Viscosity Solutions Research Articles

A finite-dimensional approximation for partial differential equations on Wasserstein space

This paper presents a finite-dimensional approximation for a class of partial differential equations on the space of probability measures. These equations are satisfied in the sense of viscosity solutions. The main result states the convergence of the viscosity solutions of the finite-dimensional PDE to the viscosity solutions of the PDE on Wasserstein space, provided that uniqueness holds for the latter, and heavily relies on an adaptation of the Barles & Souganidis monotone scheme (Barles and Souganidis, 1991) to our context, as well as on a key precompactness result for semimartingale measures. We illustrate our convergence result with the example of the Hamilton–Jacobi–Bellman and Bellman–Isaacs equations arising in stochastic control and differential games, and propose an extension to the case of path-dependent PDEs.

Quantitative characterization of bubble stability of foam concrete throughout extrusion process: From yield stress, viscosity and surface tension point of view

Foam concrete (FC) is suitable to be used as printing ink for drones in extreme environment because of its light weight, it can reduce the load of drones and improve printing efficiency. Furthermore, since the FC density and thermal insulation performance can be flexibly changed by changing the bubble content, it can be used to print functional gradient components and special-shaped insulation walls. The stability of bubbles is crucial as it directly impacts the performance of 3D printed FC (3DPFC). Here, we examined the bubble destabilization and deformation of FC throughout the mixing process, resting period prior to extrusion, and extrusion process based on three parameters, i.e., yield stress, viscosity, and surface tension. The results indicate that increasing the yield stress from 1406 Pa to 13379 Pa of the precursor leads to a decrease in bubble volume fraction after mixing from 38.26 % to 27.24 %, while increasing viscosity from 2.16 Pa s to 6.65 Pa s and decreasing surface tension from 72.4 mN/m to 33.5 mN/m are favorable for improving the sphericity of bubbles with the diameter between 300 and 800 μm in FC. In the resting stage, the yield stress of the interstitial paste is the primary factor controlling bubble stability. When the initial yield stress of the equivalent interstitial paste is 5212 Pa, the bubble volume fraction decreases by only 0.8 % within 60 min. During extrusion, high yield stress leading to bubble deformation and instability, whereas viscosity and pore solution surface tension act as sources of bubble compression resistance. There exists a suitable diameter interval for bubble pressure-bearing limit under different paste environment during extrusion.

Equivalence and regularity of weak and viscosity solutions for the anisotropic $${{\textbf {p}}}(\cdot )$$-Laplacian

Equivalence and regularity of weak and viscosity solutions for the anisotropic $${{\textbf {p}}}(\cdot )$$-Laplacian

Approximation of the HJB viscosity solutions in differential inclusion optimal control with state constraint

This paper tries to investigate a differential inclusion optimal control problem with state constraint (P) through an approximation of Hamilton–Jacobi–Bellman viscosity solutions. By invoking a duality result between the control problem (P) and the upper hull of viscosity subsolutions of the HJB equation, we prove that value functions of the discrete problems are solutions of discrete Bellman equations and converge to the value function of (P). We also establish that the optimal trajectory sequence of discrete problems converges to the solution of the optimal control problem (P). The idea behind the approximation is based on Euler polygonal arcs and Subbotin's proximal aiming condition, which allows discrete trajectories to break the state constraint, evolve in its neighbourhood and then converge to it instead of staying in the state constraint set for all time.

Existence of Viscosity Solutions for Weakly Coupled Cooperative Parabolic Systems with Fully Nonlinear Principle Part

In this paper, the existence of viscosity solutions for weakly coupled, degenerate, and cooperative parabolic systems is studied in a bounded domain. In particular, we consider the viscosity solutions of parabolic systems with fully nonlinear degenerated principal symbol and linear coupling part. The maximum principle theorem is given as well.

Comparison of Different Extraction and Purification Methods of Lime Peel Pectin and Evaluation of Physiochemical, Rheological, and Textural Properties

Background:: Pectin production has recently attracted considerable research interest due to its various applications arising from its unique characteristics. Lime peel, as a by-product of juice factories, has a promising potential for pectin extraction. Methods:: Lime peel pectin samples were extracted by three extraction processes: a) 90°C of heating for 120 min, b) 90°C of heating for 90 min and then 32 min of sonication, and c) 80°C of heating for 60 min and then 22 min of sonication. Then, they were purified either by ethanol or sodium caseinate (SC) and characterized. Results:: SC purification did not enhance the pectin yield; rather, it diminished the level of nonpectin components and resulted in purer pectin (i.e., lower sugar content). The samples purified with SC showed much lower viscosity in solution and storage modulus than the samples purified with ethanol, which could be attributed to their lower sugar content and higher pH (~ 4.6 vs. 2). However, pectin samples purified with SC had a relatively high sugar gel consistency, maybe due to protein retention in their structure as well as the preferential hydration effect of added sucrose. Conclusion:: The compact and extended conformations of the pectin sample, which was extracted by “(c) extraction method” and purified with ethanol, resulted in its high viscosity and storage modulus in solution and the highest gel consistency (16.91 Kg.s) compared to the other samples. Regardless of the conditions of extraction procedures, the purification method had a considerable impact on the pectin samples’ characteristics and potential applications.

Regularizing effect for unbounded flux-limited viscosity solutions of a discontinuous Hamilton-Jacobi equation on junction

Regularizing effect for unbounded flux-limited viscosity solutions of a discontinuous Hamilton-Jacobi equation on junction

Continuous-Time Mean Field Markov Decision Models

We consider a finite number of N statistically equal agents, each moving on a finite set of states according to a continuous-time Markov Decision Process (MDP). Transition intensities of the agents and generated rewards depend not only on the state and action of the agent itself, but also on the states of the other agents as well as the chosen action. Interactions like this are typical for a wide range of models in e.g. biology, epidemics, finance, social science and queueing systems among others. The aim is to maximize the expected discounted reward of the system, i.e. the agents have to cooperate as a team. Computationally this is a difficult task when N is large. Thus, we consider the limit for N→∞.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\rightarrow \\infty .$$\\end{document} In contrast to other papers we treat this problem from an MDP perspective. This has the advantage that we need less regularity assumptions in order to construct asymptotically optimal strategies than using viscosity solutions of HJB equations. The convergence rate is 1/N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1/\\sqrt{N}$$\\end{document}. We show how to apply our results using two examples: a machine replacement problem and a problem from epidemics. We also show that optimal feedback policies from the limiting problem are not necessarily asymptotically optimal.

Interpreting systems of continuity equations in spaces of probability measures through PDE duality

We introduce a notion of duality solution for a single or a system of transport equations in spaces of probability measures reminiscent of the viscosity solution notion for nonlinear parabolic equations. Our notion of solution by duality is, under suitable assumptions, equivalent to gradient flow solutions in case the single/system of equations has this structure. In contrast, we can deal with a quite general system of nonlinear non-local, diffusive or not, system of PDEs without any variational structure.

Stochastic optimal control problems with delays in the state and in the control via viscosity solutions and applications to optimal advertising and optimal investment problems

AbstractIn this manuscript we consider optimal control problems of stochastic differential equations with delays in the state and in the control. First, we prove an equivalent Markovian reformulation on Hilbert spaces of the state equation. Then, using the dynamic programming approach for infinite-dimensional systems, we prove that the value function is the unique viscosity solution of the infinite-dimensional Hamilton-Jacobi-Bellman equation. We apply these results to problems coming from economics: stochastic optimal advertising problems and stochastic optimal investment problems with time-to-build.

The evolution problem associated with the fractional first eigenvalue

In this paper we study the evolution problem associated with the first fractional eigenvalue. We prove that the Dirichlet problem with homogeneous boundary condition is well posed for this operator in the framework of viscosity solutions (the problem has existence and uniqueness of a solution and a comparison principle holds). In addition, we show that solutions decay to zero exponentially fast as t→∞ with a bound that is given by the first eigenvalue for this problem that we also study.

Schauder and Calderón–Zygmund type estimates for fully nonlinear parabolic equations under “small ellipticity aperture” and applications

In this manuscript, we derive some Schauder estimates for viscosity solutions to non-convex fully nonlinear second-order parabolic equations of the form: ∂tu−F(x,t,D2u)=f(x,t)inQ1=B1×(−1,0],provided that the source f and the coefficients of F are Hólder continuous functions, and F enjoys a small ellipticity aperture. Furthermore, for problems with merely bounded data, we prove that such solutions are C1,Log-Lip-regular. We also obtain Calderón-Zygmund estimates for such a class of non-convex operators. Finally, we connect our results and recent estimates for fully nonlinear models in certain solution classes.

Optimal stability results and nonlinear duality for L∞ entropy and L1 viscosity solutions

We give a new and rigorous duality relation between two central notions of weak solutions of nonlinear PDEs: entropy and viscosity solutions. It takes the form of the nonlinear dual inequality:(⋆)∫|Stu0−Stv0|φ0dx≤∫|u0−v0|Gtφ0dx,∀φ0≥0,∀u0,∀v0, where St is the entropy solution semigroup of the anisotropic degenerate parabolic equation∂tu+divF(u)=div(A(u)Du), and where we look for the smallest semigroup Gt satisfying (⋆). This amounts to finding an optimal weighted L1 contraction estimate for St. Our main result is that Gt is the viscosity solution semigroup of the Hamilton-Jacobi-Bellman equation∂tφ=supξ⁡{F′(ξ)⋅Dφ+tr(A(ξ)D2φ)}. Since weighted L1 contraction results are mainly used for possibly nonintegrable L∞ solutions u, the natural spaces behind this duality are L∞ for St and L1 for Gt. We therefore develop a corresponding L1 theory for viscosity solutions φ. But L1 itself is too large for well-posedness, and we rigorously identify the weakest L1 type Banach setting where we can have it – a subspace of L1 called Lint∞. A consequence of our results is a new domain of dependence like estimate for second order anisotropic degenerate parabolic PDEs. It is given in terms of a stochastic target problem and extends in a natural way recent results for first order hyperbolic PDEs by [N. Pogodaev, J. Differ. Equ., 2018].

Viscosity Solutions of the Eikonal Equation on the Wasserstein Space

Viscosity Solutions of the Eikonal Equation on the Wasserstein Space

Regularity of flat free boundaries for two-phase p(x)-Laplacian problems with right hand side

We consider viscosity solutions to two-phase free boundary problems for the p(x)-Laplacian with non-zero right hand side. We prove that flat free boundaries are C1,γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^{1,\\gamma }$$\\end{document}. No assumption on the Lipschitz continuity of solutions is made. These regularity results are the first ones in literature for two-phase free boundary problems for the p(x)-Laplacian and also for two-phase problems for singular/degenerate operators with non-zero right hand side. They are new even when p(x)≡p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p(x)\\equiv p$$\\end{document}, i.e., for the p-Laplacian. The fact that our results hold for merely viscosity solutions allows a wide applicability.

Near-infrared fluorescent probe for the imaging of viscosity in fatty liver mice and valuation of drug efficacy

Fatty liver disease affects at least 25 percent of the population worldwide and is a severe metabolic syndrome. Viscosity is closely related to fatty liver disease, so it is urgent to develop an effective tool for monitoring viscosity. Herein, a NIR fluorescent probe called MBC-V is developed for imaging viscosity, consisting of dimethylaniline and malonitrile-benzopyran. MBC-V is non-fluorescent in low viscosity solutions due to intramolecular rotation. In high viscosity solution, the intramolecular rotation of MBC-V is suppressed and the fluorescence is triggered. MBC-V has long emission wavelength at 720 nm and large Stokes shift about 160 nm. Moreover, MBC-V can detect changes in cell viscosity in fatty liver cells, and can image the therapeutic effects of drug in fatty liver cells. By taking advantage of NIR emission, MBC-V can be used as an imaging tool for fatty liver disease and a way to evaluate the therapeutic effect of drug for fatty liver disease.

Optimal control of stochastic differential equations with random impulses and the Hamilton–Jacobi–Bellman equation

AbstractIn this article, we study the optimal control of stochastic differential equations with random impulses. We optimize the performance index and add the influence of random impulses to the performance index with a random compensation function. Using the idea of stochastic analysis and dynamic programming principle, a new Hamilton–Jacobi–Bellman (HJB) equation is obtained, and the existence and uniqueness of its viscosity solution are proved.

Analysis of a Narrow-Stencil Finite Difference Method for Approximating Viscosity Solutions of Fully Nonlinear Second Order Parabolic PDEs

Analysis of a Narrow-Stencil Finite Difference Method for Approximating Viscosity Solutions of Fully Nonlinear Second Order Parabolic PDEs

The free boundary for a semilinear non-homogeneous Bernoulli problem

In the classical homogeneous one-phase Bernoulli-type problem, the free boundary consists of a “regular” part and a “singular” part, as Alt and Caffarelli have shown in their pioneer work (Alt and Caffarelli, 1981 [1]) that regular points are C1,γ in two-dimensions. Later, Weiss (1999) [34] first realized that in higher dimensions a critical dimension d⁎ exists so that the singularities of the free boundary can only occur when d⩾d⁎.In this paper, we consider a non-homogeneous semilinear one-phase Bernoulli-type problem, and we show that the free boundary is a disjoint union of a regular and a singular set. Moreover, the regular set is locally the graph of a C1,γ function for some γ∈(0,1). In addition, there exists a critical dimension d⁎ so that the singular set is empty if d<d⁎, discrete if d=d⁎ and of locally finite Hd−d⁎ Hausdorff measure if d>d⁎. As a byproduct, we relate the existence of viscosity solutions of a non-homogeneous problem to the Weiss-boundary adjusted energy, which provides an alternative proof to existence of viscosity solutions for non-homogeneous problems.

On a lack of stability of parametrized BV solutions to rate‐independent systems with nonconvex energies and discontinuous loads

AbstractWe consider a rate‐independent system with nonconvex energy under discontinuous external loading. The underlying space is finite‐dimensional and the loads are functions in . We investigate the stability of various solution concepts w.r.t. a sequence of loads converging weakly* in with a particular emphasis on the so‐called normalized, ‐parametrized balanced viscosity solutions. By means of three counterexamples, it is shown that common solution concepts are not stable w.r.t. weak* and even intermediate (or strict) convergence of loads in the sense that a limit of a sequence of solutions associated with these loads need not be a solution corresponding to the load in the limit. We moreover introduce a new solution concept, which is stable in this sense, but our examples show that this concept necessarily allows “solutions” that are physically meaningless.