arXiv subjects

Bogdan Maxim

Publications and source records attributed to Bogdan Maxim.

4 recordsLinked to original sources

arXiv2026

A doubly nonlinear elliptic problem with variable exponents, homogeneous Neumann boundary conditions and generalized logistic source

We study a class of doubly nonlinear elliptic problems with variable exponents and homogeneous Neumann boundary conditions arising in the time discretization, by Rothe's method, of an associated doubly nonlinear parabolic problem. A main feature of the present work is that the structural assumptions imposed on the source term are weakened: in particular, no local Lipschitz continuity with respect to the state variable is required. Instead, we work under a weaker monotonicity-type assumption involving the source term and the nonlinear function $b$. A key ingredient in the analysis is the continuity of the associated Nemytskii operators between suitable variable-exponent Lebesgue spaces.

math.AP↗

arXiv2026

A doubly nonlinear parabolic problem with variable exponents, homogeneous Neumann boundary conditions and generalized logistic source

The aim of this work is to develop a self-contained existence and uniqueness theory for a doubly nonlinear parabolic problem with variable exponents and homogeneous Neumann boundary conditions. Our approach is based on Rothe's time-discretization method and builds on the results established by the author in a previous paper. We also investigate the asymptotic behavior of the solution. All arguments are presented in full detail, so that the paper is self-contained.

math.AP↗

arXiv2025

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 $W^{1,p(x)}(Ω)$ coincides with the global minimum. The obtained existence result applies to multiple-phase elliptic problems under remarkably weak assumptions.

math.AP↗

arXiv2025

Uniqueness of the strong positive solution for a general quasilinear elliptic problem with variable exponents and homogeneous Neumann boundary conditions using a generalization of the $p(x)$-Díaz-Saa inequality

In this paper, we study a generalization of the Díaz-Saa inequality and its applications to nonlinear elliptic problems. We first present the necessary hypotheses and preliminary results before introducing an improved version of the inequality, which holds in a broader functional setting and allows applications to problems with homogeneous Neumann boundary conditions. The significance of cases where the inequality becomes an equality is also analyzed, leading to uniqueness results for certain classes of partial differential equations. Furthermore, we provide a detailed proof of a uniqueness theorem for strong positive solutions and illustrate our findings with two concrete applications: a multiple-phase problem and an elliptic quasilinear equation relevant to image processing. The paper concludes with possible directions for future research.

math.AP↗