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We establish the existence and uniqueness of the solution to the Dirichletproblem for the variable exponent $p$-Laplacian on a bounded, smooth domain$\Omega \subset {\mathbb R}^n$, where the boundary datum belongs to$W^{1,p}(\Omega)$. Our analysis considers a continuous and bounded exponent $p$satisfying $1<\inf\limits_{x\in \Omega}p(x)$ and $\sup\limits_{x\in\Omega}p(x)<\infty $, and is based on the uniform convexity of the Dirichletintegral, which is highly non trivial and in the variable exponent case is notrelated to the uniform convexity of the Sobolev norm.

Modular uniform convexity structures and applications to boundary value problems with non-standard growth