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# 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 Salón 2 vCal iCal

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.