A general quasilinear elliptic problem with variable exponents and Neumann boundary conditions for image processing

The aim of this paper is to state and prove existence and uniqueness results for a general elliptic problem with homogeneous Neumann boundary conditions, often associated with image processing tasks like denoising. The novelty is that we surpass the lack of coercivity of the Euler-Lagrange functional with an innovative technique that has at its core the idea of showing that the minimum of the energy functional over a subset of the space $W^{1,p\left(x\right)}\left(\Omega \right)$ coincides with the global minimum. The obtained existence result applies to multiple-phase elliptic problems under remarkably weak assumptions.