# Uniform Convexity in Variable Exponent Sobolev Spaces
## 1 Introduction  
This work is devoted to the study of a uniform-convexity-like property of the modular \(\rho_{p}:W^{1,p}(\Omega)\to[0,\infty)\), defined as  
\[\rho_{p}(u)=\int\limits_{\Omega}|\
abla u(x)|^{p(x)}dx, \tag{1}\]  
where \(\Omega\subset\mathbb{R}^{n}\) is a bounded domain, \(p:\Omega\to(1,\infty)\) is a measurable function and \(|\mathbf{v}|\) stands for the Euclidean norm of a vector \(\mathbf{v}\in\mathbb{R}^{n}\). We underline the fact that our work clarifies two distinct points that have so far not been covered in the literature: the consideration of the Euclidean norm \(|\
abla u|\) in the Sobolev integral above for the full range \(1<\inf\limits_{\Omega}p\) and the inclusion of the case \(\sup\limits_{\Omega}p=\infty\). As a by-product, we obtain the uniform convexity of the Sobolev-Luxemburg norm in the case where \(p\) is bounded away from \(1\) and \(\infty\). To the best of the authors' knowledge, the uniform convexity of the Sobolev norm is new for \(1<\inf\limits_{\Omega}<p<2\) ([1]). Introduced for the first time in the early 1930s, spaces of variable exponents acquired a new central role in mathematics after their emergence as the natural solution space for differential equations with non-standard growth. We refer the reader to [2, 3] for a general treatment of variable exponent spaces and their basic properties. The consideration of the Euclidean norm in (1) is of particular importance since the corresponding Dirichlet integral is Frechet differentiable and its derivative is precisely the \(p\)-Laplacian variable exponent \(\Delta_{p(x)}\). More specifically, let \(\Omega\subset\mathbb{R}^{n}\) be a domain and \(p\) be a measurable function on \(\Omega\), where \(1\leq p<\infty\). It is well known that the Luxemburg norm \(\|\cdot\|_{p}\) on the variable exponent space \(L^{p}(\Omega)\) is uniformly convex if and only if \(1<p_{-}=\inf\limits_{x\in\Omega}p(x)\leq\sup\limits_{x\in\Omega}p(x)=p_{+}<\infty\). Denote by \(W^{m,p}(\Omega)\) the usual variable exponent Sobolev space. There are certainly infinitely many equivalent norms that can be defined on \(W^{m,p}(\Omega)\). For example, if \(p\) is constant [4],\[\|u\|_{1,p}=\left(\sum\limits_{|a|=0}^{m}\|D^{a}u\|_{p}^{p}\right)^{\frac{1}{p}} \tag{2}\]  
or [2; 5]  
\[\|u\|_{1,p}=\sum\limits_{|\alpha|=0}^{m}\|D^{a}u\|_{p}, \tag{3}\]  
where \(\|\cdot\|_{p}\) stands for the \(L^{p}\) norm. The core ideas can be reduced to the case \(m=1\); thus, in the following only the space \(W^{1,p}(\Omega)\) will be considered. In general, if \(|\cdot|:\mathbb{R}^{n}\to[0,\infty)\) is a norm on \(\mathbb{R}^{n}\), the functional \(\rho_{1,p}:W^{1,p}(\Omega)\to[0,\infty]\) defined by  
\[\rho_{1,p}(u)=\int\limits_{\Omega}|u|^{p}dx+\int\limits_{\Omega}|\
abla u|^{p }dx \tag{4}\]  
is a left-continuous convex modular, and the corresponding Luxemburg norm  
\[\|u\|_{1,p}=\inf\Bigl{\{}\lambda>0:\rho_{1,p}\Bigl{(}\frac{u}{\lambda}\Bigr{)} \leq 1\Bigr{\}} \tag{5}\]  
is a norm on \(W^{1,p}(\Omega)\). Moreover, all such norms are equivalent. In particular, if \(p\) is constant on \(\Omega\), the functional \(\|\cdot\|:W^{1,p}(\Omega)\to[0,\infty]\) defined by  
\[\|u\|=\|u\|_{p}+\||\
abla u\|_{p} \tag{6}\]  
is a norm and all such norms on \(W^{1,p}(\Omega)\) are topologically equivalent. The question of uniform convexity is considerably more delicate. It is easy to prove that if the variable exponent \(p\) is bounded away from \(1\) and \(\infty\) on \(\Omega\), then (2) and (3) are uniformly convex. This can be seen, for example, by displaying a specific isometry between \(W^{1,p}(\Omega)\) (furnished with either of those norms) and \(L^{p}(\tilde{\Omega})\) for a suitable domain \(\tilde{\Omega}\) ([2; 4; 5]). For a general norm of the type (5) (in particular (6)), however, the issue is more subtle. The case \(|v|=\Bigl{(}\sum\limits_{j=1}^{n}v_{j}^{2}\Bigr{)}^{\frac{1}{2}}\) is of particular interest due to its applications in partial differential equations.