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

We establish the existence and uniqueness of the solution to the Dirichlet problem for the variable exponent $p$-Laplacian on a bounded, smooth domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, where the boundary datum belongs to $W^{1,p}(\mathrm{\Omega })$. Our analysis considers a continuous and bounded exponent $p$ satisfying $1<\underset{x\in \mathrm{\Omega }}{\inf }p(x)$ and $\underset{x\in \mathrm{\Omega }}{\sup }p(x)<\mathrm{\infty }$, and is based on the uniform convexity of the Dirichlet integral, which is highly non trivial and in the variable exponent case is not related to the uniform convexity of the Sobolev norm.