| Abstract: |
| We construct weak solutions to the $p(u)$--Laplace equation
\begin{equation*}
- \nabla\cdot \big(|\nabla u|^{p(u) -2} \nabla u\big) = f \quad \text{in } \Omega.
\end{equation*}
These problems arise in adaptive image denoising, where the diffusion depends on the solution itself.
The main challenge stems from the variable exponent, which prevents the use of fixed Sobolev spaces.
We overcome this difficulty by employing a sophisticated version of the Lipschitz truncation method, combined with a localized Minty argument.
This approach enables us to prove the existence of weak solutions under minimal regularity assumptions, without requiring the restriction $p(u) > d$, where $d$ is the space dimension.
Our result applies to a broader class of nonlinear diffusion equations and contributes to the analysis of models where the equation`s behavior may depend on the solution itself. |
|