Maximum principles and the method of moving planes for the uniformly elliptic nonlocal Bellman operator and applications
Wei Dai, Guolin Qin
备注：
In this version, we corrected the basic inequality (2.1), and modified our operator and the statements of results in Sections 2, 3 and subsection 4.4 in v2 and the published version accordingly
发表信息：
Annali di Matematica Pura ed Applicata, 200, 1085--1134 (2021)
In this paper, we establish various maximum principles and develop the method of moving planes and the sliding method (on general unbounded domains) for equations involving the uniformly elliptic nonlocal Bellman operator. As a consequence, we derive multiple applications of these maximum principles and the moving planes method. For instance, we prove symmetry, monotonicity and uniqueness results and asymptotic properties for solutions to various equations involving the uniformly elliptic nonlocal Bellman operator in bounded domains, unbounded domains, epigraph or $\mathbb{R}^{n}$. In particular, the uniformly elliptic nonlocal Monge-Ampère operator introduced by Caffarelli and Charro in \cite{CC} is a typical example of the uniformly elliptic nonlocal Bellman operator.
In this paper we provide a rate of convergence for periodic homogenization of Hamilton-Jacobi-Bellman equations with nonlocal diffusion. The result is based on the regularity of the associated effective problem, where the convexity plays a crucial role. Such regularity estimates are possible from the available literature once we provide a representation formula for the effective Hamiltonian, a result that has an independent interest.
We show that the rate of convergence of solutions of finite-difference approximations for uniformly elliptic Bellman's equations is of order at least $h^{2/3}$, where $h$ is the mesh size. The equations are considered in smooth bounded domains.
We prove a stochastic representation formula for the viscosity solution of Dirichlet terminal-boundary value problem for a degenerate Hamilton-Jacobi-Bellman integro-partial differential equation in a bounded domain. We show that the unique viscosity solution is the value function of the associated stochastic optimal control problem. We also obtain the dynamic programming principle for the associated stochastic optimal control problem in a bounded domain.
We generalize the Donsker-Varadhan minimax formula for the principal eigenvalue of a uniformly elliptic operator in nondivergence form to the first principal half-eigenvalue of a fully nonlinear operator which is concave (or convex) and positively homogeneous. Examples of such operators include the Hamilon-Jacobi-Bellman operator and the Pucci extremal operators. In the case that the two principal half-eigenvalues are not equal, we show that the measures which achieve the minimum in this formula provide a partial characterization of the solvability of the corresponding Dirichlet problem at res…
In this paper we establish periodic homogenization for Hamilton-Jacobi-Bellman (HJB) equations, associated to nonlocal operators of integro-differential type. We consider the case when the fractional diffusion has the same order as the drift term, and is weakly elliptic. The outcome of the paper is twofold. One one hand, we provide Lipschitz regularity results for weakly elliptic non-local HJB, extending the results previously obtained in [8]. On the other hand, we establish a convergence result, based on half relaxed limits and a comparison principle for the effective problem. The latter stro…
We study the well-posedness of a class of nonlocal-interaction equations on general domains $Ω\subset \mathbb{R}^d$, including nonconvex ones. We show that under mild assumptions on the regularity of domains (uniform prox-regularity), for $λ$-geodesically convex interaction and external potentials, the nonlocal-interaction equations have unique weak measure solutions. Moreover, we show quantitative estimates on the stability of solutions which quantify the interplay of the geometry of the domain and the convexity of the energy. We use these results to investigate on which domains and for which…
We derive the long time asymptotic of solutions to an evolutive Hamilton-Jacobi-Bellman equation in a bounded smooth domain, in connection with ergodic problems recently studied in \cite{bcr}. Our main assumption is an appropriate degeneracy condition on the operator at the boundary. This condition is related to the characteristic boundary points for linear operators as well as to the irrelevant points for the generalized Dirichlet problem, and implies in particular that no boundary datum has to be imposed. We prove that there exists a constant $c$ such that the solutions of the evolutive prob…
C^(sigma+alpha) regularity for fractional Hamilton-Jacobi equations, and rates of convergence in periodic homogenization
Université Paris Cité, CNRS, Sorbonne Université, Laboratoire Jacques-Louis Lions (LJLL)（巴黎西岱大学，法国国家科学研究中心，索邦大学，雅克-路易·利翁斯实验室）；O. Mayer Mathematics Institute, Romanian Academy - Iaşi Branch（罗马尼亚科学院雅西分院奥·迈尔数学研究所）；Departamento de Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción（康塞普西翁大学物理与数学院数学系）；Instituto de Matemática, Universidade Federal do Rio de Janeiro（里约热内卢联邦大学数学研究所）
In this paper we prove a nonlocal version of the celebrated Inverse Problem of Donsker and Varadhan~\cite{DV} for nonlocal elliptic operators of the form $$ \cL u = L_K u + \cB_K (h,u), $$ where $L_K$ is a uniformly elliptic nonlocal operator with smooth coefficients, and, for $h:\R^N \to \R$ smooth and bounded, $\cB_K(h,u)$ is the associated bilinear form, which can be regarded as a nonlocal transport term.
The present paper is devoted to the investigation of various properties of transition fronts in nonlocal equations in heterogeneous media of ignition type, whose existence has been established by the authors of the present paper in a previous work. It is first shown that the transition front is continuously differentiable in space with uniformly bounded and uniformly Lipschitz continuous space partial derivative. This is the first time that regularity of transition fronts in nonlocal equations is ever studied. It is then shown that the transition front is uniformly steep. Finally, asymptotic s…
We prove a $C^{1,α}$ interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to linear theory and higher regularity of a special class of nonlinear operators.
We study the well-posedness of Hamilton-Jacobi-Bellman equations on subsets of $\mathbb{R}^d$ in a context without boundary conditions. The Hamiltonian is given as the supremum over two parts: an internal Hamiltonian depending on an external control variable and a cost functional penalizing the control. The key feature in this paper is that the control function can be unbounded and discontinuous. This way we can treat functionals that appear e.g. in the Donsker-Varadhan theory of large deviations for occupation time measures. To allow for this flexibility, we assume that the internal Hamiltoni…
Let $G$ be a Carnot group. We study nonlocal diffusion equations in a domain $Ω$ of $G$ of the form $$ u_t^ε(x,t)=\int_{G}\frac{1}{ε^2}K_ε(x,y)(u^ε(y,t)-u^ε(x,t))\,dy, \qquad x\in Ω$$ with $u^ε=g(x,t)$ for $x\notinΩ$. For appropriate rescaled kernel $K_ε$ we prove that solutions $u^ε$, when $ε\rightarrow0$, uniformly approximate the solution of different local Dirichlet problem in $G$. The key tool used is the Taylor series development for a function defined on a Carnot group.
This paper deals with a parabolic-elliptic chemotaxis system with nonlocal type of source in the whole space. It's proved that the initial value problem possesses a unique global solution which is uniformly bounded. Here we identify the exponents regimes of nonlinear reaction and aggregation in such a way that their scaling and the diffusion term coincide (see Introduction). Comparing to the classical KS model (without the source term), it's shown that how energy estimates give natural conditions on the nonlinearities implying the absence of blow-up for the solution without any restriction on …
We study Mean Field Games (MFGs) driven by a large class of nonlocal, fractional and anomalous diffusions in the whole space. These non-Gaussian diffusions are pure jump Lévy processes with some $σ$-stable like behaviour. Included are $σ$-stable processes and fractional Laplace diffusion operators $(-Δ)^{\fracσ2}$, tempered nonsymmetric processes in Finance, spectrally one-sided processes, and sums of subelliptic operators of different orders. Our main results are existence and uniqueness of classical solutions of MFG systems with nondegenerate diffusion operators of order $σ\in(1,2)$. We cons…
A nonlinear Schrödinger equation (NLS) with dispersion averaged nonlinearity of saturated type is considered. Such a nonlocal NLS is of integro-differential type and it arises naturally in modeling fiber-optics communication systems with periodically varying dispersion profile (dispersion management). The associated constrained variational principle is shown to posses a ground state solution by constructing a convergent minimizing sequence through the application of a method similar to the classical concentration compactness principle of Lions. One of the obstacles in applying this variational…
In this note we give an elementary proof of the space-like real analyticity of solutions to a degenerate evolution problem that arises in the study of fractional parabolic operators of the type $(\partial_t - div_x(B(x)\nabla_x))^s$, $0<s<1$. Our primary interest is in the so-called \emph{extension variable}. We show that weak solutions that are even in such variable, are in fact real-analytic in the totality of the space variables. As an application of this result we prove the weak unique continuation property for nonlocal parabolic operators of the type above, where $B(x)$ is a uniformly ell…
In this work, we give a characterization of Lipschitz operators on spaces of $C^2(M)$ functions (also $C^{1,1}$, $C^{1,γ}$, $C^1$, $C^γ$) that obey the global comparison property-- i.e. those that preserve the global ordering of input functions at any points where their graphs may touch, often called "elliptic" operators. Here $M$ is a complete Riemannian manifold. In particular, we show that all such operators can be written as a min-max over linear operators that are a combination of drift-diffusion and integro-differential parts. In the \emph{linear} (and nonlocal) case, Courrège had charac…
In his celebrated article, Aronson established Gaussian bounds for the fundamental solution to the Cauchy problem governed by a second order divergence form operator with uniformly elliptic coefficients. We extend Aronson's proof of upper heat kernel estimates to nonlocal operators whose jumping kernel satisfies a pointwise upper bound and whose energy form is coercive. A detailed proof is given in the Euclidean space and extensions to doubling metric measure spaces are discussed.
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This paper provides a quantitative study of nonnegative solutions to nonlinear diffusion equations of porous medium-type of the form $\partial_t u + {\mathcal L}u^m=0$, $m>1$, where the operator ${\mathcal L}$ belongs to a general class of linear operators, and the equation is posed in a bounded domain $Ω\subset{\mathbb R}^N$. As possible operators we include the three most common definitions of the fractional Laplacian in a bounded domain with zero Dirichlet conditions, and also a number of other nonlocal versions. In particular, ${\mathcal L}$ can be a power of a uniformly elliptic operator …
In this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem}(or cell problem), i.e. we construct solutions of the form $λt + v(x)$. We then prove that solutions of the Cauchy problem look like those specific solutions as time goes to infinity. We face two key difficulties to carry out this classical program: (i) the fact that we handle the case of "mixed operators" for which the required ellipticity comes from a combination of the properties of th…
A nonlocal-to-local transfer principle for parabolic Calderón problems
National Cheng Kung University（国立成功大学）；National Yang Ming Chiao Tung University（国立阳明交通大学）；University of Washington（华盛顿大学）；Institute for Advanced Study, the Hong Kong University of Science and Technology（香港科技大学高等研究院）
In this paper, we prove the local-controllability to positive constant trajectories of a nonlinear system of two coupled ODE equations, posed in the one-dimensional spatial setting, with nonlocal spatial nonlinearites, and using only one localized control with a moving support. The model we deal with is derived from the well-known nonlinear reaction-diffusion Gray-Scott model when the diffusion coefficient of the first chemical species $d_u$ tends to $0$ and the diffusion coefficient of the second chemical species ${d_v}$ tends to $+ \infty$. The strategy of the proof consists in two main step…
Let $D$ be a bounded $C^2$-domain. Consider the following Dirichlet initial-boundary problem of nonlocal operators with a drift: $$ \partial_t u={\mathscr L}^{(α)}_κu+b\cdot \nabla u+f\ \mathrm{in}\ \mathbb R_+\times D,\ \ u|_{\mathbb R_+\times D^c}=0,\ u(0,\cdot)|_{D}=φ, $$ where $α\in(0,2)$ and $\mathscr L^{(α)}_κ$ is an $α$-stable-like nonlocal operator with kernel function $κ(x,z)$ bounded from above and below by positive constants, and $b:\mathbb R^d\to\mathbb R^d$ is a bounded $C^β$-function with $α+β>1$, $f: \mathbb R_+\times D\to\mathbb R$ is a $C^γ$-function in $D$ uniformly in $t$ wi…
