American Institute of Mathematical Sciences

March  2015, 14(2): 407-419. doi: 10.3934/cpaa.2015.14.407

Second order elliptic operators in $L^2$ with first order degeneration at the boundary and outward pointing drift

 1 Dipartimento di Matematica "F. Casorati”, Università di Pavia, Via Ferrata 1, 27100 Pavia 2 Dipartimento di Matematica E. De Giorgi, Università del Salento, 73100, Lecce 3 Dipartimento di Matematica "Ennio De Giorgi”, Università del Salento, C.P. 193, Lecce, I-73100 4 Department of Mathematics, Karlsruhe Institute of Technology, 76128 Karlsruhe

Received  December 2013 Revised  July 2014 Published  December 2014

We study second order elliptic operators whose diffusion coefficients degenerate at the boundary in first order and whose drift term strongly points outward. It is shown that these operators generate analytic semigroups in $L^2$ where they are equipped with their natural domain without boundary conditions. Hence, the corresponding parabolic problem can be solved with optimal regularity. In a previous work we had treated the case of inward pointing drift terms.
Citation: Simona Fornaro, Giorgio Metafune, Diego Pallara, Roland Schnaubelt. Second order elliptic operators in $L^2$ with first order degeneration at the boundary and outward pointing drift. Communications on Pure and Applied Analysis, 2015, 14 (2) : 407-419. doi: 10.3934/cpaa.2015.14.407
References:
 [1] M. Campiti and G. Metafune, Ventcel's boundary conditions and analytic semigroups, Arch. Math., 70 (1998), 377-390. doi: 10.1007/s000130050210. [2] P. Daskalopoulos and P. M. N. Feehan, Existence, uniqueness and global regularity for degenerate elliptic obstacle problems in mathematical finance, preprint, arXiv:1109.1075. [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] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, 2000. [5] C. L. Epstein and R. Mazzeo, Degenerate Diffusion Operators Arising in Population Biology, Ann. Math. Stud. 185, Princeton Univ. Press, 2013. [6] P. M. N. Feehan and C. Pop, Degenerate elliptic operators in mathematical finance and Hölder continuity for solutions to variational equations and inequalities, preprint, arXiv:1110.5594. [7] 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. Differential Equations, 256 (2014), 895-956. doi: 10.1016/j.jde.2013.08.012. [8] W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math., 55 (1952), 468-519. [9] W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 97 (1954), 1-31. [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. Differential Equations, 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] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983. doi: 10.1007/978-3-642-61798-0. [14] C. Kienzler, Flat fronts and stability for the porous medium equation, preprint, arXiv:1403.5811v1. [15] 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. [16] K.-H. Kim, Sobolev space theory of parabolic equations degenerating on the boundary of $C^1$ domains, Comm. Partial Differential Equations, 32 (2007), 1261-1280. doi: 10.1080/03605300600910449. [17] H. Koch, Non-euclidean singular integrals and the porous medium equation, Habilitation thesis (1999), www.math.uni-bonn.de/simkoch/public.html [18] J. J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of first order, Comm. Pure Appl. Math., 20 (1967), 797-872. [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] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser Verlag, 1995. doi: 10.1007/978-3-0348-9234-6. [21] A. Lunardi, Interpolation Theory, Scuola Normale Superiore Pisa, 2009. [22] G. Metafune, Analyticity for some degenerate one-dimensional evolution equation, Studia Math., 127 (1998), 251-276. [23] O. A. Oleinik and E. V. Radkevic, Second Order Equations with Non Negative Characteristic Form. Plenum Press, 1973. [24] E. M. Ouhabaz, Analysis of Heat Equations on Domains, Princeton University Press, 2005. [25] N. Shimakura, Partial Differential Operators of Elliptic Type, Amer. Math. Soc., Providence (RI), 1992. [26] K.-T. Sturm, Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math., 32 (1995), 275-312. [27] H. Tanabe, Functional Analytic Methods for Partial Diffeeential Equations, Marcel Dekker, 1997. [28] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978.

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References:
 [1] M. Campiti and G. Metafune, Ventcel's boundary conditions and analytic semigroups, Arch. Math., 70 (1998), 377-390. doi: 10.1007/s000130050210. [2] P. Daskalopoulos and P. M. N. Feehan, Existence, uniqueness and global regularity for degenerate elliptic obstacle problems in mathematical finance, preprint, arXiv:1109.1075. [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] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, 2000. [5] C. L. Epstein and R. Mazzeo, Degenerate Diffusion Operators Arising in Population Biology, Ann. Math. Stud. 185, Princeton Univ. Press, 2013. [6] P. M. N. Feehan and C. Pop, Degenerate elliptic operators in mathematical finance and Hölder continuity for solutions to variational equations and inequalities, preprint, arXiv:1110.5594. [7] 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. Differential Equations, 256 (2014), 895-956. doi: 10.1016/j.jde.2013.08.012. [8] W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math., 55 (1952), 468-519. [9] W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 97 (1954), 1-31. [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. Differential Equations, 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] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983. doi: 10.1007/978-3-642-61798-0. [14] C. Kienzler, Flat fronts and stability for the porous medium equation, preprint, arXiv:1403.5811v1. [15] 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. [16] K.-H. Kim, Sobolev space theory of parabolic equations degenerating on the boundary of $C^1$ domains, Comm. Partial Differential Equations, 32 (2007), 1261-1280. doi: 10.1080/03605300600910449. [17] H. Koch, Non-euclidean singular integrals and the porous medium equation, Habilitation thesis (1999), www.math.uni-bonn.de/simkoch/public.html [18] J. J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of first order, Comm. Pure Appl. Math., 20 (1967), 797-872. [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] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser Verlag, 1995. doi: 10.1007/978-3-0348-9234-6. [21] A. Lunardi, Interpolation Theory, Scuola Normale Superiore Pisa, 2009. [22] G. Metafune, Analyticity for some degenerate one-dimensional evolution equation, Studia Math., 127 (1998), 251-276. [23] O. A. Oleinik and E. V. Radkevic, Second Order Equations with Non Negative Characteristic Form. Plenum Press, 1973. [24] E. M. Ouhabaz, Analysis of Heat Equations on Domains, Princeton University Press, 2005. [25] N. Shimakura, Partial Differential Operators of Elliptic Type, Amer. Math. Soc., Providence (RI), 1992. [26] K.-T. Sturm, Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations, Osaka J. Math., 32 (1995), 275-312. [27] H. Tanabe, Functional Analytic Methods for Partial Diffeeential Equations, Marcel Dekker, 1997. [28] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, 1978.
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