A general quasilinear elliptic problem with variable exponents and Neumann boundary conditions for image processing
Bogdan Maxim
TL;DR
This work develops existence and uniqueness results for a broad class of quasilinear elliptic problems with variable exponent growth under homogeneous Neumann boundary conditions, motivated by image denoising. A key novelty is a variational technique that bypasses lack of coercivity by showing the energy minimum on a carefully chosen subset of $W^{1,p(x)}(\Omega)$ coincides with the global minimum, and by introducing auxiliary and perturbed problems that yield a robust solvability theory. The authors establish weak minimum/comparison principles in variable-exponent spaces, prove well-posedness and Gamma-convergence for perturbed problems, and construct minimal and maximal steady-states via a monotone operator framework. These results extend to multi-phase operators and connect to parabolic models through Rothe-type arguments, offering a solid variational foundation for image-processing PDEs with spatially inhomogeneous growth. The findings have potential implications for stable denoising algorithms and for further analytical development of quasilinear PDEs with nonstandard growth.
Abstract
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 $W^{1,p(x)}(Ω)$ coincides with the global minimum. The obtained existence result applies to multiple-phase elliptic problems under remarkably weak assumptions.