Communications on Pure and Applied Analysis (CPAA)

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

Pages: 407 - 419, Volume 14, Issue 2, March 2015      doi:10.3934/cpaa.2015.14.407

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Simona Fornaro - Dipartimento di Matematica "F. Casorati”, Università di Pavia, Via Ferrata 1, 27100 Pavia, Italy (email)
Giorgio Metafune - Dipartimento di Matematica E. De Giorgi, Università del Salento, 73100, Lecce, Italy (email)
Diego Pallara - Dipartimento di Matematica "Ennio De Giorgi”, Università del Salento, C.P. 193, Lecce, I-73100, Italy (email)
Roland Schnaubelt - Department of Mathematics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany (email)

Abstract: 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.

Keywords:  Elliptic operator, degeneration at boundary, generation result, domain, maximal regularity, gradient estimates, sums of commuting operators.
Mathematics Subject Classification:  Primary: 35K65, 35J70.

Received: December 2013;      Revised: July 2014;      Available Online: December 2014.