arXiv subjects

Melih Ucer

Publications and source records attributed to Melih Ucer.

4 recordsLinked to original sources

arXiv2026

Existence and Structure for First-Order Time-Dependent Mean-Field Games with Local Couplings

We develop a Banach-space framework for first-order time-dependent mean-field games with local couplings, using monotone operator theory and low-order $p$-Laplacian regularization to avoid high-order elliptic smoothing. Under monotonicity and power-growth assumptions, together with either a Lagrangian lower bound or strict positivity of the initial density, we prove existence of weak variational-inequality solutions by Minty's method. The constructed solutions satisfy uniform $L^\beta$ estimates on the density, $L^\alpha$ estimates on the spatial gradient of the value function, and space-time shift estimates sufficient to identify the limiting PDE system. We prove that any variational-inequality solution satisfying these bounds, regardless of how it is obtained, is a MFG solution satisfying the Hamilton--Jacobi and transport equations in the $BV$ sense. This separates the construction of VI-solutions from the verification of the PDE system, a feature not directly available in the existing stationary Banach-space framework. Finally, for each fixed density $m$, we establish a maximal value function among Hamilton--Jacobi subsolutions; every MFG value function coincides with this maximal representative on $\{m>0\}$ and initially on $\{m_0>0\}$. Under semi-strict monotonicity, the density $m$ itself is unique. Our results apply to non-separable Hamiltonians with power growth and impose no dimension restrictions.

math.AP↗

arXiv2026

Ranking Mean-Field Planning Games

This paper studies a one-dimensional Mean-Field Planning (MFP) system with a non-local, rank-based coupling. Using a potential formulation, we rewrite the system as an associated scalar partial differential equation. We prove an equivalence between classical solutions to the ranking MFP system with positive density and classical solutions to the associated potential problem, and we derive explicit reconstruction formulas. We then identify a monotonicity structure in the associated operator, which, under strict convexity assumptions, yields uniqueness of classical solutions to the associated problem and, hence, uniqueness of the ranking MFP system up to an additive constant in the value function. Finally, under superlinear growth assumptions, we exploit monotonicity to address existence in a low-regularity setting. By formulating a variational inequality for a q-Laplacian regularized operator, we apply Minty's method to establish the existence of weak solutions in the space of functions of bounded variation for a relaxed potential formulation.

math.AP↗

arXiv2025

Solving Mean-Field Games with Monotonicity Methods in Banach Spaces

This paper develops a unified framework for proving the existence of solutions to stationary first-order mean-field games (MFGs) based on the theory of monotone operators in Banach spaces. We cast the coupled MFG system as a variational inequality, overcoming the limitations of prior Hilbert-space approaches that relied on high-order regularization and typically yielded only weak solutions in the monotone operator sense. In contrast, with our low-order regularization, we obtain strong solutions. Our approach addresses the non-coercivity of the underlying MFG operator through two key regularization strategies. First, by adding a low-order $p$-Laplacian term, we restore coercivity, derive uniform a priori estimates, and pass to the limit via Minty's method. This establishes, for the first time via monotonicity methods, the existence of strong solutions for models with both standard power-growth and singular congestion, with the latter requiring a careful restriction of the operator's domain. Second, for Hamiltonians with only minimal growth hypotheses, we regularize the Hamiltonian itself via infimal convolution to prove the existence of weak solutions. Our Banach-space framework unifies and extends earlier existence results. By avoiding high-order smoothing, it not only provides a more direct theoretical path but is also ideally suited for modern numerical algorithms.

math.AP↗

arXiv2024

Regularity for Weak Solutions to First-Order Local Mean Field Games

We establish interior regularity results for first-order, stationary, local mean-field game (MFG) systems. Specifically, we study solutions of the coupled system consisting of a Hamilton-Jacobi-Bellman equation $H(x, Du, m) = 0$ and a transport equation $-\operatorname{div}(m D_pH(x, Du, m)) = 0$ in a domain $\Omega \subset \mathbb{R}^d$. Under suitable structural assumptions on the Hamiltonian $H$, without requiring monotonicity of the system, convexity of the Hamiltonian, separability in variables, or smoothness beyond basic continuity in $(p,m)$, we introduce a notion of weak solutions that allows the application of techniques from elliptic regularity theory. Our main contribution is to prove that the value function $u$ is locally H\"older continuous in $\Omega$. The proof leverages the connection between first-order MFG systems and quasilinear equations in divergence form, adapting classical techniques to handle the specific structure of MFG systems.

math.AP↗