Lebesgue solvability of equations associated to elliptic and canceling linear differential operators with measure data

• Ano de defesa: 2024
• Autor(a) principal: Biliatto, Victor Sandrin
• Orientador(a): Picon, Tiago Henrique
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Universidade Federal de São Carlos; Câmpus São Carlos
• Programa de Pós-Graduação: Programa de Pós-Graduação em Matemática - PPGM
• Palavras-chave em Português: Campos vetoriais de medida-divergência; Resolubilidade em espaços de Lebesgue; Estimativas L^1; Equações elípticas; Operadores cancelantes
• Palavras-chave em Inglês: Divergence-measure vector fields; Lebesgue solvability; L^1 estimates; Elliptic equations; Canceling operators
• Área do conhecimento CNPq: CIENCIAS EXATAS E DA TERRA::MATEMATICA::ANALISE::EQUACOES DIFERENCIAIS PARCIAIS
• Link de acesso: https://repositorio.ufscar.br/handle/20.500.14289/20335

In this thesis, we present new results on the solvability of the equation A*(x,D) f = µ for f in L^p, with complex measure data µ, associated to an elliptic linear differential operator A(x,D) of order m with variable complex coefficients. Our method is based on (m,p)-energy control of µ giving sufficient conditions for solutions when 1 ≤ p < ∞. A particular study is presented in the global setting of Lebesgue solvability for the equation A*(D) f = µ, where A(D) is a homogeneous differential operator with constant coefficients. We also obtain sufficient conditions in the limiting case p = ∞ using new L^1 (global and local) estimates on measures for elliptic and canceling operators, which are interesting on their own.