American Institute of Mathematical Sciences

Advanced
March  2021, 20(3): 1025-1038. doi: 10.3934/cpaa.2021004

An overdetermined problem associated to the Finsler Laplacian

Giulio Ciraolo 1,, and  Antonio Greco 2,

1. 

Department of Mathematics "Federigo Enriques", Università degli Studi di Milano, Italy
2. 

Department of Mathematics and Computer Science, Università degli Studi di Cagliari, Italy

* Corresponding author

Received  October 2020 Revised  November 2020 Published  January 2021

Fund Project: G. Ciraolo has been partially supported by the PRIN 2017 project "Qualitative and quantitative aspects of nonlinear PDEs" and by GNAMPA of INdAM. A. Greco has been partially supported by the research project Integro-differential Equations and Non-Local Problems, funded by Fondazione di Sardegna (2017)

We prove a rigidity result for the anisotropic Laplacian. More precisely, the domain of the problem is bounded by an unknown surface supporting a Dirichlet condition together with a Neumann-type condition which is not translation-invariant. Using a comparison argument, we show that the domain is in fact a Wulff shape. We also consider the more general case when the unknown surface is required to have its boundary on a given conical surface: in such a case, the domain of the problem is bounded by the unknown surface and by a portion of the given conical surface, which supports a homogeneous Neumann condition. We prove that the unknown surface lies on the boundary of a Wulff shape.

Keywords: Overdetermined problem, Finsler Laplacian, Dual norm, Conical domain, Comparison principle.
Mathematics Subject Classification: Primary: 35N25, 35B06; Secondary: 35R35.
Citation: Giulio Ciraolo, Antonio Greco. An overdetermined problem associated to the Finsler Laplacian. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1025-1038. doi: 10.3934/cpaa.2021004
References:
[1]

C. Bianchini and G. Ciraolo, Wulff shape characterizations in overdetermined anisotropic elliptic problems, Commun. Partial Differ. Equ., 43 (2018), 790-820.  doi: 10.1080/03605302.2018.1475488.  Google Scholar

[2]

C. Bianchini, G. Ciraolo and P. Salani, An overdetermined problem for the anisotropic capacity, Calc. Var., 55, 84 (2016). doi: 10.1007/s00526-016-1011-x.  Google Scholar

[3]

G. Bellettini and M. Paolini, Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math. J., 25 (1996), 537-566.  doi: 10.14492/hokmj/1351516749.  Google Scholar

[4]

P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control., Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser, 2004.  Google Scholar

[5]

A. Cianchi and P. Salani, Overdetermined anisotropic elliptic problems, Math. Ann., 345 (2009), 859-881.  doi: 10.1007/s00208-009-0386-9.  Google Scholar

[6]

G. Ciraolo, A. Figalli and A. Roncoroni, Symmetry results for critical anisotropic $p$-Laplacian equations in convex cones, Geom. Funct. Anal., 30 (2020), 770-803. doi: 10.1007/s00039-020-00535-3.  Google Scholar

[7]

G. Ciraolo and A. Roncoroni, Serrin's type overdetermined problems in convex cones, Calc. Var. Partial Differ. Equ., 59, 28 (2020). doi: 10.1007/s00526-019-1678-x.  Google Scholar

[8]

A. Farina and B. Kawohl, Remarks on an overdetermined boundary value problem, Calc. Var. Partial Differ. Equ., 31 (2008), 351-357.  doi: 10.1007/s00526-007-0115-8.  Google Scholar

[9]

A. Farina and E. Valdinoci, On partially and globally overdetermined problems of elliptic type, Amer. J. Math., 135 (2013), 1699-1726.  doi: 10.1353/ajm.2013.0052.  Google Scholar

[10]

E. Ferone and B. Kawohl, Remarks on a Finsler-Laplacian, Proc. Amer. Math. Soc., 137 (2009), 247-253.  doi: 10.1090/S0002-9939-08-09554-3.  Google Scholar

[11]

I. Fragalà and F. Gazzola, Partially overdetermined elliptic boundary value problems, J. Differ. Equ., 245 (2008), 1299-1322.  doi: 10.1016/j.jde.2008.06.014.  Google Scholar

[12]

I. Fragalà, F. Gazzola, J. Lamboley and M. Pierre, Counterexamples to symmetry for partially overdetermined elliptic problems, Analysis, 29 (2009), 85-93.  doi: 10.1524/anly.2009.1016.  Google Scholar

[13]

N. Garofalo and J. L. Lewis, A symmetry result related to some overdetermined boundary value problems, Amer. J. Math., 111 (1989), 9-33.  doi: 10.2307/2374477.  Google Scholar

[14]

A. Greco, Comparison principle and constrained radial symmetry for the subdiffusive $p$-Laplacian, Publ. Mat., 58 (2014), 485-498.  doi: 10.5565/PUBLMAT_58214_24.  Google Scholar

[15]

A. Greco, Symmetry around the origin for some overdetermined problems, Adv. Math. Sci. Appl., 13 (2003), 387-399.   Google Scholar

[16]

P. L. Lions and F. Pacella, Isoperimetric inequalities for convex cones, Proc. Amer. Math. Soc., 109 (1990), 477-485.  doi: 10.2307/2048011.  Google Scholar

[17]

F. Pacella and G. Tralli, Overdetermined problems and constant mean curvature surfaces in cones, Rev. Mat. Iberoam., 36 (2020), 841-867.  doi: 10.4171/rmi/1151.  Google Scholar

[18]

A. Roncoroni, A symmetry result for the $\varphi$-Laplacian in model manifolds, preprint. Google Scholar

[19]

S. Salsa, Partial Differential Equations in Action. From Modelling to Theory, 3rd ed, Springer International Publishing, 2016. doi: 10.1007/978-3-319-31238-5.  Google Scholar

[20]

R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, in Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1993. doi: 10.1017/CBO9780511526282.  Google Scholar

[21]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.  Google Scholar

[22]

G. Wang and C. Xia, A characterization of the Wulff shape by an overdetermined anisotropic PDE, Arch. Rational Mech. Anal., 199 (2011), 99-115.  doi: 10.1007/s00205-010-0323-9.  Google Scholar

show all references

References:
[1]

C. Bianchini and G. Ciraolo, Wulff shape characterizations in overdetermined anisotropic elliptic problems, Commun. Partial Differ. Equ., 43 (2018), 790-820.  doi: 10.1080/03605302.2018.1475488.  Google Scholar

[2]

C. Bianchini, G. Ciraolo and P. Salani, An overdetermined problem for the anisotropic capacity, Calc. Var., 55, 84 (2016). doi: 10.1007/s00526-016-1011-x.  Google Scholar

[3]

G. Bellettini and M. Paolini, Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math. J., 25 (1996), 537-566.  doi: 10.14492/hokmj/1351516749.  Google Scholar

[4]

P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations, and optimal control., Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser, 2004.  Google Scholar

[5]

A. Cianchi and P. Salani, Overdetermined anisotropic elliptic problems, Math. Ann., 345 (2009), 859-881.  doi: 10.1007/s00208-009-0386-9.  Google Scholar

[6]

G. Ciraolo, A. Figalli and A. Roncoroni, Symmetry results for critical anisotropic $p$-Laplacian equations in convex cones, Geom. Funct. Anal., 30 (2020), 770-803. doi: 10.1007/s00039-020-00535-3.  Google Scholar

[7]

G. Ciraolo and A. Roncoroni, Serrin's type overdetermined problems in convex cones, Calc. Var. Partial Differ. Equ., 59, 28 (2020). doi: 10.1007/s00526-019-1678-x.  Google Scholar

[8]

A. Farina and B. Kawohl, Remarks on an overdetermined boundary value problem, Calc. Var. Partial Differ. Equ., 31 (2008), 351-357.  doi: 10.1007/s00526-007-0115-8.  Google Scholar

[9]

A. Farina and E. Valdinoci, On partially and globally overdetermined problems of elliptic type, Amer. J. Math., 135 (2013), 1699-1726.  doi: 10.1353/ajm.2013.0052.  Google Scholar

[10]

E. Ferone and B. Kawohl, Remarks on a Finsler-Laplacian, Proc. Amer. Math. Soc., 137 (2009), 247-253.  doi: 10.1090/S0002-9939-08-09554-3.  Google Scholar

[11]

I. Fragalà and F. Gazzola, Partially overdetermined elliptic boundary value problems, J. Differ. Equ., 245 (2008), 1299-1322.  doi: 10.1016/j.jde.2008.06.014.  Google Scholar

[12]

I. Fragalà, F. Gazzola, J. Lamboley and M. Pierre, Counterexamples to symmetry for partially overdetermined elliptic problems, Analysis, 29 (2009), 85-93.  doi: 10.1524/anly.2009.1016.  Google Scholar

[13]

N. Garofalo and J. L. Lewis, A symmetry result related to some overdetermined boundary value problems, Amer. J. Math., 111 (1989), 9-33.  doi: 10.2307/2374477.  Google Scholar

[14]

A. Greco, Comparison principle and constrained radial symmetry for the subdiffusive $p$-Laplacian, Publ. Mat., 58 (2014), 485-498.  doi: 10.5565/PUBLMAT_58214_24.  Google Scholar

[15]

A. Greco, Symmetry around the origin for some overdetermined problems, Adv. Math. Sci. Appl., 13 (2003), 387-399.   Google Scholar

[16]

P. L. Lions and F. Pacella, Isoperimetric inequalities for convex cones, Proc. Amer. Math. Soc., 109 (1990), 477-485.  doi: 10.2307/2048011.  Google Scholar

[17]

F. Pacella and G. Tralli, Overdetermined problems and constant mean curvature surfaces in cones, Rev. Mat. Iberoam., 36 (2020), 841-867.  doi: 10.4171/rmi/1151.  Google Scholar

[18]

A. Roncoroni, A symmetry result for the $\varphi$-Laplacian in model manifolds, preprint. Google Scholar

[19]

S. Salsa, Partial Differential Equations in Action. From Modelling to Theory, 3rd ed, Springer International Publishing, 2016. doi: 10.1007/978-3-319-31238-5.  Google Scholar

[20]

R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, in Encyclopedia of Mathematics and its Applications, Cambridge University Press, 1993. doi: 10.1017/CBO9780511526282.  Google Scholar

[21]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  doi: 10.1007/BF00250468.  Google Scholar

[22]

G. Wang and C. Xia, A characterization of the Wulff shape by an overdetermined anisotropic PDE, Arch. Rational Mech. Anal., 199 (2011), 99-115.  doi: 10.1007/s00205-010-0323-9.  Google Scholar

Figure 1.  Maximizing the scalar product $ y \cdot {D \! H}_0(x) $ under the constraint $ H_0(y) = R $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The ball $ B_1(0, H_0) $ (left) is smooth, its dual (right) is not
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Finding the Euclidean norm of $ {D \! H}_0(P_\vartheta) $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Francesca Colasuonno, Fausto Ferrari. The Soap Bubble Theorem and a $ p $-Laplacian overdetermined problem. Communications on Pure & Applied Analysis, 2020, 19 (2) : 983-1000. doi: 10.3934/cpaa.2020045
[2]

Shaolin Ji, Xiaole Xue. A stochastic maximum principle for linear quadratic problem with nonconvex control domain. Mathematical Control & Related Fields, 2019, 9 (3) : 495-507. doi: 10.3934/mcrf.2019022
[3]

Monique Dauge, Thomas Ourmières-Bonafos, Nicolas Raymond. Spectral asymptotics of the Dirichlet Laplacian in a conical layer. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1239-1258. doi: 10.3934/cpaa.2015.14.1239
[4]

Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791
[5]

Changliang Zhou, Chunqin Zhou. Extremal functions of Moser-Trudinger inequality involving Finsler-Laplacian. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2309-2328. doi: 10.3934/cpaa.2018110
[6]

Timothy Blass, Rafael De La Llave, Enrico Valdinoci. A comparison principle for a Sobolev gradient semi-flow. Communications on Pure & Applied Analysis, 2011, 10 (1) : 69-91. doi: 10.3934/cpaa.2011.10.69
[7]

Yu Tian, John R. Graef, Lingju Kong, Min Wang. Existence of solutions to a multi-point boundary value problem for a second order differential system via the dual least action principle. Conference Publications, 2013, 2013 (special) : 759-769. doi: 10.3934/proc.2013.2013.759
[8]

Nicolas Augier, Ugo Boscain, Mario Sigalotti. Semi-conical eigenvalue intersections and the ensemble controllability problem for quantum systems. Mathematical Control & Related Fields, 2020, 10 (4) : 877-911. doi: 10.3934/mcrf.2020023
[9]

Ferenc Weisz. Dual spaces of mixed-norm martingale Hardy spaces. Communications on Pure & Applied Analysis, 2021, 20 (2) : 681-695. doi: 10.3934/cpaa.2020285
[10]

Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations & Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026
[11]

Shigeaki Koike, Takahiro Kosugi. Remarks on the comparison principle for quasilinear PDE with no zeroth order terms. Communications on Pure & Applied Analysis, 2015, 14 (1) : 133-142. doi: 10.3934/cpaa.2015.14.133
[12]

Xiaowei Tang, Xilin Fu. New comparison principle with Razumikhin condition for impulsive infinite delay differential systems. Conference Publications, 2009, 2009 (Special) : 739-743. doi: 10.3934/proc.2009.2009.739
[13]

Thomas Leroy. Relativistic transfer equations: Comparison principle and convergence to the non-equilibrium regime. Kinetic & Related Models, 2015, 8 (4) : 725-763. doi: 10.3934/krm.2015.8.725
[14]

Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897
[15]

Vladimir Georgiev, Koichi Taniguchi. On fractional Leibniz rule for Dirichlet Laplacian in exterior domain. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1101-1115. doi: 10.3934/dcds.2019046
[16]

Xavier Bresson, Tony F. Chan. Fast dual minimization of the vectorial total variation norm and applications to color image processing. Inverse Problems & Imaging, 2008, 2 (4) : 455-484. doi: 10.3934/ipi.2008.2.455
[17]

Bernd Kawohl, Vasilii Kurta. A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1747-1762. doi: 10.3934/cpaa.2011.10.1747
[18]

Shuxia Pan. Asymptotic spreading in a delayed dispersal predator-prey system without comparison principle. Electronic Research Archive, 2019, 27: 89-99. doi: 10.3934/era.2019011
[19]

Shanming Ji, Yutian Li, Rui Huang, Xuejing Yin. Singular periodic solutions for the p-laplacian ina punctured domain. Communications on Pure & Applied Analysis, 2017, 16 (2) : 373-392. doi: 10.3934/cpaa.2017019
[20]

Michele V. Bartuccelli, S.A. Gourley, Y. Kyrychko. Comparison and convergence to equilibrium in a nonlocal delayed reaction-diffusion model on an infinite domain. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1015-1026. doi: 10.3934/dcdsb.2005.5.1015

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (76)
  • HTML views (174)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences