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Hitoshi Ishii

Publications and source records attributed to Hitoshi Ishii.

At least 19 recordsLinked to original sources
arXiv2026

Balance between degenerate elliptic operators and coercive Hamiltonians

For $p>1$, we consider the boundary value problem for fully nonlinear degenerate elliptic equations $-\lambda_i(D^2u)+|Du|^p+\gamma u=f(x)$ in bounded domains with Dirichlet or boundary blow-up conditions; here $\lambda_i(D^2u)$ denotes the $i$-th eigenvalue of the Hessian. We study existence and nonexistence of solutions together with the asymptotic behaviour of the solutions when $\gamma$ goes to zero. A priori Lipschitz estimates play an important role. The interplay between the operator's degeneracy and the superlinear growth of the Hamiltonian gives rise to phenomena that are very different depending on which of the two terms dominates, e.g. the ergodic dichotomy takes place only when $i=N$, while new phenomena arise for $i<N$ in which case, under mild conditions, solutions that blow up even in just one point do not exist, and conditions on the size of $f$ must be imposed for the existence of solutions to the Dirichlet problem with homogeneous boundary condition.

math.AP↗
arXiv2025

The vanishing discount problem for nonlocal Hamilton-Jacobi equations

We establish a convergence result for the vanishing discount problem in the context of nonlocal HJ equations. We consider a fairly general class of discounted first-order and convex HJ equations which incorporate an integro-differential operator posed on the $d$-dimensional torus, and we show that the solutions converge to a specific critical solution as the discount factor tends to zero. Our approach relies on duality techniques for nonlocal convex HJ equations, building upon Hahn-Banach separation theorems to develop a generalized notion of Mather measure. The results are applied to a specific class of convex and superlinear Hamiltonians.

math.AP↗
arXiv2025

Liouville type results for the fractional truncated Laplacians in a half-space

Existence issues of viscosity supersolutions in the half-space $\mathbb R^N_+$, for a class of fully nonlinear integral equations involving the fractional truncated Laplacians and a power-like nonlinearity in the unknown function, are addressed in this paper, the aim being to obtain estimates on the threshold exponents separating the existence from the nonexistence regimes.

math.AP↗
arXiv2024

Fully nonlinear elliptic PDEs in thin domains with oblique boundary condition

In this preprint we consider fully nonlinear equations in thin domains with oblique boundary condition, finding some new phenomena, in particular the limit equation contains "new terms" of the second, first and zeroth order which don't have an equivalent in the Neumann case treated in our previous work arXiv:2404.19577. The classical laplacian problem with Neumann boundary condition, goes back to the well known result of Hale and Raugel (1992).

math.AP↗
arXiv2022

Spoken Dialogue Strategy Focusing on Asymmetric Communication with Android Robots

Humans are easily conscious of small differences in an android robot's (AR's) behaviors and utterances, resulting in treating the AR as not-human, while ARs treat us as humans. Thus, there exists asymmetric communication between ARs and humans. In our system at Dialogue Robot Competition 2022, this asymmetry was a considerable research target in our dialogue strategy. For example, tricky phrases such as questions related to personal matters and forceful requests for agreement were experimentally used in AR's utterances. We assumed that these AR phrases would have a reasonable chance of success, although humans would likely hesitate to use the phrases. Additionally, during a five-minute dialogue, our AR's character, such as its voice tones and sentence expressions, changed from mechanical to human-like type in order to pretend to tailor to customers. The characteristics of the AR developed by our team, DSML-TDU, are introduced in this paper.

cs.RO↗
arXiv2022

Propagation of minima for nonlocal operators

In this paper we state some sharp maximum principle, i.e. we characterize the geometry of the sets of minima for supersolutions of equations involving the $k$-\emph{th fractional truncated Laplacian} or the $k$-\emph{th fractional eigenvalue} which are fully nonlinear integral operators whose nonlocality is somehow $k$-dimensional.

math.AP↗
arXiv2022

The vanishing discount problem for monotone systems of Hamilton-Jacobi equations: a counterexample to the full convergence

In recent years there has been intense interest in the vanishing discount problem for Hamilton-Jacobi equations. In the case of the scalar equation, B. Ziliotto has recently given an example of the Hamilton-Jacobi equation having non-convex Hamiltonian in the gradient variable, for which the full convergence of the solutions does not hold as the discount factor tends to zero. We give here an explicit example of nonlinear monotone systems of Hamilton-Jacobi equations having convex Hamiltonians in the gradient variable, for which the full convergence of the solutions fails as the discount factor goes to zero.

math.AP↗
arXiv2022

An example in the vanishing discount problem for monotone systems of Hamilton-Jacobi equations

In recent years, there have been many contributions to the vanishing discount problem for Hamilton-Jacobi equations. In the case of the scalar equation, B. Ziliotto [Convergence of the solutions of the discounted Hamilton-Jacobi equation: a counterexample. J. Math. Pures Appl. (9) 128 (2019), 330-338] has shown an example of the Hamilton-Jacobi equation having non-convex Hamiltonian in the gradient variable, for which the full convergence of the solutions does not hold as the discount factor tends to zero. We give an example of the nonlinear monotone system of Hamilton-Jacobi equations having convex Hamiltonians in the gradient variable, for which the whole family convergence of the solutions does not hold.

math.AP↗
arXiv2021

Nonlinear Neumann problems for fully nonlinear elliptic PDEs on a quadrant

We consider the nonlinear Neumann problem for fully nonlinear elliptic PDEs on a quadrant. We establish a comparison theorem for viscosity sub and supersolutions of the nonlinear Neumann problem. The crucial argument in the proof of the comparison theorem is to build a $C^{1,1}$ test function which takes care of the nonlinear Neumann boundary condition. A similar problem has been treated on a general $n$-dimensional orthant by Biswas, Ishii, Subhamay, and Wang [SIAM J. Control Optim. 55 (2017), pp. 365--396], where the functions ($H_i$ in the main text) describing the boundary condition are required to be positively one-homogeneous, and the result in this paper removes the positive homogeneity in two-dimension. An existence result for solutions is also presented.

math.AP↗
arXiv2021

Hamilton-Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown

We study the Hamilton-Jacobi equations $H(x,Du,u)=0$ in $M$ and $\partial u/\partial t +H(x,D_xu,u)=0$ in $M\times(0,\infty)$, where the Hamiltonian $H=H(x,p,u)$ depends Lipschitz continuously on the variable $u$. In the framework of the semicontinuous viscosity solutions due to Barron-Jensen, we establish the comparison principle, existence theorem, and representation formula as value functions for extended real-valued, lower semicontinuous solutions for the Cauchy problem. We also establish some results on the long-time behavior of solutions for the Cauchy problem and classification of solutions for the stationary problem.

math.AP↗
arXiv2020

The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling

We establish a convergence theorem for the vanishing discount problem for a weakly coupled system of Hamilton-Jacobi equations. The crucial step is the introduction of Mather measures and their relatives for the system, which we call respectively viscosity Mather and Green-Poisson measures. This is done by the convex duality and the duality between the space of continuous functions on a compact set and the space of Borel measures on it. This is part 1 of our study of the vanishing discount problem for systems, which focuses on the linear coupling, while part 2 will be concerned with nonlinear coupling.

math.AP↗
arXiv2020

The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 2: Nonlinear coupling

We study the vanishing discount problem for a nonlinear monotone system of Hamilton-Jacobi equations. This continues the first author's investigation on the vanishing discount problem for a monotone system of Hamilton-Jacobi equations. As in Part 1, we introduce by the convex duality Mather measures and their analogues for the system, which we call respectively Mather and Green-Poisson measures, and prove a convergence theorem for the vanishing discount problem. Moreover, we establish an existence result for the ergodic problem.

math.AP↗
arXiv2020

Discrete approximation of the viscous HJ equation

We consider a stochastic discretization of the stationary viscous Hamilton Jacobi equation on the flat d dimensional torus, associated with a Hamiltonian, convex and superlinear in the momentum variable. We show that each discrete problem admits a unique continuous solution on the torus, up to additive constants. By additionally assuming a technical condition on the associated Lagrangian, we show that each solution of the viscous Hamilton Jacobi equation is the limit of solutions of the discrete problems, as the discretization step goes to zero.

math.AP↗
arXiv2019

Averaging of Hamilton-Jacobi equations over Hamiltonian flows

We study the asymptotic behavior of solutions to the Dirichlet problem for Hamilton-Jacobi equations with large drift terms, where the drift terms are given by the Hamiltonian vector fields of Hamiltonian $H$. This is an attempt to understand the averaging effect for fully nonlinear degenerate elliptic equations. In this work, we restrict ourselves to the case of Hamilton-Jacobi equations. The second author has already established averaging results for Hamilton-Jacobi equations with convex Hamiltonians ($G$ below) under the classical formulation of the Dirichlet condition. Here we treat the Dirichlet condition in the viscosity sense, and establish an averaging result for Hamilton-Jacobi equations with relatively general Hamiltonian $G$.

math.AP↗
arXiv2019

Positivity sets of supersolutions of degenerate elliptic equations and the strong maximum principle

We investigate positivity sets of nonnegative supersolutions of the fully nonlinear elliptic equations $F(x,u,Du,D^2u)=0$ in $Ω$, where $Ω$ is an open subset of ${\mathbb R}^N$, and the validity of the strong maximum principle for $F(x,u,Du,D^2u)=f$ in $Ω$, with $f\in\text{C}(Ω)$ being nonpositive. We obtain geometric characterizations of positivity sets $\left\{x\inΩ\,:\, u(x)>0\right\}$ of nonnegative supersolutions $u$ and establish the strong maximum principle under some geometric assumption on the set $\left\{x\inΩ\,:\, f(x)=0\right\}$.

math.AP↗
arXiv2019

The Vanishing Discount problem for Hamilton-Jacobi Equations in the Euclidean Space

We study the asymptotic behavior of the solutions to a family of discounted Hamilton Jacobi equations, posed in the Euclidean N dimensional space, when the discount factor goes to zero. The ambient space being noncompact, we introduce an assumption implying that the Aubry set is compact and there is no degeneracy at infinity. Our approach is to deal not with a single Hamiltonian and Lagrangian but with the whole space of generalized Lagrangians, and then to define via duality minimizing measures associated to both the corresponding ergodic and discounted equations. The asymptotic result follows from convergence properties of these measures with respect to the narrow topology. We use as duality tool a separation theorem in locally convex Hausdorff spaces, we use the strict topology in the space of the bounded generalized Lagrangians as well.

math.AP↗