# 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 & 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.  doi: 10.1007/s000130050210.  Google Scholar [2] P. Daskalopoulos and P. M. N. Feehan, Existence, uniqueness and global regularity for degenerate elliptic obstacle problems in mathematical finance,, preprint, ().   Google Scholar [3] P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation,, J. Amer. Math. Soc., 11 (1998), 899.  doi: 10.1090/S0894-0347-98-00277-X.  Google Scholar [4] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Springer-Verlag, (2000).   Google Scholar [5] C. L. Epstein and R. Mazzeo, Degenerate Diffusion Operators Arising in Population Biology,, Ann. Math. Stud. 185, (2013).   Google Scholar [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, ().   Google Scholar [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.  doi: 10.1016/j.jde.2013.08.012.  Google Scholar [8] W. Feller, The parabolic differential equations and the associated semi-groups of transformations,, Ann. of Math., 55 (1952), 468.   Google Scholar [9] W. Feller, Diffusion processes in one dimension,, Trans. Amer. Math. Soc., 97 (1954), 1.   Google Scholar [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.  doi: 10.1016/j.matpur.2007.02.001.  Google Scholar [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.  doi: 10.1016/j.jde.2011.09.017.  Google Scholar [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.  doi: 10.1016/j.jmaa.2013.01.030.  Google Scholar [13] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar [14] C. Kienzler, Flat fronts and stability for the porous medium equation,, preprint, ().   Google Scholar [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.  doi: 10.1090/S0002-9947-99-02349-1.  Google Scholar [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.  doi: 10.1080/03605300600910449.  Google Scholar [17] H. Koch, Non-euclidean singular integrals and the porous medium equation,, Habilitation thesis (1999), (1999).   Google Scholar [18] J. J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of first order,, Comm. Pure Appl. Math., 20 (1967), 797.   Google Scholar [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, (2004), 65.  doi: 10.1007/978-3-540-44653-8_2.  Google Scholar [20] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Birkhäuser Verlag, (1995).  doi: 10.1007/978-3-0348-9234-6.  Google Scholar [21] A. Lunardi, Interpolation Theory,, Scuola Normale Superiore Pisa, (2009).   Google Scholar [22] G. Metafune, Analyticity for some degenerate one-dimensional evolution equation,, Studia Math., 127 (1998), 251.   Google Scholar [23] O. A. Oleinik and E. V. Radkevic, Second Order Equations with Non Negative Characteristic Form., Plenum Press, (1973).   Google Scholar [24] E. M. Ouhabaz, Analysis of Heat Equations on Domains,, Princeton University Press, (2005).   Google Scholar [25] N. Shimakura, Partial Differential Operators of Elliptic Type,, Amer. Math. Soc., (1992).   Google Scholar [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.   Google Scholar [27] H. Tanabe, Functional Analytic Methods for Partial Diffeeential Equations,, Marcel Dekker, (1997).   Google Scholar [28] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland, (1978).   Google Scholar

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##### References:
 [1] M. Campiti and G. Metafune, Ventcel's boundary conditions and analytic semigroups,, Arch. Math., 70 (1998), 377.  doi: 10.1007/s000130050210.  Google Scholar [2] P. Daskalopoulos and P. M. N. Feehan, Existence, uniqueness and global regularity for degenerate elliptic obstacle problems in mathematical finance,, preprint, ().   Google Scholar [3] P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation,, J. Amer. Math. Soc., 11 (1998), 899.  doi: 10.1090/S0894-0347-98-00277-X.  Google Scholar [4] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Springer-Verlag, (2000).   Google Scholar [5] C. L. Epstein and R. Mazzeo, Degenerate Diffusion Operators Arising in Population Biology,, Ann. Math. Stud. 185, (2013).   Google Scholar [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, ().   Google Scholar [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.  doi: 10.1016/j.jde.2013.08.012.  Google Scholar [8] W. Feller, The parabolic differential equations and the associated semi-groups of transformations,, Ann. of Math., 55 (1952), 468.   Google Scholar [9] W. Feller, Diffusion processes in one dimension,, Trans. Amer. Math. Soc., 97 (1954), 1.   Google Scholar [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.  doi: 10.1016/j.matpur.2007.02.001.  Google Scholar [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.  doi: 10.1016/j.jde.2011.09.017.  Google Scholar [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.  doi: 10.1016/j.jmaa.2013.01.030.  Google Scholar [13] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer-Verlag, (1983).  doi: 10.1007/978-3-642-61798-0.  Google Scholar [14] C. Kienzler, Flat fronts and stability for the porous medium equation,, preprint, ().   Google Scholar [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.  doi: 10.1090/S0002-9947-99-02349-1.  Google Scholar [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.  doi: 10.1080/03605300600910449.  Google Scholar [17] H. Koch, Non-euclidean singular integrals and the porous medium equation,, Habilitation thesis (1999), (1999).   Google Scholar [18] J. J. Kohn and L. Nirenberg, Degenerate elliptic-parabolic equations of first order,, Comm. Pure Appl. Math., 20 (1967), 797.   Google Scholar [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, (2004), 65.  doi: 10.1007/978-3-540-44653-8_2.  Google Scholar [20] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Birkhäuser Verlag, (1995).  doi: 10.1007/978-3-0348-9234-6.  Google Scholar [21] A. Lunardi, Interpolation Theory,, Scuola Normale Superiore Pisa, (2009).   Google Scholar [22] G. Metafune, Analyticity for some degenerate one-dimensional evolution equation,, Studia Math., 127 (1998), 251.   Google Scholar [23] O. A. Oleinik and E. V. Radkevic, Second Order Equations with Non Negative Characteristic Form., Plenum Press, (1973).   Google Scholar [24] E. M. Ouhabaz, Analysis of Heat Equations on Domains,, Princeton University Press, (2005).   Google Scholar [25] N. Shimakura, Partial Differential Operators of Elliptic Type,, Amer. Math. Soc., (1992).   Google Scholar [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.   Google Scholar [27] H. Tanabe, Functional Analytic Methods for Partial Diffeeential Equations,, Marcel Dekker, (1997).   Google Scholar [28] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators,, North-Holland, (1978).   Google Scholar
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