Comparison Principle For Viscosity Solutions Research Articles

Large deviations for slow-fast processes on connected complete Riemannian manifolds

We consider a class of slow-fast processes on a connected complete Riemannian manifold M. The limiting dynamics as the scale separation goes to ∞ is governed by the averaging principle. Around this limit, we prove large deviation principles with an action-integral rate function for the slow process by nonlinear semigroup methods together with Hamilton–Jacobi-Bellman (HJB) equation techniques.Our main innovation is solving the comparison principle for viscosity solutions for the HJB equation on M and the construction of a variational viscosity solution for the non-smooth Hamiltonian, which lies at the heart of deriving the action integral representation for the rate function.

Hölder estimates for viscosity solutions of nonlocal equations with variable-order fractional Laplace term

We consider a class of linear integro-differential equations with variable-order fractional Laplacian. Under sharp assumptions on the variable-order α(x,y), we prove the local Hölder continuity of bounded viscosity solutions. In particular, for the continuous right-hand side f, we show that weak solutions are viscosity solutions. Furthermore, we prove a comparison principle for viscosity solutions when the function f is strictly away from zero.

A comparison principle for semilinear Hamilton–Jacobi–Bellman equations in the Wasserstein space

The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton–Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the 2-Wasserstein distance in the doubling of variables argument, which is done by introducing a further entropy penalization that ensures that the relevant optima are achieved at positive, Lipschitz continuous densities with finite Fischer information. This allows to prove uniqueness and stability of viscosity solutions in the class of bounded Lipschitz continuous (with respect to the 1-Wasserstein distance) functions. The result does not appeal to a mean field control formulation of the equation, and, as such, applies to equations with nonconvex Hamiltonians and measure-dependent volatility. For convex Hamiltonians that derive from a potential, we prove that the value function associated with a suitable mean-field optimal control problem with nondegenerate idiosyncratic noise is indeed the unique viscosity solution.

Comparison principles for viscosity solutions of elliptic branches of fully nonlinear equations independent of the gradient

The validity of the comparison principle in variable coefficient fully nonlinear gradient free potential theory is examined and then used to prove the comparison principle for fully nonlinear partial differential equations which determine a suitable potential theory. The approach combines the notions of proper elliptic branches inspired by Krylov ^{[<span class="xref"><a href="#b18" ref-type="bibr">18</a></span>]} with the <i>monotonicity-duality method</i> initiated by Harvey and Lawson ^{[<span class="xref"><a href="#b12" ref-type="bibr">12</a></span>]}. In the variable coefficient nonlinear potential theory, a special role is played by the <i>Hausdorff continuity of the proper elliptic map</i> Θ which defines the potential theory. In the applications to nonlinear equations defined by an operator <i>F</i>, structural conditions on <i>F</i> will be determined for which there is a correspondence principle between Θ-<i>subharmonics/superharmonics</i> and <i>admissible viscosity sub and supersolutions</i> of the nonlinear equation and for which comparison for the equation follows from the associated compatible potential theory. General results and explicit models of interest in differential geometry will be examined. Examples of improvements with respect to existing results on comparison principles will be given.

Coupling Lévy measures and comparison principles for viscosity solutions

We prove new comparison principles for viscosity solutions of nonlinear integro-differential equations. The operators to which the method applies include but are not limited to those of Lévy–Itô type. The main idea is to use an optimal transport map to couple two different Lévy measures and use the resulting coupling in a doubling of variables argument.

Bubbles in assets with finite life

We study the speculative value of a finitely lived asset when investors disagree and short sales are limited. In this case, investors are willing to pay a speculative value for the resale option they obtain when they acquire the asset. Using martingale arguments, we characterize the equilibrium speculative value as a solution to a fixed point problem for a monotone operator $$\mathbb F$$ . A Dynamic Programming Principle applies and is used to show that the minimal solution to the fixed-point problem is a viscosity solution of a naturally associated (non-local) obstacle problem. Combining the monotonicity of the operator $${\mathbb {F}}$$ and a comparison principle for viscosity solutions to the obstacle problem we obtain several comparison of solution results. We also use a characterization of the exercise boundary of the obstacle problem to study the effect of an increase in the costs of transactions on the value of the bubble and on the volume of trade, and in particular to quantify the effect of a small transaction (Tobin) tax.

Equivalence of weak and viscosity solutions to the $${\texttt {p}}(x)$$ p ( x ) -Laplacian in Carnot groups

We show the equivalence of weak and viscosity solutions to the $${\texttt {p}}(x)$$-Laplacian in Carnot groups, under certain reasonable restrictions on the function $${\texttt {p}}(x)$$ and on the viscosity solution. We also obtain a comparison principle for viscosity solutions. This then implies that viscosity solutions to Dirichlet problems are unique.

Approximation of General Facets by Regular Facets with Respect to Anisotropic Total Variation Energies and Its Application to Crystalline Mean Curvature Flow

AbstractWe show that every bounded subset of a euclidean space can be approximated by a set that admits a certain vector field, the so‐called Cahn‐Hoffman vector field, that is subordinate to a given anisotropic metric and has a square‐integrable divergence. More generally, we introduce a concept of facets as a kind of directed sets, and show that they can be approximated in a similar manner.We use this approximation to construct test functions necessary to prove the comparison principle for viscosity solutions of the level set formulation of the crystalline mean curvature flow that were recently introduced by the authors. As a consequence, we obtain the well‐posedness of the viscosity solutions in an arbitrary dimension, which extends the validity of the result in the previous paper.© 2018 Wiley Periodicals, Inc.

Remarks on the comparison principle for quasilinear PDE with no zeroth order terms

A comparison principle for viscosity solutions of second-order quasilinear elliptic partial di erential equations with no zeroth order terms is shown. A di erent transformation from that of Barles and Busca in [3] is adapted to enable us to deal with slightly more general equations.

The Dirichlet problem for fully nonlinear elliptic equations non-degenerate in a fixed direction

We prove a comparison principle for viscosity solutions of a fully nonlinear equation satisfying a condition of non-degeneracy in a fixed direction. We apply these results to prove that a continuous solution of the corresponding Dirichlet problem exists. To obtain the existence of barrier functions and well-posedness, we find suitable explicit assumptions on the domain and on the ellipticity constants of the operator.

A Comparison Principle for Hamilton–Jacobi Equations Related to Controlled Gradient Flows in Infinite Dimensions

We develop new comparison principles for viscosity solutions of Hamilton–Jacobi equations associated with controlled gradient flows in function spaces as well as the space of probability measures. Our examples are optimal control of Ginzburg–Landau and Fokker–Planck equations. They arise in limit considerations of externally forced non-equilibrium statistical mechanics models, or through the large deviation principle for interacting particle systems. Our approach is based on two key ingredients: an appropriate choice of geometric structure defining the gradient flow, and a free energy inequality resulting from such gradient flow structure. The approach allows us to handle Hamiltonians with singular state dependency in the nonlinear term, as well as Hamiltonians with a state space which does not satisfy the Radon–Nikodym property. In the case where the state space is a Hilbert space, the method simplifies existing theories by avoiding the perturbed optimization principle.

Comparison Principle for Viscosity Solutions of Fully Nonlinear, Degenerate Elliptic Equations

We provide structural assumptions under which comparison principles hold for viscosity solutions of fully nonlinear degenerate elliptic equations. These assumptions appear to be more traceable and easier to verify than previously known ones.

Generalized Cone Comparison Principle for Viscosity Solutions of the Aronsson Equation and Absolute Minimizers

For any non-negative, quasi-convex Hamiltonian H ∈ C 2(R n ), we consider the L ∞-functional F(u, ·) = ‖H(∇ u)‖ L ∞(·) over . We introduce the notion of comparison with generalized cones (abbreviated CGC) and prove that CGC, viscosity solutions of the Aronsson equation, and absolute minimizers of F(·) are equivalent. This extends an earlier result by Crandall et al. (2001).

On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations

We prove some comparison principles for viscosity solutions of fully nonlinear degenerate elliptic equations that satisfy some conditions of partial non-degeneracy instead of the usual uniform ellipticity or strict monotonicity. These results are applied to the well-posedness of the Dirichlet problem under suitable conditions at the characteristic points of the boundary. The examples motivating the theory are operators of the form of sum of squares of vector fields plus a nonlinear first order Hamiltonian and the Pucci operator over the Heisenberg group.

Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients

In this paper we prove the comparison principle for viscosity solutions of second order, degenerate elliptic pdes with a discontinuous, inhomogeneous term having discontinuities on Lipschitz surfaces. It is shown that appropriate sub and supersolutions $u,v$ of a Dirichlet type boundary value problem satisfy $u\leq v$ in $\Omega$. In particular, continuous viscosity solutions are unique. We also give examples of existence results and apply the comparison principle to prove convergence of approximations.

An application of [formula omitted] approximation to comparison principles for viscosity solutions of curvature equations

We prove the comparison principle for viscosity super- and sub-solutions of degenerate prescribed curvature using C 1 , 1 approximation.

A Free Boundary Problem with Curvature

In this paper we are interested in a free boundary problem with a motion law involving the mean curvature term of the free boundary. Viscosity solutions are introduced as a notion of global-time solutions past singularities. We show the comparison principle for viscosity solutions, which yields the existence of minimal and maximal solutions for given initial data. We also prove uniqueness of the solution for several classes of initial data and discuss the possibility of nonunique solutions.

Comparison principles for viscosity solutions of elliptic equations via fuzzy sum rule

A comparison principle for viscosity sub- and super-solutions of second order elliptic partial differential equations is derived using the “fuzzy sum rule” of non-smooth calculus. This method allows us to weaken the assumptions made on the function F when the equation F ( x , u , D u , D 2 u ) = 0 is under consideration.

A note on comparison principles for viscosity solutions of fully nonlinear second order partial differential equations

A note on comparison principles for viscosity solutions of fully nonlinear second order partial differential equations

Comparison Principles and Pointwise Estimates for Viscosity Solutions of Nonlinear Elliptic Equations

We prove comparison principles for viscosity solutions of nonlinear second order, uniformly elliptic equations, which extend previous results of P. L. Lions, R. Jensen and H. Ishii. Some basic pointwise estimates for classical solutions are also extended to continuous viscosity solutions.