Generations of semigroups for elliptic operators in \'L POT. 2\' (\'OMEGA\') and \'C IND. 0(\'OMEGA\')

Author(s)

Pedro David Huillca Leva

• Document type: Master's Dissertation
• Press: São Carlos.
• Institution: Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB)
• Defense date: 2014-03-18

Examining board members

Alexandre Nolasco de Carvalho; Francisco Odair Vieira de Paiva; Ederson Moreira dos Santos

• Advisor: Alexandre Nolasco de Carvalho

Abstract

In this work we study the generation of semigroups by elliptic operators in two spaces. Firstly we study the generation of semigroup in the space \'L POT. 2\' (OMEGA) for elliptic operators of order 2m with \'OMEGA\' regular domain. More precisely, if \'OMEGA\' is a bounded domain with \\PARTIAL OMEGA\' \'IT BELONGS\' \'C POT. 2m\', L (x, D) = \\ sigma INF.ALPHA \'> or =\' 2m, \'a IND. alpha\' ( x) \'D POT alpha\' is an elliptic differential operator of order 2m, with \'a IND. alpha\' \' \'IT BELONGS\' \'C POT. j\' (OMEGA), j = max {0, [\'ALPHA\'] - m}, and A : D (A) \'THIS CONTAINED\' \'L POT. 2\' (OMEGA) \'ARROW\' \'L POT. 2\' (OMEGA) is linear operator given or D(A) = \'H POT. 2m\' (OMEGA) \'INTERSECTION\' \'H POT. m INF. 0 (OMEGA) (Au) (x) = L (x,D) u then -A generates a holomorphic \'C IND. 0\'-semigroup in \'L POT. 2\'.(OMEGA). Secondly we study the generation of semigroup in \'C IND. 0\' (OMEGA) = {u \'IT BELONGS\' (c INF. O\' (OMEGA BAR) : \'u [IND. \\partial omega\' = 0} for elliptic operators of second order with \'OMEGA\' satisfying a geometric property. That is, if \'OMEGA\' \'IT BELONGS\' \'R POT. n\' (n > or = 2) is a bounded domain that satisfies the uniform exterior cone condition, L is the elliptic operator given by Lu : = - \\SIGMA SUP. n INF. i,j = 1\' \'a IND. i, j\' \'D IND. ij \' u + \\SIGMA SUP n INF. j=1\' \'b IND j D IND j\' u + cu with real coefficients \'a IND. ij, \'b IND. j\' , c satisfying \'b ind. j\' \'IT BELONGS\' \' L POT. INFTY\' (omega), j = 1, ..., n, c \'it belongs\' \'L POT. INFTY\' (OMEGA), \'c > or =\' 0, \'\'a IND. ij \'IT BELONGS\' C (OMNEGA BAR) \'INTERSECTION\' (OMEGA), and \'A IND. 0\' is part of L in \'C IND. 0\'(OMEGA), that is, D (\'A IND. 0\') = {u \'IT BELONGS\' \'C IND. 0\' (OMEGA) INTERSECTION \'W POT. 2, n IND. loc (OMEGA)} \'A IND. 0u\' = Lu, then - \'A IND. 0\' generates a bounded holomorphic \'C IND. 0\'-semigroup on \'C IND. 0\' (OMEGA) (AU)

• FAPESP's process: 12/16814-4 - Semigroups Generated by Elliptic Operators on Co
• Grantee: Pedro David Huillca Leva
• Support Opportunities: Scholarships in Brazil - Master