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Usted está aquí: Inicio / Actividades / Seminarios / Seminario de Ecuaciones Diferenciales No Lineales (SEDNOL) / Actividades del Seminario de Ecuaciones Diferenciales No Lineales / Uniform Continuity and Brézis-Lieb Type Splitting for Superposition Operators in Sobolev Space

Uniform Continuity and Brézis-Lieb Type Splitting for Superposition Operators in Sobolev Space

Ponente: Nils Ackermann
Institución: IM-UNAM
Tipo de Evento: Investigación
Cuándo 11/02/2016
de 10:00 a 11:00
Dónde Salón 2
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Denote by \(\mathcal{F}\) a superposition (or Nemyckii-)
operator induced by a continuous function
\(f\colon\mathbb{R}\to\mathbb{R}\) that satisfies a polynomial growth
condition with exponent \(\mu>0\).  If \(\nu\ge 1\) is such that \(\mu\nu>2\)
and that the Sobolev embedding of \(H^1(\mathbb{R}^N)\) in
\(L^{\mu\nu}(\mathbb{R}^N)\) is locally compact we prove that
\(\mathcal{F}\colon H^1(\mathbb{R}^N)\to L^\nu(\mathbb{R}^N)\) is
uniformly continuous on any bounded subset of \(H^1(\mathbb{R}^N)\).  This
result implies a variant of the Brézis-Lieb Lemma that applies to more
general nonlinear superposition operators within this range of growth
exponents.  In particular, no convexity or Hölder continuity
assumptions are imposed on \(f\), in contrast to previously known results.