June  2022, 21(6): 2115-2145. doi: 10.3934/cpaa.2022052

Multi-dimensional degenerate operators in $L^p$-spaces

1. 

Dipartimento di Matematica "Felice Casorati", Università degli Studi di Pavia, 27100, Pavia, Italy

2. 

Dipartimento di Matematica e Fisica "Ennio De Giorgi", Università del Salento, C.P.193, 73100, Lecce, Italy

3. 

Department of Mathematics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany

*Corresponding author

Received  November 2021 Revised  January 2022 Published  June 2022 Early access  March 2022

Fund Project: D.P. is associated to I.N.F.N. (Sezione di Lecce) and is member of G.N.A.M.P.A. of the Italian Istituto Nazionale di Alta Matematica (INdAM) and has been partially supported by the PRIN 2015 MIUR project 2015233N54

This paper is concerned with second-order elliptic operators whose diffusion coefficients degenerate at the boundary in first order. In this borderline case, the behavior strongly depends on the size and direction of the drift term. Mildly inward (or outward) pointing and strongly outward pointing drift terms were studied before. Here we treat the intermediate case equipped with Dirichlet boundary conditions, and show generation of an analytic positive $ C_0 $-semigroup. The main result is a precise description of the domain of the generator, which is more involved than in the other cases and exhibits reduced regularity compared to them.

Citation: Simona Fornaro, Giorgio Metafune, Diego Pallara, Roland Schnaubelt. Multi-dimensional degenerate operators in $L^p$-spaces. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2115-2145. doi: 10.3934/cpaa.2022052
References:
[1]

W. ArendtG. Metafune and D. Pallara, Schrödinger operators with unbounded drift, J. Operat. Theor., 55 (2006), 185-211. 

[2]

P. Daskalopoulos and P. M. N. Feehan, $C^{1, 1}$ regularity for degenerate elliptic obstacle problems, J. Differ. Equ., 260 (2016), 5043–5074. doi: 10.1016/j. jde. 2015.11.037.

[3]

P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation, J. Amer. Math. Soc., 11, (1998), 899–965. doi: 10.1090/S0894-0347-98-00277-X.

[4]

R. Denk, M. Hieber and J. Prüss, R-boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type, Memoirs of the American Mathematical Society, 2003. doi: 10.1090/memo/0788.

[5]

H. Dong, T. Phan and H.V. Tran, Degenerate linear parabolic equations in divergence form on the upper half space, arXiv: 2107.08033.

[6]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.

[7] C. L. Epstein and R. Mazzeo, Degenerate Diffusion Operators Arising in Population Biology, Princeton Univ. Press, Princeton (NJ), 2013.  doi: 10.1515/9781400846108.
[8]

P. M. N. Feehan and C. Pop, Schauder a priori estimates and regularity of solutions to boundary-degenerate elliptic linear second-order partial differential equations, J. Differ. Equ., 256 (2014), 895–956. doi: 10.1016/j. jde. 2013.08.012.

[9]

P. M. N. Feehan and C. Pop, Boundary-degenerate elliptic operators and Hölder continuity for solutions to variational equations and inequalities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1075–1129. doi: 10.1016/j. anihpc. 2016.07.005.

[10]

S. Fornaro, G. Metafune, D. Pallara and J. Prüss, $L^p$-theory for some elliptic and parabolic problems with first order degeneracy at the boundary, J. Math. Pures Appl., 87 (2007), 367–393. doi: 10.1016/j. matpur. 2007.02.001.

[11]

S. Fornaro, G. Metafune, D. Pallara and R. Schnaubelt, Degenerate operators of Tricomi type in $L^p$-spaces and in spaces of continuous functions, J. Differ. Equ., 252 (2012), 1182–1212. doi: 10.1016/j. jde. 2011.09.017.

[12]

S. Fornaro, G. Metafune, D. Pallara and R. Schnaubelt, One-dimensional degenerate operators in $L^p$-spaces, J. Math. Anal. Appl., 402 (2013), 308–318. doi: 10.1016/j. jmaa. 2013.01.030.

[13]

S. Fornaro, G. Metafune, D. Pallara and R. Schnaubelt, Second order elliptic operators in $L^2$ with first order degeneration at the boundary and outward pointing drift, Commun. Pure Appl. Anal., 14 (2015), 407–419. doi: 10.3934/cpaa. 2015.14.407.

[14]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[15]

C. Kienzler, H. Koch and J. L. Vázquez, Flatness implies smoothness for solutions of the porous medium equation, Calc. Var. Partial Differ. Equ., 57 (2018), 42 pp. doi: 10.1007/s00526-017-1296-4.

[16]

J. U. Kim, An $L^p$ a priori estimate for the Tricomi equation in the upper half–space, Trans. Amer. Math. Soc., 351 (1999), 4611–4628. doi: 10.1090/S0002-9947-99-02349-1.

[17]

K. H. Kim, Sobolev space theory of parabolic equations degenerating on the boundary of $C^1$ domain, Commun. Partial Differ. Equ., 32 (2007), 1261–1280. doi: 10.1080/03605300600910449.

[18]

H. Koch, Non-Euclidean Singular Integrals and the Porous Medium Equation, Habilitation thesis, 1999.

[19]

P. C. Kunstmann and L. Weis, Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus, in Functional Analytic Methods for Evolution Equations (eds. M. Iannelli, R. Nagel and S. Piazzera), Springer-Verlag, (2004), 65–311. doi: 10.1007/978-3-540-44653-8_2.

[20]

G. Metafune, L. Negro and C. Spina, Sharp kernel estimates for elliptic operators with second-order discontinuous coefficients, J. Evol. Equ., 18 (2018), 467–514. doi: 10.1007/s00028-017-0408-0.

[21]

G. MetafuneL. Negro and C. Spina, $L^p$ estimates for the Caffarelli Silvestre extension operators, J. Differ. Equ., 316 (2022), 290-345.  doi: 10.1016/j.jde.2022.01.049.

[22]

G. Metafune, M. Sobajima and C. Spina, Kernel estimates for elliptic operators with second-order discontinuous coefficients, J. Evol. Equ., 17 (2017), 485–522. doi: 10.1007/s00028-016-0355-1.

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[24]

J. Prüss, On second-order elliptic operators with complete first-order boundary degeneration and strong outward drift, Arch. Math. (Basel), 108 (2017), 301–311. doi: 10.1007/s00013-016-0992-1.

[25]

L. Weis, A new approach to maximal $L_p$-regularity, in Evolution Equations and Their Applications in Physical and Life Sciences (eds. G. Lumer and L. Weis), Marcel Dekker, (2001), 195–214.

show all references

References:
[1]

W. ArendtG. Metafune and D. Pallara, Schrödinger operators with unbounded drift, J. Operat. Theor., 55 (2006), 185-211. 

[2]

P. Daskalopoulos and P. M. N. Feehan, $C^{1, 1}$ regularity for degenerate elliptic obstacle problems, J. Differ. Equ., 260 (2016), 5043–5074. doi: 10.1016/j. jde. 2015.11.037.

[3]

P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation, J. Amer. Math. Soc., 11, (1998), 899–965. doi: 10.1090/S0894-0347-98-00277-X.

[4]

R. Denk, M. Hieber and J. Prüss, R-boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type, Memoirs of the American Mathematical Society, 2003. doi: 10.1090/memo/0788.

[5]

H. Dong, T. Phan and H.V. Tran, Degenerate linear parabolic equations in divergence form on the upper half space, arXiv: 2107.08033.

[6]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000.

[7] C. L. Epstein and R. Mazzeo, Degenerate Diffusion Operators Arising in Population Biology, Princeton Univ. Press, Princeton (NJ), 2013.  doi: 10.1515/9781400846108.
[8]

P. M. N. Feehan and C. Pop, Schauder a priori estimates and regularity of solutions to boundary-degenerate elliptic linear second-order partial differential equations, J. Differ. Equ., 256 (2014), 895–956. doi: 10.1016/j. jde. 2013.08.012.

[9]

P. M. N. Feehan and C. Pop, Boundary-degenerate elliptic operators and Hölder continuity for solutions to variational equations and inequalities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1075–1129. doi: 10.1016/j. anihpc. 2016.07.005.

[10]

S. Fornaro, G. Metafune, D. Pallara and J. Prüss, $L^p$-theory for some elliptic and parabolic problems with first order degeneracy at the boundary, J. Math. Pures Appl., 87 (2007), 367–393. doi: 10.1016/j. matpur. 2007.02.001.

[11]

S. Fornaro, G. Metafune, D. Pallara and R. Schnaubelt, Degenerate operators of Tricomi type in $L^p$-spaces and in spaces of continuous functions, J. Differ. Equ., 252 (2012), 1182–1212. doi: 10.1016/j. jde. 2011.09.017.

[12]

S. Fornaro, G. Metafune, D. Pallara and R. Schnaubelt, One-dimensional degenerate operators in $L^p$-spaces, J. Math. Anal. Appl., 402 (2013), 308–318. doi: 10.1016/j. jmaa. 2013.01.030.

[13]

S. Fornaro, G. Metafune, D. Pallara and R. Schnaubelt, Second order elliptic operators in $L^2$ with first order degeneration at the boundary and outward pointing drift, Commun. Pure Appl. Anal., 14 (2015), 407–419. doi: 10.3934/cpaa. 2015.14.407.

[14]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.

[15]

C. Kienzler, H. Koch and J. L. Vázquez, Flatness implies smoothness for solutions of the porous medium equation, Calc. Var. Partial Differ. Equ., 57 (2018), 42 pp. doi: 10.1007/s00526-017-1296-4.

[16]

J. U. Kim, An $L^p$ a priori estimate for the Tricomi equation in the upper half–space, Trans. Amer. Math. Soc., 351 (1999), 4611–4628. doi: 10.1090/S0002-9947-99-02349-1.

[17]

K. H. Kim, Sobolev space theory of parabolic equations degenerating on the boundary of $C^1$ domain, Commun. Partial Differ. Equ., 32 (2007), 1261–1280. doi: 10.1080/03605300600910449.

[18]

H. Koch, Non-Euclidean Singular Integrals and the Porous Medium Equation, Habilitation thesis, 1999.

[19]

P. C. Kunstmann and L. Weis, Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus, in Functional Analytic Methods for Evolution Equations (eds. M. Iannelli, R. Nagel and S. Piazzera), Springer-Verlag, (2004), 65–311. doi: 10.1007/978-3-540-44653-8_2.

[20]

G. Metafune, L. Negro and C. Spina, Sharp kernel estimates for elliptic operators with second-order discontinuous coefficients, J. Evol. Equ., 18 (2018), 467–514. doi: 10.1007/s00028-017-0408-0.

[21]

G. MetafuneL. Negro and C. Spina, $L^p$ estimates for the Caffarelli Silvestre extension operators, J. Differ. Equ., 316 (2022), 290-345.  doi: 10.1016/j.jde.2022.01.049.

[22]

G. Metafune, M. Sobajima and C. Spina, Kernel estimates for elliptic operators with second-order discontinuous coefficients, J. Evol. Equ., 17 (2017), 485–522. doi: 10.1007/s00028-016-0355-1.

[23]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[24]

J. Prüss, On second-order elliptic operators with complete first-order boundary degeneration and strong outward drift, Arch. Math. (Basel), 108 (2017), 301–311. doi: 10.1007/s00013-016-0992-1.

[25]

L. Weis, A new approach to maximal $L_p$-regularity, in Evolution Equations and Their Applications in Physical and Life Sciences (eds. G. Lumer and L. Weis), Marcel Dekker, (2001), 195–214.

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