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

An overdetermined problem associated to the Finsler Laplacian

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  March 2021 Early access  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.

Citation: Giulio Ciraolo, Antonio Greco. An overdetermined problem associated to the Finsler Laplacian. Communications on Pure and 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[12]

I. FragalàF. GazzolaJ. Lamboley and M. Pierre, Counterexamples to symmetry for partially overdetermined elliptic problems, Analysis, 29 (2009), 85-93.  doi: 10.1524/anly.2009.1016.

[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.

[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.

[15]

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

[16]

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

[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.

[18]

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

[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.

[20]

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

[21]

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

[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.

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[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.

[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.

[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.

[12]

I. FragalàF. GazzolaJ. Lamboley and M. Pierre, Counterexamples to symmetry for partially overdetermined elliptic problems, Analysis, 29 (2009), 85-93.  doi: 10.1524/anly.2009.1016.

[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.

[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.

[15]

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

[16]

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

[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.

[18]

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

[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.

[20]

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

[21]

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

[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.

Figure 1.  Maximizing the scalar product $ y \cdot {D \! H}_0(x) $ under the constraint $ H_0(y) = R $
Figure 2.  The ball $ B_1(0, H_0) $ (left) is smooth, its dual (right) is not
Figure 3.  Finding the Euclidean norm of $ {D \! H}_0(P_\vartheta) $
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