doi: 10.3934/dcdss.2022144
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Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups

1. 

Dipartimento di Matematica e Fisica "Ennio De Giorgi", Università del Salento, and INFN, Sezione di Lecce, Ex Collegio Fiorini - Via per Arnesano - Lecce, Italy

2. 

Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università degli Studi di Modena e Reggio Emilia, Via Campi 213/b, 41125 Modena, Italy

*Corresponding author: Sergio Polidoro

We dedicate this paper to Jerry Goldstein on the occasion of his 80th birthday.

Received  April 2022 Revised  June 2022 Early access August 2022

Fund Project: The first author is supported by the grant of Gruppo Nazionale per l'Analisi Matematica, la Probabilitá e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and by INFN, Seione di Lecce. The second author is supported by the grant of Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with Hölder continuous coefficients. The kernels appearing in the integrals are supported on the level and superlevel sets of the fundamental solution relative the adjoint differential operator. We then extend the aforementioned formulas to some subelliptic operators on Carnot groups. In this case we rely on the theory of finite perimeter sets on stratified Lie groups.

Citation: Diego Pallara, Sergio Polidoro. Mean value formulas for classical solutions to some degenerate elliptic equations in Carnot groups. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022144
References:
[1]

L. Ambrosio, Fine properties of sets of finite perimeter in doubling metric measure spaces, Set-Valued Anal., 10 (2002), 111-128.  doi: 10.1023/A:1016548402502.

[2]

L. Ambrosio and M. Scienza, Locality of the perimeter in Carnot groups and chain rule, Ann. Mat. Pura Appl., 189 (2010), 661-678.  doi: 10.1007/s10231-010-0130-9.

[3]

A. BonfiglioliE. Lanconelli and F. Uguzzoni, Fundamental solutions for non-divergence form operators on stratified groups, Trans. Amer. Math. Soc., 356 (2004), 2709-2737.  doi: 10.1090/S0002-9947-03-03332-4.

[4]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub–Laplacians, Springer Monographs in Mathematics, 2007.

[5]

M. BramantiM. Miranda and D. Pallara, Two characterization of $BV$ functions on Carnot groups via the heat semigroup, Int. Math. Res. Not., 17 (2012), 3845-3876.  doi: 10.1093/imrn/rnr170.

[6]

L. CapognaD. Danielli and N. Garofalo, The geometric Sobolev embedding for vector fields and the isoperimetric inequality, Comm. Anal. Geom., 2 (1994), 203-215.  doi: 10.4310/CAG.1994.v2.n2.a2.

[7]

G. Citti, N. Garofalo and E. Lanconelli, Harnack's inequality for sum of squares of vector fields plus a potential, Amer. J. Math., 115 (1993), 699–734. doi: 10.2307/2375077.

[8]

E. B. Fabes and N. Garofalo, Mean value properties of solutions to parabolic equations with variable coefficients, J. Math. Anal. Appl., 121 (1987), 305-316.  doi: 10.1016/0022-247X(87)90249-6.

[9]

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207.  doi: 10.1007/BF02386204.

[10]

B. FranchiR. Serapioni and F. Serra Cassano, Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields, Houston J. Math., 22 (1996), 859-890. 

[11]

B. FranchiR. Serapioni and F. Serra Cassano, Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups, Comm. Anal. Geom., 11 (2003), 909-944.  doi: 10.4310/CAG.2003.v11.n5.a4.

[12]

B. FranchiR. Serapioni and F. Serra Cassano, On the strcucture of finite perimeter sets in Carnot step 2 groups, J. Geom. Anal., 13 (2003), 421-466.  doi: 10.1007/BF02922053.

[13]

N. Garofalo and E. Lanconelli, Asymptotic behavior of fundamental solutions and potential theory of parabolic operators with variable coefficients, Math. Ann., 283 (1989), 211-239.  doi: 10.1007/BF01446432.

[14]

N. Garofalo and E. Lanconelli, Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type, Trans. Amer. Math. Soc., 321 (1990), 775-792.  doi: 10.1090/S0002-9947-1990-0998126-5.

[15]

H. Kalf, On E. E. Levi's method of constructing a fundamental solution for second-order elliptic equations, Rend. Circ. Mat. Palermo, 41 (1992), 251-294.  doi: 10.1007/BF02844669.

[16]

A. E. Kogoj and E. Lanconelli, An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations, Mediterr. J. Math., 1 (2004), 51-80.  doi: 10.1007/s00009-004-0004-8.

[17]

V. Magnani, A new differentiation, shape of the unit ball, and perimeter measure, Indiana Univ. Math. J., 66 (2017), 183-204.  doi: 10.1512/iumj.2017.66.6007.

[18]

V. Magnani, Towards a theory of area in homogeneous groups, Calc. Var. Partial Diff. Eq., 58 (2019), Paper No. 91, 39 pp. doi: 10.1007/s00526-019-1539-7.

[19]

E. Malagoli, D. Pallara and S. Polidoro, Mean value formulas for classical solutions to uniformly parabolic equations in divergence form with non-smooth coefficients, Math. Nachr., to appear.

[20] F. Morgan, Geometric Measure Theory. A Beginner's Guide, $3^{rd}$ ed., Academic Press, 2000. 
[21]

D. Pallara and S. Polidoro, Mean value formulas for classical solutions to subelliptic equations in stratified Lie groups, to appear.

[22]

B. Pini, Sulle equazioni a derivate parziali, lineari del secondo ordine in due variabili, di tipo parabolico, Ann. Mat. Pura Appl., 32 (1951), 179-204.  doi: 10.1007/BF02417958.

[23]

S. Polidoro, On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type, Le Matematiche, 49 (1994), 53-105. 

[24]

F. Serra Cassano, Some topics of geometric measure theory in Carnot groups, in: Geometry, Analysis and Dynamics on Sub-Riemannian Manifolds, Volume I, D. Barilari, U. Boscain, M. Sigalotti eds., EMS 2016, 1–121.

[25]

S. R. S. Varadhan, On the behavior of the fundamental solution of the heat equation with variable coefficients, Comm. Pure Appl. Math., 20 (1967), 431-455.  doi: 10.1002/cpa.3160200210.

[26]

S. R. S. Varadhan, Diffusion processes in a small time interval, Comm. Pure Appl. Math., 20 (1967), 659-685.  doi: 10.1002/cpa.3160200404.

[27]

N. A. Watson, A theory of subtemperatures in several variables, Proc. London Math. Soc., 26 (1973), 385-417.  doi: 10.1112/plms/s3-26.3.385.

show all references

References:
[1]

L. Ambrosio, Fine properties of sets of finite perimeter in doubling metric measure spaces, Set-Valued Anal., 10 (2002), 111-128.  doi: 10.1023/A:1016548402502.

[2]

L. Ambrosio and M. Scienza, Locality of the perimeter in Carnot groups and chain rule, Ann. Mat. Pura Appl., 189 (2010), 661-678.  doi: 10.1007/s10231-010-0130-9.

[3]

A. BonfiglioliE. Lanconelli and F. Uguzzoni, Fundamental solutions for non-divergence form operators on stratified groups, Trans. Amer. Math. Soc., 356 (2004), 2709-2737.  doi: 10.1090/S0002-9947-03-03332-4.

[4]

A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, Stratified Lie Groups and Potential Theory for Their Sub–Laplacians, Springer Monographs in Mathematics, 2007.

[5]

M. BramantiM. Miranda and D. Pallara, Two characterization of $BV$ functions on Carnot groups via the heat semigroup, Int. Math. Res. Not., 17 (2012), 3845-3876.  doi: 10.1093/imrn/rnr170.

[6]

L. CapognaD. Danielli and N. Garofalo, The geometric Sobolev embedding for vector fields and the isoperimetric inequality, Comm. Anal. Geom., 2 (1994), 203-215.  doi: 10.4310/CAG.1994.v2.n2.a2.

[7]

G. Citti, N. Garofalo and E. Lanconelli, Harnack's inequality for sum of squares of vector fields plus a potential, Amer. J. Math., 115 (1993), 699–734. doi: 10.2307/2375077.

[8]

E. B. Fabes and N. Garofalo, Mean value properties of solutions to parabolic equations with variable coefficients, J. Math. Anal. Appl., 121 (1987), 305-316.  doi: 10.1016/0022-247X(87)90249-6.

[9]

G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207.  doi: 10.1007/BF02386204.

[10]

B. FranchiR. Serapioni and F. Serra Cassano, Meyers-Serrin type theorems and relaxation of variational integrals depending on vector fields, Houston J. Math., 22 (1996), 859-890. 

[11]

B. FranchiR. Serapioni and F. Serra Cassano, Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups, Comm. Anal. Geom., 11 (2003), 909-944.  doi: 10.4310/CAG.2003.v11.n5.a4.

[12]

B. FranchiR. Serapioni and F. Serra Cassano, On the strcucture of finite perimeter sets in Carnot step 2 groups, J. Geom. Anal., 13 (2003), 421-466.  doi: 10.1007/BF02922053.

[13]

N. Garofalo and E. Lanconelli, Asymptotic behavior of fundamental solutions and potential theory of parabolic operators with variable coefficients, Math. Ann., 283 (1989), 211-239.  doi: 10.1007/BF01446432.

[14]

N. Garofalo and E. Lanconelli, Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type, Trans. Amer. Math. Soc., 321 (1990), 775-792.  doi: 10.1090/S0002-9947-1990-0998126-5.

[15]

H. Kalf, On E. E. Levi's method of constructing a fundamental solution for second-order elliptic equations, Rend. Circ. Mat. Palermo, 41 (1992), 251-294.  doi: 10.1007/BF02844669.

[16]

A. E. Kogoj and E. Lanconelli, An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations, Mediterr. J. Math., 1 (2004), 51-80.  doi: 10.1007/s00009-004-0004-8.

[17]

V. Magnani, A new differentiation, shape of the unit ball, and perimeter measure, Indiana Univ. Math. J., 66 (2017), 183-204.  doi: 10.1512/iumj.2017.66.6007.

[18]

V. Magnani, Towards a theory of area in homogeneous groups, Calc. Var. Partial Diff. Eq., 58 (2019), Paper No. 91, 39 pp. doi: 10.1007/s00526-019-1539-7.

[19]

E. Malagoli, D. Pallara and S. Polidoro, Mean value formulas for classical solutions to uniformly parabolic equations in divergence form with non-smooth coefficients, Math. Nachr., to appear.

[20] F. Morgan, Geometric Measure Theory. A Beginner's Guide, $3^{rd}$ ed., Academic Press, 2000. 
[21]

D. Pallara and S. Polidoro, Mean value formulas for classical solutions to subelliptic equations in stratified Lie groups, to appear.

[22]

B. Pini, Sulle equazioni a derivate parziali, lineari del secondo ordine in due variabili, di tipo parabolico, Ann. Mat. Pura Appl., 32 (1951), 179-204.  doi: 10.1007/BF02417958.

[23]

S. Polidoro, On a class of ultraparabolic operators of Kolmogorov-Fokker-Planck type, Le Matematiche, 49 (1994), 53-105. 

[24]

F. Serra Cassano, Some topics of geometric measure theory in Carnot groups, in: Geometry, Analysis and Dynamics on Sub-Riemannian Manifolds, Volume I, D. Barilari, U. Boscain, M. Sigalotti eds., EMS 2016, 1–121.

[25]

S. R. S. Varadhan, On the behavior of the fundamental solution of the heat equation with variable coefficients, Comm. Pure Appl. Math., 20 (1967), 431-455.  doi: 10.1002/cpa.3160200210.

[26]

S. R. S. Varadhan, Diffusion processes in a small time interval, Comm. Pure Appl. Math., 20 (1967), 659-685.  doi: 10.1002/cpa.3160200404.

[27]

N. A. Watson, A theory of subtemperatures in several variables, Proc. London Math. Soc., 26 (1973), 385-417.  doi: 10.1112/plms/s3-26.3.385.

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