<p>We investigate positivity sets of nonnegative supersolutions of the fully nonlinear elliptic equations <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis x comma u comma upper D u comma upper D squared u right-parenthesis equals 0">
<mml:semantics>
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>x</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>u</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>D</mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mi>D</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mi>u</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
<mml:annotation encoding="application/x-tex">F(x,u,Du,D^2u)=0</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega">
<mml:semantics>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:annotation encoding="application/x-tex">\Omega</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, where <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega">
<mml:semantics>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:annotation encoding="application/x-tex">\Omega</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> is an open subset of <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript upper N">
<mml:semantics>
<mml:msup>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="double-struck">R</mml:mi>
</mml:mrow>
<mml:mi>N</mml:mi>
</mml:msup>
<mml:annotation encoding="application/x-tex">\mathbb {R}^N</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, and the validity of the strong maximum principle for <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F left-parenthesis x comma u comma upper D u comma upper D squared u right-parenthesis equals f">
<mml:semantics>
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>x</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>u</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>D</mml:mi>
<mml:mi>u</mml:mi>
<mml:mo>,</mml:mo>
<mml:msup>
<mml:mi>D</mml:mi>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mi>u</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mi>f</mml:mi>
</mml:mrow>
<mml:annotation encoding="application/x-tex">F(x,u,Du,D^2u)=f</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> in <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega">
<mml:semantics>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:annotation encoding="application/x-tex">\Omega</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>, with <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f element-of normal upper C left-parenthesis normal upper Omega right-parenthesis">
<mml:semantics>
<mml:mrow>
<mml:mi>f</mml:mi>
<mml:mo>∈<!-- ∈ --></mml:mo>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:mi mathvariant="normal">C</mml:mi>
</mml:mrow>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:mo stretchy="false">)</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">f\in \mathrm {C}(\Omega )</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> being nonpositive. We obtain geometric characterizations of positivity sets <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet x element-of normal upper Omega colon u left-parenthesis x right-parenthesis greater-than 0 EndSet">
<mml:semantics>
<mml:mrow>
<mml:mo fence="false" stretchy="false">{</mml:mo>
<mml:mi>x</mml:mi>
<mml:mo>∈<!-- ∈ --></mml:mo>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:mspace width="thinmathspace" />
<mml:mo>:</mml:mo>
<mml:mspace width="thinmathspace" />
<mml:mi>u</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>x</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>&gt;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo fence="false" stretchy="false">}</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\{x\in \Omega \,:\, u(x)&gt;0\}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> of nonnegative supersolutions <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u">
<mml:semantics>
<mml:mi>u</mml:mi>
<mml:annotation encoding="application/x-tex">u</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> and establish the strong maximum principle under some geometric assumption on the set <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartSet x element-of normal upper Omega colon f left-parenthesis x right-parenthesis equals 0 EndSet">
<mml:semantics>
<mml:mrow>
<mml:mo fence="false" stretchy="false">{</mml:mo>
<mml:mi>x</mml:mi>
<mml:mo>∈<!-- ∈ --></mml:mo>
<mml:mi mathvariant="normal">Ω<!-- Ω --></mml:mi>
<mml:mspace width="thinmathspace" />
<mml:mo>:</mml:mo>
<mml:mspace width="thinmathspace" />
<mml:mi>f</mml:mi>
<mml:mo stretchy="false">(</mml:mo>
<mml:mi>x</mml:mi>
<mml:mo stretchy="false">)</mml:mo>
<mml:mo>=</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo fence="false" stretchy="false">}</mml:mo>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\{x\in \Omega \,:\, f(x)=0\}</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>.</p>

In this paper we investigate the validity and the consequences of the maximum principle for degenerate elliptic operators whose higher order term is the sum of k eigenvalues of the Hessian. In particular we shed some light on some very unusual phenomena due to the degeneracy of the operator. We prove moreover Lipschitz regularity results and boundary estimates under convexity assumptions on the domain. As a consequence we obtain the existence of solutions of the Dirichlet problem and of principal eigenfunctions.

We study two asymptotic problems for the Langevin equation with variable friction coefficient. The first is the small mass asymptotic behavior, known as the Smoluchowski-Kramers approximation, of the Langevin equation with strictly positive variable friction. The second result is about the limiting behavior of the solution when the friction vanishes in regions of the domain. Previous works on this subject considered one dimensional settings with the conclusions based on explicit computations.

In [17] (Part 1 of this series), we have introduced a variational approach to studying the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations in a torus. We develop this approach further here to handle boundary value problems. In particular, we establish new representation formulas for solutions of discount problems, critical values, and use them to prove convergence results for the vanishing discount problems. (C) 2016 Elsevier Masson SAS. All rights reserved.

We develop a variational approach to the vanishing discount problem for fully nonlinear, degenerate elliptic, partial differential equations. Under mild assumptions, we introduce viscosity Mather measures for such partial differential equations, which are natural extensions of the Mather measures. Using the viscosity Mather measures, we prove that the whole family of solutions v(lambda) of the discount problem with the factor lambda &gt; 0 converges to a solution of the ergodic problem as lambda -&gt; 0. (C) 2016 Elsevier Masson SAS. All rights reserved.

Motivated by a control problem of a certain queueing network we consider a control problem where the dynamics is constrained in the nonnegative orthant R-+(d) of the d-dimensional Euclidean space and controlled by the reflections at the faces/boundaries. We define a discounted value function associated to this problem and show that the value function is a viscosity solution to a certain HJB equation in R-+(d) with nonlinear Neumann type boundary condition. Under certain conditions, we also characterize this value function as the unique solution to this HJB equation.

This is the second part of our work on metastability results for parabolic equations with drift. The aim is to present a self-contained study, using partial differential equations methods, of the metastability properties of the solutions to quasi-linear parabolic equations with drift, and to obtain results similar to those in Freidlin and Koralov [6, 8].

We consider the ergodic (or additive eigenvalue) problem for the Neumann- type boundary-value problem for Hamilton-Jacobi equations and the corresponding discounted problems. Denoting by u(lambda) the solution of the discounted problem with discount factor lambda &gt; 0, we establish the convergence of the whole family {u(lambda)} lambda &gt; 0 to a solution of the ergodic problem as.. 0, and give a representation formula for the limit function via the Mather measures and Peierls function. As an interesting by-product, we introduce Mather measures associated with Hamilton-Jacobi equations with the Neumann- type boundary conditions. These results are variants of the main results in a recent paper by Davini et al., who study the same convergence problem on smooth compact manifolds without boundary.

This is a continuation of Ikoma and Ishii (Ann Inst H Poincar, Anal Non Lin,aire 29:783-812, 2012) and we study the eigenvalue problem for fully nonlinear elliptic operators, positively homogeneous of degree one, on finite intervals or balls. In the multi-dimensional case, we consider only radial eigenpairs. Our eigenvalue problem has a general first-order boundary condition which includes, as a special case, the Dirichlet, Neumann and Robin boundary conditions. Given a nonnegative integer n, we prove the existence and uniqueness, modulo multiplication of the eigenfunction by a positive constant, of an eigenpair whose eigenfunction, as a radial function in the multi-dimensional case, has exactly n zeroes. When an eigenfunction has n zeroes, we call the corresponding eigenvalue of nth order. Furthermore, we establish results concerning comparison of two eigenvalues, characterizations of nth order eigenvalues via differential inequalities, the maximum principle for the boundary value problem in connection with the principal eigenvalue, and existence of a solution having n zeroes, as a radial function in the multi-dimensional case, of the boundary value problem with an inhomogeneous term.

We study the exponentially long-time behavior of solutions to linear uniformly parabolic equations that are small perturbations of transport equations with vector fields having a globally stable (attractive) equilibrium in the domain. The result is that the solutions converge to a constant, which is either the initial value at the stable point or the boundary value at the minimum of the associated quasi-potential. Problems of this type were considered by Freidlin and Wentzell and Freidlin and Koralov, using probabilistic arguments related to the theory of large deviations. Our approach, which is self-contained, relies entirely on pde arguments, and is flexible to the extent that allows us to study a class of semilinear equations of similar structure. This note also prepares the ground for the forthcoming Part II of this work where we consider general quasilinear problems.

We study the dynamical boundary value problem for Hamilton-Jacobi equations of the eikonal type with a small parameter. We establish two results concerning the asymptotic behavior of solutions of the Hamilton-Jacobi equations: one concerns with the convergence of solutions as the parameter goes to zero and the other with the large-time asymptotics of solutions of the limit equation. (C) 2014 The Authors. Mathematische Nachrichten published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim.

We introduce a new PDE approach to establishing the large time asymptotic behavior of solutions of Hamilton-Jacobi equations, which modifies and simplifies the previous ones (Barles et al. in Arch Ration Mech Anal 204(2):515-558, 2012; Barles and Souganidis in SIAM J Math Anal 31(4):925-939, 2000), under a refined "strict convexity" assumption on the Hamiltonians. Not only such "strict convexity" conditions generalize the corresponding requirements on the Hamiltonians in Barles and Souganidis (SLAM J Math Anal 31(4):925-939, 2000), but also one of the most refined our conditions covers the situation studied in Namah and Roquejoffre (Commun Partial Differ Equ 24(5-6):883-893, 1999).

We present an introduction to the theory of viscosity solutions of first-order partial differential equations and a review on the optimal control/dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations, with the Neumann boundary condition. This article also includes some of basics of mathematical analysis related to the optimal control/dynamical approach for easy accessibility to the topics.

We study the eigenvalue problem for positively homogeneous, of degree one, elliptic ODE on finite intervals and PDE on balls. We establish the existence and completeness results for principal and higher eigenpairs, i.e., pairs of an eigenvalue and its corresponding eigenfunction. (c) 2012 Elsevier Masson SAS. All rights reserved.

In this paper, we consider minimization formulas which arise typically in optimal control and weak KAM theory for Hamilton-Jacobi equations. Given a minimization formula, we define a uniqueness set for the formula, which replaces the original region of minimization without changing its values. Our goal is to provide a necessary and sufficient condition that a given set be a uniqueness set. We also provide a characterization of the existence of a minimal uniqueness set with respect to set inclusion.

In this article, we study the large time behavior of solutions of first-order Hamilton-Jacobi Equations set in a bounded domain with nonlinear Neumann boundary conditions, including the case of dynamical boundary conditions. We establish general convergence results for viscosity solutions of these Cauchy-Neumann problems by using two fairly different methods: the first one relies only on partial differential equations methods, which provides results even when the Hamiltonians are not convex, and the second one is an optimal control/dynamical system approach, named the "weak KAM approach", which requires the convexity of Hamiltonians and gives formulas for asymptotic solutions based on Aubry-Mather sets.

In this note we present a unified approach, based on pde methods, for the study of averaging principles for (small) stochastic perturbations of Hamiltonian flows in two space dimensions. Such problems were introduced by Freidlin and Wentzell and have been the subject of extensive study in the last few years using probabilistic arguments. When the Hamiltonian flow has critical points, it exhibits complicated behavior near the critical points under a small stochastic perturbation. Asymptotically the slow (averaged) motion takes place on a graph. The issues are to identify both the equations on the sides and the boundary conditions at the vertices of the graph. Our approach is very general and applies also to degenerate anisotropic elliptic operators which could not be considered using the previous methodology. (C) 2011 Elsevier Inc. All rights reserved.

We study the long-time asymptotic behavior of solutions u of the Hamilton-Jacobi equation u(tau)(x, t) + H(x, Du(x, t)) = 0 in Omega x (0, infinity), where Omega is a bounded open subset of R(n), with Hamiltonian H = H(x, p) being convex and coercive in p, and establish the uniform convergence of u to an asymptotic solution as t -&gt; infinity.

We study convex Hamilton-Jacobi equations H(x, Du) = 0 and u(t) + H(x, Du) = 0 in a bounded domain Omega of R-n with the Neumann type boundary condition D(gamma)u = g in the viewpoint of weak KAM theory, where gamma is a vector field on the boundary partial derivative Omega pointing a direction oblique to partial derivative Omega. We establish the stability under the formations of infimum and of convex combinations of subsolutions of convex Hamilton-Jacobi equations, some comparison and existence results for convex and coercive Hamilton-Jacobi equations with the Neumann type boundary condition as well as existence results for the Skorokhod problem. We define the Aubry set associated with the Neumann type boundary problem and establish some properties of the Aubry set including the existence results for the "calibrated" extremals for the corresponding action functional (or variational problem). (C) 2010 Elsevier Masson SAS. All rights reserved.

Let Omega subset of R(n) be a bounded domain, and let 1 &lt; p &lt; 8 and sigma &lt; p. We study the nonlinear singular integral equation
M[u](x) = f(0)(x) in Omega
with the boundary condition u = g(0) on partial derivative Omega, where f(0) is an element of C(&lt;(Omega)over bar&gt;) and g(0) is an element of C(partial derivative Omega) are given functions and M is the singular integral operator given by
M[u](x) = p. v. integral(B(0, rho(x))) p - sigma/vertical bar z vertical bar(n+sigma) vertical bar u(x + z) - u(x)vertical bar(p) (2)(u(x + z) - u(x)) dz,
with some choice of rho is an element of C(Omega) having the property, 0 &lt; rho(x) &lt;= dist (x, partial derivative Omega). We establish the solvability (well-posedness) of this Dirichlet problem and the convergence uniform on &lt;(Omega)over bar&gt; as sigma -&gt; p, of the solution us of the Dirichlet problem to the solution u of the Dirichlet problem for the p-Laplace equation nu Delta(p)u = f(0) in Omega with the Dirichlet condition u = g(0) on partial derivative Omega, where the factor nu is a positive constant (see (7.2)).

We investigate the long-time behavior of viscosity solutions of Hamilton-Jacobi equations in R(n) with convex and coercive Hamiltonians and give three general criteria for the convergence of solutions to asymptotic solutions as time goes to infinity. We apply the criteria to obtain more specific sufficient conditions for the convergence to asymptotic solutions and then examine them with examples. We take a dynamical approach, based on tools from weak KAM theory such as extremal curves, Aubry sets and representation formulas for solutions, for these investigations.

We show firstly the equivalence between existence of a periodic solution of the Hamilton-Jacobi equation 'formula presented' is a bounded domain of 'formula presented', with the Dirichlet boundary condition 'formula presented' and that of a subsolution of the stationary problem 'formula presented' under the assumptions that the function 'formula presented' is periodic in t and H is coercive. Here 'formula presented' denotes the average of f over the period. This proposition is a variant of a recent result for 'formula presented' due to Bostan-Namah, and we give a different and simpler approach to such an equivalence. Secondly, we establish that any periodic solution u(x, t) of the problem, ut + H(x, Du) = 0 in 'formula presented' and 'formula presented', is constant in t on the Aubry set for H. Here H is assumed to be convex, coercive and strictly convex in a sense.

We discuss the recent developments related to the large-time asymptotic behavior of solutions of Hamilton-Jacobi equations.

We study the large time behavior of solutions of the Cauchy problem for the Hamilton-Jacobi equation u(t) + H(x, Du) = 0 in R-n x (0, infinity), where H(x, p) is continuous on R-n x R-n and convex in p. We establish a general convergence result for viscosity solutions u(x, t) of the Cauchy problem as t -&gt; infinity . (C) 2007 Elsevier Masson SAS. All rights reserved.

We study the long time behavior of viscosity solutions of the Cauchy problem for Hamilton-Jacobi equations in n. We prove that if the Hamiltonian H(x, p) is coercive and strictly convex in a mild sense in p and upper semi-periodic in x, then any solution of the Cauchy problem "converges" to an asymptotic solution for any lower semi-almost periodic initial function.

We establish general representation formulas for solutions of Hamilton-Jacobi equations with convex Hamiltonians. In order to treat representation formulas on general domains, we introduce a notion of ideal boundary similar to the Martin boundary [21] in potential theory. We apply such representation formulas to investigate maximal solutions, in certain classes of functions, of Hamilton-Jacobi equations. Part of the results in this paper has been announced in [22].

We study the long time behavior of solutions of the Cauchy problem for semilinear parabolic equations with the Ornstein-Uhlenbeck operator in R-N . The long time behavior in the main results is stated with help of the corresponding to ergodic problem, which complements, in the case of unbounded domains, the recent developments on long time behaviors of solutions of (viscous) Hamilton-Jacobi equations due to Namah (1996), Namah and Roquejoffre (1999), Roquejoffre (1998), Fathi (1998), Barles and Souganidis (2000, 2001). We also establish existence and uniqueness results for solutions of the Cauchy problem and ergodic problem for semilinear parabolic equations with the Ornstein-Uhlenbeck operator.

We study the asymptotic behavior of the viscosity solution of the Cauchy problem for the Hamilton-Jacobi equation ut + alpha x (.) Du + H(Du) = f(x) in R-n X (0, infinity), where alpha is a positive constant and H is a convex function on Rn, and establish a convergence result for the viscosity solution u(x, t) as t -&gt; infinity.

We construct a discrete stochastic approximation of a convexified Gauss curvature flow of boundaries of bounded open sets in an anisotropic external field. We also show that a weak solution to the PDE which describes the motion of a bounded open set is unique and is a viscosity solution of it. © 2004 Society for Industrial and Applied Mathematics.

We show that the convergence, as p --&gt; infinity, of the solution u(p) of the Dirichlet problem for -Delta p(u)(x) = f(x) in a bounded domain Omega subset of R-n with zero-Dirichlet boundary condition and with continuous f in the following cases: (i) one-dimensional case, radial cases; (ii) the case of no balanced family; and (iii) two cases with vanishing integral. We also give some properties of the maximizers for the functional integral(Omega) f(x)v(x) dx in the space of functions v is an element of C((Omega) over bar) boolean AND W-1,W-infinity (Omega) satisfying v\(theta Omega) = 0 and parallel to Dv parallel to(L infinity(Omega)) &lt;= 1.

We establish comparison and existence theorems of viscosity solutions of the initial-boundary value problem for some singular degenerate parabolic partial differential equations with nonlinear oblique derivative boundary conditions. The theorems cover the capillary problem for the mean curvature flow equation and apply to more general Neumann-type boundary problems for parabolic equations in the level set approach to motion of hypersurfaces with velocity depending on the normal direction and curvature. (C) 2004 Elsevier Ltd. All rights reserved.

We establish comparison and existence theorems of viscosity solutions of the initial-boundary value problem for some singular degenerate parabolic partial differential equations with nonlinear oblique derivative boundary conditions. The theorems cover the capillary problem for the mean curvature flow equation and apply to more general Neumann-type boundary problems for parabolic equations in the level set approach to motion of hypersurfaces with velocity depending on the normal direction and curvature. (C) 2004 Elsevier Ltd. All rights reserved.

We show the existence of weak solutions to the partial differential equation which describes the motion by R-curvature in R-d, by the continuum limit of a class of infinite particle systems. We also show that weak solutions of the partial differential equation are viscosity solutions and give the uniqueness result on both weak and viscosity solutions.

We construct a discrete stochastic approximation of a convexified Gauss curvature flow of boundaries of bounded open sets in an anisotropic external field. We also show that a weak solution to the PDE which describes the motion of a bounded open set is unique and is a viscosity solution of it.

We study the geometric evolution of a nonconvex stone by the wearing process via the partial differential equation methods. We use the so-called level set approach to this geometric evolution of a set. We establish a comparison theorem, an existence theorem, and some stability properties of solutions of the partial differential equation arising in this level set approach, and define the flow of a set by the wearing process via the level set approach.

We show the existence of weak solutions to the partial differential equation which describes the motion by R-curvature in R-d, by the continuum limit of a class of infinite particle systems. We also show that weak solutions of the partial differential equation are viscosity solutions and give the uniqueness result on both weak and viscosity solutions.

We study the geometric evolution of a nonconvex stone by the wearing process via the partial differential equation methods. We use the so-called level set approach to this geometric evolution of a set. We establish a comparison theorem, an existence theorem, and some stability properties of solutions of the partial differential equation arising in this level set approach, and define the flow of a set by the wearing process via the level set approach.

We study the relaxation of Hamilton-Jacobi equations. The relaxation in our terminology is the following phenomenon: the pointwise supremum over a certain collection of subsolutions, in the almost everywhere sense, of a Hamilton-Jacobi equation yields a viscosity solution of the ``convexified'' Hamilton-Jacobi equation. This phenomenon has recently been observed in [13] in eikonal equations. We show in this paper that this relaxation is a common phenomenon for a wide range of Hamilton-Jacobi equations.

We study the relaxation of Hamilton-Jacobi equations. The relaxation in our terminology is the following phenomenon: the pointwise supremum over a certain collection of subsolutions, in the almost everywhere sense, of a Hamilton-Jacobi equation yields a viscosity solution of the ``convexified'' Hamilton-Jacobi equation. This phenomenon has recently been observed in [13] in eikonal equations. We show in this paper that this relaxation is a common phenomenon for a wide range of Hamilton-Jacobi equations.

Let Omega be an open bounded subset of Rn and f a continuous function on Omega satisfying f(x) &gt; 0 for all x is an element of Omega. We consider the maximization problem for the integral fOmega f(x)u(x) dx over all Lipschitz continuous functions u subject to the Dirichlet boundary condition u = 0 on partial derivativeOmega and to the gradient constraint of the form H(Du(x)) less than or equal to 1, and prove that the supremum is 'achieved' by the viscosity solution of H(Du(x)) = 1 in Omega and u = 0 on partial derivativeOmega, where H denotes the convex envelope of H. This result is applied to an asymptotic problem, as p --&gt; infinity, for quasi-minimizers of the integral
integral(Omega)[1/p H(Du(x))(p) - f(x)u(x)] dx.
An asymptotic problem as k --&gt; infinity for
inf integral(Omega)[k dist(Du(x),K) - f(x)u(x)] dx
is also considered, where the infimum is taken all over u is an element of W-0(1,1)(Omega) and the set K is given by {xi \ H (xi) less than or equal to 1}.

Let Omega be an open bounded subset of Rn and f a continuous function on Omega satisfying f(x) &gt; 0 for all x is an element of Omega. We consider the maximization problem for the integral fOmega f(x)u(x) dx over all Lipschitz continuous functions u subject to the Dirichlet boundary condition u = 0 on partial derivativeOmega and to the gradient constraint of the form H(Du(x)) less than or equal to 1, and prove that the supremum is 'achieved' by the viscosity solution of H(Du(x)) = 1 in Omega and u = 0 on partial derivativeOmega, where H denotes the convex envelope of H. This result is applied to an asymptotic problem, as p --&gt; infinity, for quasi-minimizers of the integral
integral(Omega)[1/p H(Du(x))(p) - f(x)u(x)] dx.
An asymptotic problem as k --&gt; infinity for
inf integral(Omega)[k dist(Du(x),K) - f(x)u(x)] dx
is also considered, where the infimum is taken all over u is an element of W-0(1,1)(Omega) and the set K is given by {xi \ H (xi) less than or equal to 1}.

We study the asymptotic behavior of solutions of the Cauchy problem for a functional partial differential equation with a small parameter as the parameter tends to zero. We establish a convergence theorem in which the limit problem is identified with the Cauchy problem for a nonlinear parabolic partial differential equation. We also present comparison and existence results for the Cauchy problem for the functional partial differential equation and the limit problem.

A two-dimensional random crystalline algorithm was proposed for Gauss curvature flow. Gauss curvature flow of smooth closed convex hypersurfaces in Rd+1 was defined. A discrete-time approximation scheme for Gauss curvature flow was briefly described.

We propose and study a random crystalline algorithm (a discrete approximation) of the Gauss curvature flow of smooth simple closed convex curves in R-2 as a stepping stone to the full understanding of such phenomena as the wearing process of stones on a beach.

We investigate, via the dynamic programming approach, optimal control problems of infinite horizon with state constraint, where the state X-t is given as a solution of a controlled stochastic differential equation and the state constraint is described either by the condition that X-t is an element of (G) over bar for all t &gt; 0 or by the condition that X-t is an element of G for all t &gt; 0, where G be a given open subset of R-N. Under the assumption that for each z is an element of partial derivativeG there exists a, G A, where A denotes the control set, such that the diffusion matrix sigma (x, a) vanishes for a = a(z) and for x is an element of partial derivativeG in a neighborhood of z and the drift vector b(x, a) directs inside of G at z for a = az and x = z as well as some other mild assumptions, we establish the unique existence of a continuous viscosity solution of the state constraint problem for the associated Hamilton-jacobi-Bellman equation, prove that the value functions V associated with the constraint (G) over bar, V-r of the relaxed problem associated with the constraint (G) over bar, and V-o associated with the constraint G, satisfy in the viscosity sense the state constraint problem, and establish Holder regularity results for the viscosity solution of the state constraint problem.

We investigate, via the dynamic programming approach, optimal control problems of infinite horizon with state constraint, where the state X-t is given as a solution of a controlled stochastic differential equation and the state constraint is described either by the condition that X-t is an element of (G) over bar for all t &gt; 0 or by the condition that X-t is an element of G for all t &gt; 0, where G be a given open subset of R-N. Under the assumption that for each z is an element of partial derivativeG there exists a, G A, where A denotes the control set, such that the diffusion matrix sigma (x, a) vanishes for a = a(z) and for x is an element of partial derivativeG in a neighborhood of z and the drift vector b(x, a) directs inside of G at z for a = az and x = z as well as some other mild assumptions, we establish the unique existence of a continuous viscosity solution of the state constraint problem for the associated Hamilton-jacobi-Bellman equation, prove that the value functions V associated with the constraint (G) over bar, V-r of the relaxed problem associated with the constraint (G) over bar, and V-o associated with the constraint G, satisfy in the viscosity sense the state constraint problem, and establish Holder regularity results for the viscosity solution of the state constraint problem.

We formulate the wearing process of a nonconvex stone in terms of partial differential equations (PDEs). We establish a comparison theorem, an existence theorem, and some stability properties of solutions of this PDE.

We show that the algorithm considered by Ishii [GAKUTO Internat. Ser. Math. Sci. Appl. 5, Gakkotosho, Tokyo, 1995, pp. 111-127] and Ishii, Pires, and Souganidis [J. Math. Soc. Japan, 50 (1999), pp. 267-308] can be applied to motion by mean curvature with right-angle boundary condition.

We show that the algorithm considered by Ishii [GAKUTO Internat. Ser. Math. Sci. Appl. 5, Gakkotosho, Tokyo, 1995, pp. 111-127] and Ishii, Pires, and Souganidis [J. Math. Soc. Japan, 50 (1999), pp. 267-308] can be applied to motion by mean curvature with right-angle boundary condition.

The paper proves that the Dirichlet problem for the first-order Hamilton-Jacobi equation in an open subset of R-n
H (x, u, D(x ')u) = 0 in Omega, u = g on partial derivative Omega,
where D(x ')u is the partial gradient of the scalar function u with respect to the first n ' variables (n ' less than or equal to n), has a viscosity solution which is unique a.e. When applied to the periodic homogenization of Hamilton-Jacobi equations in a perforated set, the result yields the a.e. convergence of the solutions of the problem at scale epsilon as epsilon --&gt; 0 to the solution of the homogenized Hamilton-Jacobi equation.

We extend and clarify some of observations due to Barren and Jensen concerning the relation between subdifferentials and superdifferentials of a function and extend the comparison principle far semicontinuous solutions of Hamilton-Jacobi equations with convex Hamiltonians to that in infinite-dimensional Hilbert spaces.

The paper proves that the Dirichlet problem for the first-order Hamilton-Jacobi equation in an open subset of R-n
H (x, u, D(x ')u) = 0 in Omega, u = g on partial derivative Omega,
where D(x ')u is the partial gradient of the scalar function u with respect to the first n ' variables (n ' less than or equal to n), has a viscosity solution which is unique a.e. When applied to the periodic homogenization of Hamilton-Jacobi equations in a perforated set, the result yields the a.e. convergence of the solutions of the problem at scale epsilon as epsilon --&gt; 0 to the solution of the homogenized Hamilton-Jacobi equation.

We formulate the wearing process of a nonconvex stone in terms of partial differential equations (PDEs). We establish a comparison theorem, an existence theorem, and some stability properties of solutions of this PDE.

We give a characterization of the existence of bounded solutions for Hamilton-Jacobi equations in ergodic control problems with state-constraint. This result is applied to the reexamination of the counterexample given in [5] concerning the existence of solutions for ergodic control problems in infinite-dimensional Hilbert spaces and also establishing results on effective Hamiltonians in periodic homogenization of Hamilton-Jacobi equations.

We present a method of constructing E-optimal controls in the feedback form for state constraint problems.
Our method is as follows: We first find feedback laws directly from the associated Hamilton-Jacobi-Bellman equation and an approximation of the value function by the inf-convolution. We then construct piece-wise constant controls so that corresponding cost functionals approximate the value function of state constraint problems. (C) 2000 Editions scientifiques et medicales Elsevier SAS.

We study an invariance property for a controlled stochastic differential equation and give a few of its characterizations in connection with the corresponding Hamilton-Jacobi-Bellman equation.

We give a characterization of the existence of bounded solutions for Hamilton-Jacobi equations in ergodic control problems with state-constraint. This result is applied to the reexamination of the counterexample given in [5] concerning the existence of solutions for ergodic control problems in infinite-dimensional Hilbert spaces and also establishing results on effective Hamiltonians in periodic homogenization of Hamilton-Jacobi equations.

We present a method of constructing E-optimal controls in the feedback form for state constraint problems.
Our method is as follows: We first find feedback laws directly from the associated Hamilton-Jacobi-Bellman equation and an approximation of the value function by the inf-convolution. We then construct piece-wise constant controls so that corresponding cost functionals approximate the value function of state constraint problems. (C) 2000 Editions scientifiques et medicales Elsevier SAS.

We study an invariance property for a controlled stochastic differential equation and give a few of its characterizations in connection with the corresponding Hamilton-Jacobi-Bellman equation.

We study the convergence of general threshold dynamics type approximation schemes to hypersurfaces moving with normal velocity depending on the normal direction and the curvature tensor. We also present results about the asymptotic shape of fronts propagating by threshold dynamics. Our results generalize and extend models introduced in the theories of cellular automaton and motion by mean curvature.

We study the convergence of general threshold dynamics type approximation schemes to hypersurfaces moving with normal velocity depending on the normal direction and the curvature tensor. We also present results about the asymptotic shape of fronts propagating by threshold dynamics. Our results generalize and extend models introduced in the theories of cellular automaton and motion by mean curvature.

We provide a rigorous interpretation of the level set approach to certain nonlocal geometric motions modelling etching effects in manufacture. The shadowing of certain parts of a surface by other parts gives rise to a nonlocal Hamilton-Jacobi type PDE, with a multivalued Hamiltonian. We also show that deposition effects do not fall within the conventional level set framework, and accordingly must be reinterpreted for numerical implementation.

We provide a rigorous interpretation of the level set approach to certain nonlocal geometric motions modelling etching effects in manufacture. The shadowing of certain parts of a surface by other parts gives rise to a nonlocal Hamilton-Jacobi type PDE, with a multivalued Hamiltonian. We also show that deposition effects do not fall within the conventional level set framework, and accordingly must be reinterpreted for numerical implementation.

The first-order Hamilton-Jacobi-Bellman equation associated with the state constraint problem for optimal control is studied. Instead of the boundary condition which Soner introduced, a new and appropriate boundary condition for the PDE is proposed. The uniqueness and Lipschitz continuity of viscosity solutions for the boundary value problem are obtained.

In this note we study the generalized motion of noncompact hyper-surfaces with normal velocity depending on the normal direction and the curvature tensor. This work extends the by-now-classical works of Evans and Spruck (for mean curvature) and Chen, Giga and Goto (for general motions with sublinear curvature dependence), because it allows general dependence on the curvature tensor. It also allows a general treatment of the generalized evolution including noncompact hypersurfaces. A number of results regarding no interior, convexity, etc. are also presented.

In this note we study the generalized motion of noncompact hyper-surfaces with normal velocity depending on the normal direction and the curvature tensor. This work extends the by-now-classical works of Evans and Spruck (for mean curvature) and Chen, Giga and Goto (for general motions with sublinear curvature dependence), because it allows general dependence on the curvature tensor. It also allows a general treatment of the generalized evolution including noncompact hypersurfaces. A number of results regarding no interior, convexity, etc. are also presented.

We establish uniqueness or comparison results for a class of Hamilton-Jacobi equations and give characterizations of maximal solutions of Hamilton-Jacobi equations. The results are applied to characterizing value functions of exit time problems in optimal control.

We establish uniqueness or comparison results for a class of Hamilton-Jacobi equations and give characterizations of maximal solutions of Hamilton-Jacobi equations. The results are applied to characterizing value functions of exit time problems in optimal control.

In this paper we consider stochastic differential equations with reflecting boundary conditions for domains that might have corners and for which the allowed directions of reflection at a point on the boundary of the domain are possibly oblique. The main results are strong existence and uniqueness for solutions of such equations. A key ingredient is a family of relatively regular functions appropriate to the given domain and directions of reflection. Two cases are treated in the paper. In the first case the direction of reflection is single valued and varies smoothly, and the main new feature is that the boundary of the domain may be nonsmooth. In the second case the domain is taken to be the intersection of a finite number of domains with relatively smooth boundary, and at the resulting corner points more than one oblique direction is allowed.

In this paper we consider stochastic differential equations with reflecting boundary conditions for domains that might have corners and for which the allowed directions of reflection at a point on the boundary of the domain are possibly oblique. The main results are strong existence and uniqueness for solutions of such equations. A key ingredient is a family of relatively regular functions appropriate to the given domain and directions of reflection. Two cases are treated in the paper. In the first case the direction of reflection is single valued and varies smoothly, and the main new feature is that the boundary of the domain may be nonsmooth. In the second case the domain is taken to be the intersection of a finite number of domains with relatively smooth boundary, and at the resulting corner points more than one oblique direction is allowed.

A weak formulation for an interface dynamics coupled with a diffusion equation is introduced. A global-in-time weak solution is constructed for an arbitrary initial data under a periodic boundary condition. The result applies to the interface equation obtained as a certain singular limit of some reaction-diffusion systems including the activator-inhibitor model.

The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions.

A weak formulation for an interface dynamics coupled with a diffusion equation is introduced. A global-in-time weak solution is constructed for an arbitrary initial data under a periodic boundary condition. The result applies to the interface equation obtained as a certain singular limit of some reaction-diffusion systems including the activator-inhibitor model.

The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions.

We show the effectiveness of viscosity-solution methods in asymptotic problems for second-order elliptic partial differential equations (PDEs) with a small parameter. Our stress here is on the point that the methods, based on stability results [3], [16], apply without hard PDE calculations. We treat two examples from [11] and [23]. Moreover, we generalize the results to those for Hamilton-Jacobi-Bellman equations with a small parameter.

By an appropriate modification of the viscosity solution concept, we introduce a notion solution of a PDE that is applicable, among others, to the Bellman equation and first general classes of optimal control problems with the only restriction on a payoff functional that the stopping time is bounded by a fixed number T. We consider this PDE on the attainable set from OMEGA-0, a set Of given initial conditions. We prove both existence and uniqueness results for optimal control problems. The approach is illustrated with several examples and comments.

We show the effectiveness of viscosity-solution methods in asymptotic problems for second-order elliptic partial differential equations (PDEs) with a small parameter. Our stress here is on the point that the methods, based on stability results [3], [16], apply without hard PDE calculations. We treat two examples from [11] and [23]. Moreover, we generalize the results to those for Hamilton-Jacobi-Bellman equations with a small parameter.

By an appropriate modification of the viscosity solution concept, we introduce a notion solution of a PDE that is applicable, among others, to the Bellman equation and first general classes of optimal control problems with the only restriction on a payoff functional that the stopping time is bounded by a fixed number T. We consider this PDE on the attainable set from OMEGA-0, a set Of given initial conditions. We prove both existence and uniqueness results for optimal control problems. The approach is illustrated with several examples and comments.

We prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations. One of them extends some recent uniqueness results by Jensen. Some establish the uniqueness of solutions for second‐order Isaacs' equations and hence include the uniqueness results for Bellman equations by P.‐L. Lions. Our comparison results apply even for discontinuous solutions and so Perron's method readily yields the existence of continuous solutions. Copyright © 1989 Wiley Periodicals, Inc., A Wiley Company

We present a new, direct proof of the uniqueness theorem for a class of Hamilton-Jacobi equations including the eikonal equation in geometric optics. © 1987 American Mathematical Society.

Recently, Crandall and Lions introduced the notion of viscosity solution for Hamilton–Jacobi equations to settle the uniqueness problem of generalized solutions of Hamilton–Jacobi equations. The existence of viscosity solutions of Hamilton–Jacobi equations was established under the same hypotheses on the Hamiltonians as those for the uniqueness of viscosity solutions. Thus, the chapter presents the Hamiltonian as a max or max–min of linear functions of p, and proves the uniform continuity of the value function of the associated optimal control or differential game problem. © 1985, Elsevier Inc. All rights reserved.