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Master's Dissertation
DOI
10.11606/D.55.2014.tde-28042014-095009
Document
Author
Full name
Pedro David Huillca Leva
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2014
Supervisor
Committee
Carvalho, Alexandre Nolasco de (President)
Paiva, Francisco Odair Vieira de
Santos, Ederson Moreira dos
Title in Portuguese
Geração de semigrupos por operadores elípticos em L POT. 2 (OMEGA) e C INF. 0 (OMEGA)
Keywords in Portuguese
Condição do cone exterior uniforme
Geração de semigrupos
Operadores elípticos
Semigrupos holomorfos
Abstract in Portuguese
Neste trabalho estudaremos a geração do semigrupos por operadores elípticos em dois espaços. Em primeiro lugar estudaremos a geração de semigrupo no espaço 'L POT.2' ('OMEGA') por operadores elípticos de ordem 2m com 'OMEGA' suficientemente regular. Mais precisamente, se 'OMEGA' é um domínio limitado com 'PARTIAL OMEGA' de classe 'C POT. 2m,' L (x;D) = 'SIGMA' / ['alpha'] '< ou =' 'a IND. alpha' (x) 'D POT. alpha' é um operador diferencial elíptico de ordem 2m, com 'a IND. alpha' 'PERTENCE' ' 'C POT.j' ('OMEGA'), j = max {0, ['alpha'] - m}, e A : D(A) 'ESTÁ CONTIDO' EM 'L POT. 2 ('OMEGA') 'SETA' ' L POT. 2 ('OMEGA') é o operador linear dado por D(A) = 'H POT. 2m' ('OMEGA') 'H POT. m INF. 0' ('OMEGA'), (Au)(x) = L (x;D)u; então -A gera um 'C IND. 0'-semigrupo holomorfo em 'L POT.2' ('OMEGA'). ). Em segundo lugar estudaremos a geração de semigrupo em 'C IND. 0'('OMEGA") = ) = {u 'PERTENCE A' C ('OMEGA' 'BARRA") : u['PARTIAL omega' = 0} por operadores elípticos de ordem 2 com 'OMEGA' satisfazendo uma propriedade geométrica. Mais precisamente, se 'OMEGA' ESTA CONTIDO EM' 'R POT. n' (n '> ou =' 2) é um domínio limitado que satisfaz a condição de cone exterior uniforme, L é o operador Lu := - \\SIGMA SUP n INF. i,j = 1' 'a IND. ij 'D IND. ij u + '\SIGMA SUP. n IND. j=1 'b IND. j' u + cu com coeficientes reais 'a IND. ij' , 'b IND. j' , c que satisfazem 'b IND. j ' 'PERTENCE A' 'L POT. INFTY' ('OMEGA') , j = 1, ..., n, c 'PERTENCE A ' 'L POT> INFTY' (OMEGA), c '> ou =' 0, 'a IND. ij' 'PERTECE A' C(' OMEGA BARRA)' ' INTERSECCAO' 'L POT. INFTY' (OMEGA),e 'A IND. 0' é parte de L em 'C IND. 0' ("OMEGA'), isto é, D('A IND. 0') = {u 'PERTENCE A' 'C IND. 0' ('OMEGA') 'INTERSECÇÂO' 'W POT. 2, n INF. loc' ('OMEGA') : Lu 'PERTENCE A' 'C IND. 0' ('OMEGA')' 'A IND. 0' u = Lu, então -'A IND. 0' gera um 'C IND. 0-semigrupo holomorfo limitado em 'C IND. 0' ('OMEGA')
Title in English
Generations of semigroups for elliptic operators in 'L POT. 2' ('OMEGA') and 'C IND. 0('OMEGA')
Keywords in English
Condition of uniform exterior cone
Elliptic operators
generation of semigroups
Holomorphic semigroups
Abstract in English
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)
 
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Publishing Date
2014-04-29
 
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