In this work we consider viscosity solutions to second order partial differential equations on Riemannian manifolds. We prove maximum principles for solutions to Dirichlet problem on a compact Riemannian manifold with boundary. Using a different method, we generalize maximum principles of Omori and Yau to a viscosity v…
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48 results for “Viscosity method”
New proofs for curvature problems using a viscosity approach.
problem Constant rank theorems for curvature problems in compact and non-compact settings.
method Viscosity approach to prove constant rank theorems for curvature problems.
result Generalization of a differential inequality for subtrace.
Paper studies viscosity solutions in unique Martinet spaces.
problem Properties of viscosity solutions in Martinet spaces.
method Established properties and proved uniqueness of solutions.
result Uniqueness of viscosity solutions in Martinet spaces.
Paper finds unique viscosity solution to complex control problems.
problem Complex stochastic control problems with singular terminal state constraints.
method Establishes existence of unique nonnegative continuous viscosity solution using novel comparison principle.
result Unique viscosity solution to HJB equation for linear-quadratic control problems.
This is the content of the lectures given by the author at the winter school KAWA3 held at the University of Barcelona in 2012 from January 30 to February 3. The main goal was to give an account of viscosity techniques and to apply them to degenerate Complex Monge-Ampère equations following recent works of P. Eyssidieu…
Proves existence and uniqueness of viscosity solutions to complex Hessian equations on compact Hermitian manifolds.
problem Existence and uniqueness of viscosity solutions to complex Hessian equations.
method Proves existence and uniqueness using viscosity solutions and determinant domination conditions.
result Viscosity solutions exist and are unique under certain conditions.
Smooth solutions found for a specific type of Yamabe problem.
problem Regularity of viscosity solutions to the -Yamabe problem in the negative cone.
method Analysis of Lipschitz viscosity solutions with specific assumptions.
result Existence and smoothness of solutions away from a negligible set.
Study shows various weak solutions to complex flows match, proving viscosity equals pluripotential.
problem Comparing weak solutions to complex Monge-Ampère flows.
method Examined various notions of weak subsolutions and showed they coincide.
result Viscosity solution equals pluripotential solution.
The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.
Convex solutions to a specific equation are smooth when the phase is smooth enough.
problem Regularity of solutions to the Lagrangian mean curvature equation.
method Showed regularity for convex solutions under Hölder continuity conditions on the phase.
result Convex viscosity solutions are regular if the Lagrangian phase is Hölder continuous.
Real analytic solutions found for special Lagrangian equation.
problem Analyzing convex solutions of the special Lagrangian equation.
method Interior regularity established for convex viscosity solutions.
result All convex solutions are real analytic in the interior.
We establish the estimates of modulus of continuity for viscosity solutions of nonlinear evolution equations on manifolds, extending previous work of B. Andrews and J. Clutterbuck for regular solutions on manifolds \cite{AC3} and the first author's recent work for viscosity solutions in Euclidean spaces \cite{me1}.
Model predicts viscosity of multicomponent systems efficiently.
problem Expensive experimental viscosity measurements in various industries.
method Artificial neural networks trained on a database of chemical systems and temperatures.
result Model Viskositas provides more accurate predictions with lower errors, variability, and outliers.
New equations reveal viscosity from boundary measurements.
problem Determine viscosity from boundary measurements for incompressible fluids.
method Equivalent new system of elliptic equations, Dirichlet-to-Neumann map analysis.
result Dirichlet-to-Neumann map uniquely determines viscosity and its derivatives on the boundary.
Establishes unique weak solution to complex flow on smooth manifolds.
problem Existence of weak solutions to complex flow equations.
method Applying viscosity theory to prove existence of unique solution.
result Unique global weak solution exists for smooth manifolds.
We present a viscosity approach to the min-max construction of closed geodesics on compact Riemannian manifolds of arbitrary dimension. We also construct counter-examples in dimension and to the -regularity in the convergence procedure. Furthermore, we prove the lower semi-continuity of the index o…
We develop an alternative approach to Degenerate complex Monge-Ampère equations on compact Kähler manifolds based on the concept of viscosity solutions and compare systematically viscosity concepts with pluripotential theoretic ones. We generalize to the Kähler case a theorem due to Dinew and Zhang in the projective ca…
Viscosity solutions are suitable notions in the study of nonlinear PDEs justified by estimates established via the maximum principle or the comparison principle. Here we prove that the isoperimetric profile functions of Riemannian manifolds with Ricci lower bound are viscosity super-solutions of some nonlinear differen…
In this paper, we investigate the moduli of continuity for viscosity solutions of a wide class of nonsingular quasilinear evolution equations and also for the level set mean curvature flow, which is an example of singular degenerate equations. We prove that the modulus of continuity is a viscosity subsolution of some o…
This paper develops a novel numerical method for pricing American options in a two-asset jump-diffusion model.
problem Pricing American options under correlated two-asset jump-diffusion models using finite difference methods often fails to preserve monotonicity and accurately discretize jump integrals.
method Introduces a novel monotone integration scheme to solve 2-D Partial Integro-Differential Equations (PIDEs) efficiently and accurately.
result The proposed method ensures convergence to the viscosity solution of the variational inequality and is both -stable and consistent.
Study on Tukey depth in machine learning using Hamilton-Jacobi equations.
problem Understanding Tukey depth in machine learning applications.
method Derive necessary conditions for Tukey depth in continuum limit, formulating them as a Hamilton-Jacobi equation.
result Prove existence and uniqueness of viscosity solutions for the derived equation, which bounds Tukey depth.
We extend the stochastic Perron method to analyze the framework of stochastic target games, in which one player tries to find a strategy such that the state process almost surely reaches a given target no matter which action is chosen by the other player. Within this framework, our method produces a viscosity sub-solut…
Establishes regularity of conformal harmonic maps.
problem Ensuring smoothness of harmonic maps.
method Viscosity techniques for stationary varifolds.
result Weakly conformal maps are smooth if stationary.
Paper aims to minimize ruin probability in insurance companies using Sparre Andersen model.
problem Minimizing ruin probability in insurance companies with Sparre Andersen surplus process.
method Markovization of the surplus process, investigation of value function's regularity, dynamic programming principle, and comparison of viscosity solutions.
result The value function is the unique constrained viscosity solution to the Hamilton-Jacobi-Bellman equation.
Generic level sets in mean curvature flow are BV solutions.
problem Understanding the behavior of level sets in mean curvature flow.
method Using the framework of sets of finite perimeter and distributional solutions, the paper extends Evans and Spruck's work.
result Generic level sets are distributional solutions with optimal energy dissipation rate.
We study viscosity solutions to complex hessian equations. In the local case, we consider a bounded domain in the standard Kähler form in and Under some suitable conditions on , we prove that the equation $(dd^c \varphi)^m\wedgeβ^{n-m}=F(x,\varphi)β^n,\ \f=…
Quadratic Hessian equationmath.AP
Abstract: Survey on quadratic Hessian equations, their properties, and open problems.
problem Understanding quadratic Hessian equations and their solutions.
method Survey and review of existing research.
result Survey of entire solutions, viscosity solutions, and Hessian estimates.
In this paper, we adapt stochastic Perron's method to analyze a stochastic target problem with unbounded controls in a jump diffusion set-up. With this method, we construct a viscosity sub-solution and super-solution to the associated Hamiltonian-Jacobi-Bellman (HJB) equations. Under comparison principles, uniqueness o…
Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.
problem Understanding moduli of continuity for fully nonlinear parabolic equations.
method Proving moduli of continuity of viscosity solutions are subsolutions of one-dimensional parabolic equations.
result Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations with bounded initial data.
Comparison principles and Lipschitz regularity for some nonlinear degenerate elliptic equationsmath.AP
Proves Lipschitz continuity for solutions of certain nonlinear elliptic equations.
problem Interior Lipschitz regularity for continuous viscosity solutions of nonlinear degenerate elliptic equations.
method Establishes interior Lipschitz regularity using a weak form of the strong comparison principle.
result Weak form of the strong comparison principle (principle of propagation of touching points) for specific operators.
This paper concerns the continuous time mean-variance portfolio selection problem with a special nonlinear wealth equation. This nonlinear wealth equation has a nonsmooth coefficient and the dual method developed in [6] does not work. We invoke the HJB equation of this problem and give an explicit viscosity solution of…
We observe that the comparison result of Barles-Biton-Ley for viscosity solutions of a class of nonlinear parabolic equations can be applied to a geometric fully nonlinear parabolic equation which arises from the graphic solutions for the Lagrangian mean curvature flow.
Proves smoothness and estimates for special Lagrangian solutions with semi-convexity.
problem Smoothness and estimates for special Lagrangian solutions.
method Viscosity solutions, smoothness, interior derivative estimates, sharpness of conditions.
result New Liouville theorem and effective Hessian estimates for special Lagrangian solutions.
Study values American passport options in an exponential Lévy model.
problem Valuing an exotic derivative called the American passport option.
method Derived pricing equation using dynamic programming principle and proved viscosity solution.
result Option value is a viscosity solution of variational inequality and is convex.
Study uses viscosity solutions to solve control problems involving measure-valued martingales.
problem Stochastic control problems with measure-valued martingale state processes.
method Viscosity solution approach exploiting structural properties of MVM processes.
result Value function is the unique viscosity solution to the HJB equation.
New method generates critical points for complex functionals.
problem Proving the Willmore conjecture in complex geometric settings.
method Minmax hierarchies and fibrations for critical points.
result New proof of the Willmore conjecture.
The paper studies limits of flows on Kähler surfaces, proving convergence to solutions of equations.
problem Analyzing limits of flows on Kähler surfaces and their convergence to solutions of equations.
method Using a property of limits of viscosity subsolutions.
result Proves convergence of flows to weak solutions of the Monge-Ampère equation.
Study on Kähler-Ricci flow on manifolds with big canonical bundle.
problem Analyzing convergence of Kähler-Ricci flow on manifolds of general type.
method Normalized Kähler-Ricci flow, viscosity theory for degenerate complex Monge-Ampère flows.
result Convergence to singular Kähler-Einstein metric in canonical class.
We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.
Adapts viscosity method for free boundary minimal surfaces.
problem Finding min-max free boundary minimal surfaces.
method Adapting Rivière's viscosity method to free boundary conditions.
result Min-max value is sum of branched minimal immersions.
Optimal reinsurance strategies for multi-line insurance companies.
problem Choosing the best dynamic reinsurance policies for multi-line insurance companies.
method Characterized the optimal survival function as the unique nondecreasing viscosity solution of the HJB equation, solved numerically using the finite difference method.
result Provided proof of convergence of numerical solution to the survival probability function.
Study on solutions to complex equations, proving strong comparison and Liouville theorems.
problem Analyzing continuous viscosity solutions to fully nonlinear elliptic equations.
method Proving strong comparison principle and Hopf Lemma for (non-uniformly) elliptic equations.
result Liouville theorem for entire solutions, showing they are either constants or standard bubbles.
In this paper, we study the valuation of American type derivatives in the stochastic volatility model of Barndorff-Nielsen and Shephard (2001). We characterize the value of such derivatives as the unique viscosity solution of an integral-partial differential equation when the payoff function satisfies a Lipschitz condi…
Deep Galerkin Method estimates value function for mean-field control problem.
problem Optimal control of agents with average welfare as the objective.
method Apply DGM to estimate value function and distribution evolution.
result Neural network approximations converge to analytical solution.
Study proves solutions to nonlocal operator obstacle problems, including financial models.
problem Existence, uniqueness, and regularity of viscosity solutions to obstacle problems with nonlocal operators.
method Proved existence, uniqueness, and regularity using viscosity solutions; provided sufficient conditions for Hölder and Lipschitz continuity.
result Viscosity solutions for nonlocal operators in financial models match option prices.
We present the min-max construction of critical points of the area using penalization arguments. Precisely, for any immersion of a closed surface into a given closed manifold, we add to the area Lagrangian a term equal to the norm of the second fundamental form of the immersion times a "viscosity" parameter. …
The study proves lower semi-continuity of the index for minimal surfaces using viscosity methods.
problem Proving lower semi-continuity of the index for minimal surfaces.
method Using Hilbert manifold structure and viscous approximation of the area.
result Proves lower semi-continuity of the index for minimal surfaces.
Investor aims to meet financial goals with deadlines and target amounts, considering stock trading costs.
problem Goal-based portfolio selection with fixed transaction costs.
method Stochastic Perron's method to show value function is unique viscosity solution to quasi-variational inequalities. Existence of optimal strategy established.
result Optimal trading strategy differs significantly from frictionless case, revealing complex regions and strategies.