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Jeongmin Han

Publications and source records attributed to Jeongmin Han.

11 recordsLinked to original sources
arXiv2025

Regularity for fully nonlinear elliptic equations in generalized Orlicz spaces

In this paper, we establish an optimal global Calder\'{o}n-Zygmund type estimate for the viscosity solution to the Dirichlet boundary problem of fully nonlinear elliptic equations with possibly nonconvex nonlinearities. We prove that the Hessian of the solution is as integrable as the nonhomogeneous term in the setting of a given generalized Orlicz space even when the nonlinearity is asymptotically convex with respect to the Hessian of the solution.

math.AP↗
arXiv2025

Consistency of tug-of-war type operators on random data clouds

In this paper, we study a tug-of-war type operator on geometric graphs and its associated Dirichlet problem on a random data cloud. Specifically, we analyze the convergence of the value functions as the number of data points increases and the step size of the game shrinks. This analysis reveals the connection between our tug-of-war type operator and the corresponding model problem. A key ingredient in establishing this result is the consistency of the operator.

math.AP↗
arXiv2025

Tug-of-war games related to $p$-Laplace type equations with zeroth order terms

In this paper, we investigate a class of tug-of-war games that incorporate a constant payoff discount rate at each turn. The associated model problems are $p$-Laplace type partial differential equations with zeroth-order terms. We establish existence, uniqueness, and regularity results for the corresponding game value functions. Furthermore, we explore properties of the solutions to the model PDEs, informed by the analysis of the underlying games.

math.AP↗
arXiv2024

Tug-of-war games associated with boundary value problems involving derivatives

In this paper, we study a certain type of noisy tug-of-war game which can be regarded as an interpretation of a certain type of boundary value problem for the normalized $p$-Laplace equation, where $1<p<2$. More precisely, we will investigate the boundary regularity of the value function to the game and its convergence to a viscosity solution of the model problem.

math.AP↗
arXiv2024

Tug-of-war games related to oblique derivative boundary value problems with the normalized $p$-Laplacian

In this paper, we are concerned with game-theoretic interpretations to the following oblique derivative boundary value problem \begin{align*} \left\{ \begin{array}{ll} \Delta_{p}^{N}u=0 & \textrm{in $ \Omega$,}\\ \langle \beta , Du \rangle + \gamma u = \gamma G & \textrm{on $ \partial \Omega$,}\\ \end{array} \right. \end{align*} where $\Delta_{p}^{N}$ is the normalized $p$-Laplacian. This problem can be regarded as a generalized version of the Robin boundary value problem for the Laplace equations. We construct several types of stochastic games associated with this problem by using `shrinking tug-of-war'. For the value functions of such games, we investigate the properties such as existence, uniqueness, regularity and convergence.

math.AP↗
arXiv2024

Dynamic programming principle in cost-efficient sequential design: optimal update scheduling under cost constraints

We study sequential cost-efficient design in a situation where each update of covariates involves a fixed time cost typically considerable compared to a single measurement time. The problem arises from parameter estimation in switching measurements on superconducting Josephson junctions which are components needed in quantum computers and other superconducting electronics. In switching measurements, a sequence of current pulses is applied to the junction and a binary voltage response is observed. The measurement requires a very low temperature that can be kept stable only for a relatively short time, and therefore it is essential to use an efficient design. We use the dynamic programming principle from the mathematical theory of optimal control to solve the optimal update times. Specifically, we give a formulation for $D$-optimal experimental design based on the dynamic programming principle with a binary response model. Our simulations demonstrate the cost-efficiency compared to the previously used methods.

stat.ME↗
arXiv2023

Game-theoretic approach to Hölder regularity for PDEs involving eigenvalues of the Hessian

We prove a local Hölder estimate with an exponent $0<δ<\frac 12$ for solutions of the dynamic programming principle $$u^\varepsilon (x) =\sum_{j=1}^n α_j\inf_{\dim(S)=j}\sup_{\substack{v\in S\\ |v|=1}}\frac{ u^\varepsilon (x + \varepsilon v) + u^\varepsilon (x - \varepsilon v)}{2}.$$ The proof is based on a new coupling idea from game theory. As an application, we get the same regularity estimate for viscosity solutions of the PDE $$\sum_{i=1}^n α_iλ_i(D^2u)=0,$$ where $λ_1(D^2 u)\leq\cdots\leq λ_n(D^2 u)$ are the eigenvalues of the Hessian.

math.AP↗
arXiv2021

Time-dependent tug-of-war games and normalized parabolic $p$-Laplace equations

This paper concerns value functions of time-dependent tug-of-war games. We first prove the existence and uniqueness of value functions and verify that these game values satisfy a dynamic programming principle. Using the arguments in the proof of existence of game values, we can also deduce asymptotic behavior of game values when $T \to \infty$. Furthermore, we investigate boundary regularity for game values. Thereafter, based on the regularity results for value functions, we deduce that game values converge to viscosity solutions of the normalized parabolic $p$-Laplace equation.

math.AP↗
arXiv2021

$L^{p}$-estimates for the Hessians of solutions to fully nonlinear parabolic equations with oblique boundary conditions

We study fully nonlinear parabolic equations in nondivergence form with oblique boundary conditions. An optimal and global Calderón-Zygmund estimate is obtained by proving that the Hessian of the viscosity solution to the oblique boundary problem is as integrable as the nonhomogeneous term in $L^{p}$ spaces under minimal regularity requirement on the nonlinear operator, the boundary data and the boundary of the domain.

math.AP↗