Figure 1.
Maximizing the scalar product $ y \cdot {D \! H}_0(x) $ under the constraint $ H_0(y) = R $
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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 |
- Abstract
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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 and Applied Analysis,
2021, 20
(3)
: 1025-1038.
doi: 10.3934/cpaa.2021004
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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. 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. |
| [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. 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. |
| [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 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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