Traces for fractional Sobolev spaces with variable exponents

Document Type: Original Article

Authors

1 U Buenos Aires

2 Universidad de Buenos Aires Facultad de Ciencias Exactas y Naturales Depto Matematica Ciudad Universitaria, pab 1, Buenos Aires, Argentina

Abstract

In this note we prove a trace theorem in fractional spaces with variable exponents. To be more precise, we show that if $p\colon\overline{\Omega }\times \overline{\Omega } \rightarrow (1,\infty )$ and $q\colon\partial \Omega
\rightarrow (1,\infty )$ are continuous functions such that
\[ \frac{(n-1)p(x,x)}{n-sp(x,x)}>q(x) \qquad \mbox{ in } \partial \Omega \cap \{x\in\overline{\Omega}\colon n-sp(x,x) >0\}, \]
then the inequality
    $$
        \Vert f\Vert _{\scriptstyle  L^{q(\cdot)}(\partial \Omega )}
        \leq C \left\{ \Vert f\Vert _{\scriptstyle L^{\bar{p}(\cdot)}(\Omega )}+
        [f]_{s,p(\cdot,\cdot)} \right\}
    $$
holds. Here $\bar{p}(x)=p(x,x)$ and $\lbrack f\rbrack_{s,p(\cdot,\cdot)} $ denotes the fractional seminorm with variable exponent, that is given by
    \begin{equation*}
        \lbrack f\rbrack_{s,p(\cdot,\cdot)} :=
        \inf \left\{ \lambda >0\colon
        \int_{\Omega}\int_{\Omega }\frac{|f(x)-f(y)|^{p(x,y)}}{\lambda ^{p(x,y)}
        |x-y|^{n+sp(x,y)}}dxdy<1\right\}
    \end{equation*}
and $\Vert f\Vert _{\scriptstyle L^{q(\cdot)}(\partial \Omega )}$ and $\Vert f\Vert _{\scriptstyle L^{\bar{p}(\cdot)}(\Omega )}$ are the usual Lebesgue norms with variable exponent.

Keywords