Revista Matematica Iberoamericana 32, 1425–1446 (2016)\nStollmann, P.: A dual characterization of length spaces with application to Dirichlet metric spaces. Stud. Math. 198, 221–233 (2010)\nSturm, K.T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and \\(L^p\\)-Liouville properties. J. Rein. Angew. Math. 456, 173–196 (1994)\nSturm, K.T.: Is a diffusion process determined by its intrinsic metric? Chaos Solitons Fractals 8, 1855–1860 (1997)\nWang, C.Y.: The Aronsson equation for absolute minimizers of \\(L^\\infty \\) functionals associated with vector fields satisfying Hörmander’s conditions. Trans. AMS. 359, 91–113 (2007)",{"VOID":1055},"10.1007\u002Fs00526-023-02484-9","2024-05-27T10:45:01.162+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs00526-023-02484-9",[1059,1083],{"id":1060,"sortIndex":21,"researcher":20,"roles":1061,"affiliations":1062,"properties":1080,"displayName":1082,"givenName":20,"familyName":20},"05a6169e-da3a-446a-b589-3c8b15cc9532",[225],[1063,1071],{"id":1064,"sortIndex":21,"affiliation":1065,"properties":20},"bbe0246e-594f-4c9e-be39-25ad9e561ca2",{"id":1064,"createTime":20,"updateTime":20,"relativeEntities":1066,"slug":20,"properties":1067,"entityType":20,"verifyStatus":20,"verifyTime":20,"verifyNote":20,"languages":20,"translateLanguages":20,"viewCount":20,"url":20,"parentIds":1070,"statistic":20},[],{"title":1068},{"VI":1069},"School of Mathematical Science, Beihang University, Beijing, People’s Republic of 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this article we study the validity of the Whitney \n                  \n                    \n                  \n                  $$C^1$$\n                  \n                    \n                  \n                 extension property for horizontal curves in sub-Riemannian manifolds that satisfy a first-order Taylor expansion compatibility condition. We first consider the equiregular case, where we show that the extension property holds true whenever a suitable non-singularity property holds for the endpoint map on the Carnot groups obtained by nilpotent approximation. We then discuss the case of sub-Riemannian manifolds with singular points and we show that all step-2 manifolds satisfy the \n                  \n                    \n                  \n                  $$C^1$$\n                  \n                    \n                  \n                 extension property. We conclude by showing that the \n                  \n                    \n                  \n                  $$C^1$$\n                  \n                    \n                  \n                 extension property implies a Lusin-like approximation theorem for horizontal curves on sub-Riemannian manifolds.",{"EN":1164},"On the Whitney extension property for continuously differentiable horizontal curves in sub-Riemannian manifolds",{"VOID":1166},"[]",{"VOID":1168},"Agrachev, A.A., Barilari, D., Boscain, U.: Introduction to Riemannian and Sub-Riemannian geometry. Preprint SISSA (2016)\nAgrachev, A.A., Boarotto, F., Lerario, A.: Homotopically invisible singular curves. Calc. Var. Partial Differ. Equ. 56(4), 105 (2017)\nAgrachev, A.A., Gamkrelidze, R.V.: Exponential representation of flows and a chronological enumeration. Mat. Sb. (N.S.) 107(149), 467–532 (1978)\nAgrachëv, A.A., Sarychev, A.V.: Filtrations of a Lie algebra of vector fields and the nilpotent approximation of controllable systems. Dokl. Akad. Nauk SSSR 295(4), 777–781 (1987)\nAgrachev, A.A., Sarychev, A.V.: Abnormal sub-Riemannian geodesics: morse index and rigidity. Ann. Inst. Henri Poincaré Anal. Non Linéaire 13(6), 635–690 (1996)\nAgrachev, A.A., Sachkov, Y.L.: Control Theory From the Geometric Viewpoint, Volume 87 of Encyclopaedia of Mathematical Sciences, vol. 87. Springer, Berlin (2004). ( Control Theory and Optimization, II)\nBarilari, D., Boscain, U., Sigalotti, M. (ed.): Geometry, analysis and dynamics on sub-Riemannian manifolds. EMS Series of Lectures in Mathematics, Vol. 1 & 2. Lecture notes from the IHP Trimester held at the Institut Henri Poincaré, Paris and from the CIRM Summer School “Sub-Riemannian Manifolds: From Geodesics to Hypoelliptic Diffusion” held in Luminy, Fall (2014). European Mathematical Society (EMS), Zürich (2016)\nBoscain, U., Charlot, G., Ghezzi, R., Sigalotti, M.: Lipschitz classification of almost-Riemannian distances on compact oriented surfaces. J. Geom. Anal. 23(1), 438–455 (2013)\nBellaïche A. (1996) The tangent space in sub-Riemannian geometry. In: Bellaïche A., Risler JJ. (eds.) Sub-Riemannian Geometry. Progress in Mathematics, vol. 144, Birkhäuser Basel\nBryant, R.L., Hsu, L.: Rigidity of integral curves of rank \\(2\\) distributions. Invent. Math. 114(2), 435–461 (1993)\nBonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie Groups and Potential Theory for Their Sub-Laplacians. Springer Monographs in Mathematics. Springer, Berlin (2007)\nBianchini, R.M., Stefani, G.: Graded approximations and controllability along a trajectory. SIAM J. Control Optim. 28(4), 903–924 (1990)\nFranchi, B., Serapioni, R., Serra Cassano, F.: Rectifiability and perimeter in the Heisenberg group. Math. Ann. 321(3), 479–531 (2001)\nGrušin, V.V.: A certain class of hypoelliptic operators. Mat. Sb. (N.S.) 83(125), 456–473 (1970)\nJean, F.: Control of Nonholonomic Systems: From Sub-Riemannian Geometry to Motion Planning. SpringerBriefs in Mathematics. Springer, Cham (2014)\nJuillet, N., Sigalotti, M.: Pliability, or the Whitney extension theorem for curves in Carnot groups. Anal. PDE 10, 1637–1661 (2017)\nLe Donne, E., Speight, G.: Lusin approximation for horizontal curves in step 2 Carnot groups. Calc. Var. Partial Differ. Equ. 55(5), 111 (2016)\nRifford, L.: Sub-Riemannian Geometry and Optimal Transport. SpringerBriefs in Mathematics. Springer, Cham (2014)\nSerra Cassano, F.: Some topics of geometric measure theory in carnot groups. In: Barilari, D., Boscain, U., Sigalotti, M. (eds.) Geometry, Analysis and Dynamics on Sub-Riemannian Manifolds. EMS Series of Lectures in Mathematics, Vol. 1, pp. vi+324. European Mathematical Society (EMS), Zürich (2016)\nSpeight, G.: Lusin approximation and horizontal curves in Carnot groups. Rev. Mat. Iberoam. 32(4), 1423–1444 (2016)\nSussmann, H.J.: Some properties of vector field systems that are not altered by small perturbations. J. Differ. Equ. 20(2), 292–315 (1976)\nTrélat, E.: Some properties of the value function and its level sets for affine control systems with quadratic cost. J. Dyn. Control Syst. 6(4), 511–541 (2000)\nTrélat, E.: Contrôle optimal. Mathématiques Concrètes. [Concrete Mathematics]. Vuibert, Paris. Théorie & applications. [Theory and applications] (2005)\nVodopyanov, S.K.: Differentiability of curves in the category of Carnot manifolds. Dokl. Math. 74(2), 686–691 (2006)\nVodopyanov, S.K., Pupyshev, I.M.: Whitney-type theorems on the extension of functions on Carnot groups. Sib. Mat. J. 47(4), 731–752 (2006)\nZimmerman, S.: The Whitney extension theorem for \\({C}^1\\), horizontal curves in the Heisenberg group. J. Geom. Anal. 28, 61–83 (2018). https:\u002F\u002Fdoi.org\u002F10.1007\u002Fs12220-017-9807-2",{"VOID":1170},"10.1007\u002Fs00526-018-1336-8","2024-06-26T09:32:57.283+00:00","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002Fs00526-018-1336-8",[1174,1189],{"id":1175,"sortIndex":21,"researcher":20,"roles":1176,"affiliations":1177,"properties":1186,"displayName":1188,"givenName":20,"familyName":20},"bd70a4da-147b-4d96-8418-725848e39720",[225],[1178],{"id":1179,"sortIndex":21,"affiliation":1180,"properties":20},"51f41453-6c38-4133-8459-7edc68d8f9db",{"id":1179,"createTime":20,"updateTime":20,"relativeEntities":1181,"slug":20,"properties":1182,"entityType":20,"verifyStatus":20,"verifyTime":20,"verifyNote":20,"languages":20,"translateLanguages":20,"viewCount":20,"url":20,"parentIds":1185,"statistic":20},[],{"title":1183},{"VI":1184},"CMAP, École Polytechnique, CNRS, Inria, Université Paris-Saclay, Palaiseau, France",[],{"title":1187},{"VI":1188},"Ludovic 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S.V.: The effect of singularities of the potential energy on the integrability of mechanical systems. Prikl. Mat. Mekh. 48(3), 356–362 (1984)","https:\u002F\u002Fwww.sciencedirect.com\u002Fscience\u002Farticle\u002Fpii\u002F002189288490128X",{"doi":1376},"10.1016\u002F0021-8928(84)90128-x",{"id":20,"text":1378,"url":20,"identifiers":1379},"Bolotin, S.V.: Nonintegrability of the problem of \\(n\\) centers for \\(n>2\\). Vestnik Moskov. Univ. Ser. I Mat. Mekh. 3, 65–68 (1984)",{},{"id":20,"text":1381,"url":20,"identifiers":1382},"Bolotin, S.V.: Homoclinic orbits of geodesic flows on surfaces. Russ. J. Math. Phys. 1(3), 275–288 (1993)",{},{"id":20,"text":1384,"url":20,"identifiers":1385},"Bolotin, S.V., Kozlov, V.V.: Topological approach to the generalized \\(n\\)-centre problem. Uspekhi Mat. Nauk 72(3(435)), 65–96 (2017)",{},{"id":549,"text":1387,"url":551,"identifiers":1388},"Bolotin, S.V., Kozlov, V.V.: Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom. Izv. Ross. Akad. Nauk Ser. 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Syst. 38(2), 566–582 (2018)",{"doi":553},{"id":20,"text":1402,"url":1403,"identifiers":1404},"Contreras, G., Iturriaga, R.: Global minimizers of autonomous lagrangians. https:\u002F\u002Fwww.cimat.mx\u002F~gonzalo\u002Flibro\u002Flagrangians.pdf (2000)","https:\u002F\u002Fwww.cimat.mx\u002F~gonzalo\u002Flibro\u002Flagrangians.pdf",{},{"id":549,"text":1406,"url":551,"identifiers":1407},"da Luz, A., Maderna, E.: On the free time minimizers of the Newtonian \\(N\\)-body problem. Math. Proc. Cambrid. Philos. Soc. 156(2), 209–227 (2014)",{"doi":553},{"id":20,"text":1409,"url":1410,"identifiers":1411},"Fathi, A.: Weak kam theorem in lagrangian dynamics preliminary version 10. https:\u002F\u002Fwww.math.u-bordeaux.fr\u002F~pthieull\u002FRecherche\u002FKamFaible\u002FPublications\u002FFathi2008_01.pdf (2008)","https:\u002F\u002Fwww.math.u-bordeaux.fr\u002F~pthieull\u002FRecherche\u002FKamFaible\u002FPublications\u002FFathi2008_01.pdf",{},{"id":549,"text":1413,"url":551,"identifiers":1414},"Gordon, W.B.: A minimizing property of Keplerian orbits. Am. J. Math. 99(5), 961–971 (1977)",{"doi":553},{"id":549,"text":1416,"url":551,"identifiers":1417},"Klein, M., Knauf, A.: Chaotic motion in Coulombic potentials. Mathematical Physics, X (Leipzig, 1991), pp. 308–312. Springer, Berlin (1992)",{"doi":553},{"id":1419,"text":1420,"url":1421,"identifiers":1422},"cb3f5025-c54a-41f7-8079-3357a4b7589c","Knauf, A.: Ergodic and topological properties of Coulombic periodic potentials. Commun. Math. Phys. 110(1), 89–112 (1987)","https:\u002F\u002Flink.springer.com\u002Farticle\u002F10.1007\u002FBF01209018",{"doi":1423},"10.1007\u002FBF01209018",{"id":1425,"text":1426,"url":1427,"identifiers":1428},"d808f782-fe9e-4e49-9486-8f622045ea90","Knauf, A.: The \\(n\\)-centre problem of celestial mechanics for large energies. J. Eur. Math. Soc. (JEMS) 4(1), 1–114 (2002)","https:\u002F\u002Fems.press\u002Fdoi\u002F10.1007\u002Fs100970100037",{"doi":1429},"10.1007\u002Fs100970100037",{"id":549,"text":1431,"url":551,"identifiers":1432},"Knauf, A., Taimanov, I.A.: Integrability of the \\(n\\)-center problem at high energies. Dokl. Akad. Nauk 397(1), 20–22 (2004)",{"doi":553},{"id":1434,"text":1435,"url":1436,"identifiers":1437},"6c3149d9-cc6f-4e8b-be51-963c67118ee7","Mañé, R.: Lagrangian flows: the dynamics of globally minimizing orbits. Bol. Soc. Brasil. Mat. 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Anal. 213(3), 981–991 (2014)",{"doi":553},{"id":549,"text":1452,"url":551,"identifiers":1453},"Rabinowitz, P.H.: Heteroclinics for a reversible Hamiltonian system. Ergod. Theory Dyn. Syst. 14(4), 817–829 (1994)",{"doi":553},{"id":549,"text":1455,"url":551,"identifiers":1456},"Rabinowitz, P.H.: Connecting orbits for a reversible Hamiltonian system. Ergod. Theory Dyn. Syst. 20(6), 1767–1784 (2000)",{"doi":553},{"id":549,"text":1458,"url":551,"identifiers":1459},"Soave, N., Terracini, S.: Symbolic dynamics for the \\(N\\)-centre problem at negative energies. Discret. Contin. Dyn. Syst. 32(9), 3245–3301 (2012)",{"doi":553},{"id":549,"text":1461,"url":551,"identifiers":1462},"Yu, G.: Periodic solutions of the planar \\(N\\)-center problem with topological constraints. Discret. Contin. Dyn. 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We study the nonlinear singular integral equation\n                  \n                    \n                  \n                  $$ M[u](x) = f_0(x)\\quad {\\rm in}\\,\\Omega$$\n                with the boundary condition u = g\n                        0 on ∂Ω, where \n                  \n                    \n                  \n                  $${f_0\\in C(\\overline\\Omega)}$$\n                 and \n                  \n                    \n                  \n                  $${g_0\\in C(\\partial\\Omega)}$$\n                 are given functions and M is the singular integral operator given by\n                  \n                    \n                  \n                  $$M[u](x)={\\rm p.v.} \\int\\limits_{B(0,\\rho(x))} \\frac{p-\\sigma}{|z|^{n+\\sigma}}|u(x+z)-u(x)|^{p-2} (u(x+z)-u(x))\\,{\\rm dz},$$\n                with some choice of \n                  \n                    \n                  \n                  $${\\rho\\in C(\\overline\\Omega)}$$\n                 having the property, 0 \u003C ρ(x) ≤ dist (x, ∂Ω). We establish the solvability (well-posedness) of this Dirichlet problem and the convergence uniform on \n                  \n                    \n                  \n                  $${\\overline\\Omega}$$\n                , as σ → p, of the solution u\n                        \n                  σ\n                 of the Dirichlet problem to the solution u of the Dirichlet problem for the p-Laplace equation νΔ\n                  p\n                \n                        u = f\n                        0 in Ω with the Dirichlet condition u = g\n                        0 on ∂Ω, where the factor ν is a positive constant (see (7.2)).",{"EN":1473},"A class of integral equations and approximation of p-Laplace equations",{"VOID":1475},"[\"1590521608562041355\"]",{"VOID":1477},"Andreu F., Mazón J.M., Rossi J.D., Toledo J.: A nonlocal p-Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions. SIAM J. Math. Anal. 40(5), 1815–1851 (2008)\nAndreu F., Mazón J.M., Rossi J.D., Toledo J.: A nonlocal p-Laplacian evolution equation with Neumann boundary conditions. J. Math. Pures Appl. 90, 201–227 (2008)\nBarles G., Chasseigne E., Imbert C.: Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(3), 567–585 (2008)\nBarles G., Imbert C.: Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(3), 567–585 (2008)\nCrandall M.G., Ishii H., Lions P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)\nCaffarelli L., Silvestre L.: Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math. 62(5), 597–728 (2009)\nDiBenedetto E.: C 1+α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7(8), 827–850 (1983)\nDuistermaat, J.J., Kolk, J.A.C.: Multidimensional real analysis. II. Integration, Cambridge Studies in Advanced Mathematics, 87. Cambridge University Press, Cambridge (2004)\nForcadel N., Imbert C., Monneau R.: Homogenization of the dislocation dynamics and of some particle systems with two-body interactions. Discrete Contin. Dyn. Syst. 23(3), 785–826 (2009)\nIshii, H., Matsumura, H.: Non-local Hamilton–Jacobi equations arising in dislocation dynamics, to appear in Z. Anal. Anwendungen\nIshii H., Souganidis P.E.: Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor. Tohoku Math. J. (2) 47(2), 227–250 (1995)\nJuutinen P., Lindqvist P., Manfredi J.J.: On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation. SIAM J. Math. Anal. 33(3), 699–717 (2001)\nLewis J.L.: Regularity of the derivatives of solutions to certain degenerate elliptic equations. Indiana Univ. 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