Lipschitz Continuous Solutions Research Articles

Global Existence for the Confined Muskat Problem

In this paper we show global existence of the Lipschitz continuous solution for the stable Muskat problem with finite depth (confined) and initial data satisfying some smallness conditions relating the amplitude, the slope, and the depth. The cornerstone of the argument is that, for these small initial data, both the amplitude and the slope remain uniformly bounded for all positive times. We notice that, for some of these solutions, the slope can grow but it remains bounded. This is very different from the infinite deep case, where the slope of the solutions satisfy a maximum principle. Our work generalizes a previous result where the depth is infinite.

Large time behavior for some nonlinear degenerate parabolic equations

We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton–Jacobi–Bellman equations. Defining Σ as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside Σ and, on Σ, the Hamiltonian is either strictly convex or satisfies an assumption similar of the one introduced by Barles–Souganidis (2000) for first-order Hamilton–Jacobi equations. This latter assumption allows to deal with equations with nonconvex Hamiltonians. We can also release the uniform parabolic requirement outside Σ. As a consequence, we prove the convergence of some everywhere degenerate second-order equations.

UNIQUENESS RESULTS FOR DIAGONAL HYPERBOLIC SYSTEMS WITH LARGE AND MONOTONE DATA

We study the uniqueness of solutions to diagonal hyperbolic systems in one spatial dimension and we present two uniqueness results. First, we establish a global existence and uniqueness theorem for continuous solutions to strictly hyperbolic systems. Second, we establish a global existence and uniqueness theorem for Lipschitz continuous solutions to hyperbolic systems that need not be strictly hyperbolic. Furthermore, an application is presented for one-dimensional flows in isentropic gas dynamics.

Uniqueness and Approximation of a Photometric Shape-from-Shading Model

We deal with an inverse problem where we want to determine the surface of an object using the information contained in two or more pictures which correspond to different light conditions. In particular, we will examine the case where the light source direction varies between the pictures, and we will show how this additional information allows us to obtain a uniqueness result solving the well-known convex/concave ambiguity of the shape-from-shading problem. We will prove a uniqueness result for weak (Lipschitz continuous) solutions that improves previous results in [R. Kozera, Appl. Math. Comput., 44 (1991), pp. 1--103] and [R. Onn and A. Bruckstein, Int. J. Comput. Vision, 5 (1990), pp. 105--113]. We also propose some approximation schemes for the numerical solution of this problem and analyze the properties of two approximation schemes: an upwind finite difference scheme and a semi-Lagrangian scheme. Finally, we present some numerical tests on smooth and nonsmooth surfaces coming from virtual and real i...

Approximation of an optimal value of a Bolza functional

In this article, we show that under reasonable assumptions every Lipschitz-continuous solution to a Hamilton–Jacobi inequality approximates with a priori known error the optimal value of a respective Bolza functional and that such approximation is stable. The solutions of Hamilton–Jacobi variational inequalities can be easily obtained by well-known numerical methods as approximate solutions of Hamilton–Jacobi equations resulting from related Bolza functionals. The main strength of this approach lies in the fact that both precise solution to the Hamilton–Jacobi PDE and the distance between that solution and its numerical approximation need not be known in order to solve the original Bolza problem.

Nonsmooth hypersurfaces with smooth Levi curvature

In R3, Lipschitz continuous viscosity solutions to the K-prescribed Levi curvature equation are smooth and strictly pseudoconvex if K is smooth and strictly positive; see [5]. We show here that in R2n+1, n>1, a similar result does not hold; that is, we prove the existence in Cn+1, n>1, of nonsmooth pseudoconvex hypersurfaces with smooth Levi curvature.

Lipschitz Continuous Viscosity Solutions for a Class of Fully Nonlinear Equations on Lie Groups

In this paper, we prove existence and uniqueness of Lipschitz continuous viscosity solutions for Dirichlet problems involving a class a fully non-linear operators on Lie groups. In particular, we consider the elementary symmetric functions of the eigenvalues of the Hessian built with left-invariant vector fields.

The initial-boundary value problem of hyperbolic integro-differential systems of nonlinear balance laws

This research explores the initial-boundary value problem for the 2×2 hyperbolic systems of balance laws whose sources are the time-dependent and contain the integral of unknowns. Perturbed Riemann and boundary Riemann problems are provided to account for the time-dependence of sources. Their approximation solutions are constructed by modified Lax’s method. In addition, we introduce a new version of Glimm scheme (GGS) and study its stability which is proved by the wave interaction estimates in a dissipativity assumption. With the consistency of GGS, the existence of a global weak solution satisfying the entropy inequality is then achieved. Finally the Lipschitz continuous solution to the problem is established by the weak convergence of the residual.

Lipschitz continuous solutions to the Cauchy problem for quasi-linear hyperbolic systems

Lipschitz continuous solutions to the Cauchy problem for 1-D first order quasilinear hyperbolic systems are considered. Based on the methods of approximation and integral equations, the author gives two definitions of Lipschitz solutions to the Cauchy problem and proves the existence and uniqueness of solutions.

Local existence of Lipschitz-continuous solutions of systems of nonlinear functional equations with iterated deviations

Some conditions which guarantee the local existence of Lipschitz-continuous solutions of some systems of nonlinear functional equations with complicated deviations dependent of unknown functions are presented.

New perturbation methods for nonlinear parabolic problems

We develop methods aimed at deriving regularity results for solutions to nonlinear degenerate parabolic equations and systems via local perturbation; as a consequence we obtain, in a unified way, Lipschitz continuity of solutions under weak parabolicity assumptions, and gradient continuity results in borderline cases. Nonlinear Schauder estimates as those of Misawa (2002) [29] are recovered and extended to more general settings.

Global entropy solutions to a class of quasi-linear wave equations with large time-oscillating sources

This research explores the Cauchy problem for a class of quasi-linear wave equations with time dependent sources. It can be transformed into the Cauchy problem of hyperbolic integro-differential systems of nonlinear balance laws. We introduce the generalized Glimm scheme in new version and study its stability which is proved by Glimm-type interaction estimates in a dissipativity assumption. The generalized solutions to the perturbed Riemann problems, the building blocks of generalized Glimm scheme, are constructed by Riemann problem method modeled on the source free equations. The global existence for the Lipschitz continuous solutions and weak solutions to the systems is established by the consistency of scheme and the weak convergence of source. Finally, the weak solutions are also the entropy solutions which satisfy the entropy inequality.

On the global existence for the Muskat problem

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an L^2(\mathbb R) maximum principle, in the form of a new "log'' conservation law which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance \|f\|_1 \le 1/5 . Previous results of this sort used a small constant \epsilon \ll1 which was not explicit. Lastly, we prove a global existence result for Lipschitz continuous solutions with initial data that satisfy \|f_0\|_{L^\infty}<\infty and \|\partial_x f_0\|_{L^\infty}<1 . We take advantage of the fact that the bound \|\partial_x f_0\|_{L^\infty}<1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law.

On Nonlinear Volterra Integral Equations with State Dependent Delays in Several Variables

Lipschitz continuous solutions of nonlinear Volterra integral equations with state dependent delays in several variables are investigated. The results are based on a comparison method and the Banach fixed point principle.

On viscosity solutions of Hamilton-Jacobi equations

We consider the Dirichlet problem for Hamilton-Jacobi equations and prove existence, uniqueness and continuous dependence on boundary data of Lipschitz continuous maximal viscosity solutions.

Global solutions for initial–boundary value problem of quasilinear wave equations

This work investigates the existence of globally Lipschitz continuous solutions to a class of initial–boundary value problem of quasilinear wave equations. Applying the Lax's method and generalized Glimm's method, we construct the approximate solutions of initial–boundary Riemann problem near the boundary layer and perturbed Riemann problem away from the boundary layer. By showing the weak convergence of residuals for the approximate solutions, we establish the global existence for the derivatives of solutions and obtain the existence of global Lipschitz continuous solutions of the problem.

Globally Lipschitz continuous solutions to a class of quasilinear wave equations

This work investigates the existence of globally Lipschitz continuous solutions to a class of Cauchy problem of quasilinear wave equations. Applying Lax's method and generalized Glimm's method, we construct the approximate solutions of the corresponding perturbed Riemann problem and establish the global existence for the derivatives of solutions. Then, the existence of global Lipschitz continuous solutions can be carried out by showing the weak convergence of residuals for the source term of equation.

A homotopy approach to solving the inverse mean curvature flow

In \(\mathbb{R}^{n}\), n < 7, we treat the quasilinear, degenerate parabolic initial and boundary value problem which is the natural parabolic extension of Huisken and Ilmanen’s weak inverse mean curvature flow (IMCF). We prove long time existence and partial uniqueness of Lipschitz continuous weak solutions u(x,t) and show C 1,α-regularity for the sets ∂{x| u(x,t) < z }. Our approach offers a new approximation for weak solutions of the IMCF starting from a class of interesting and easily obtainable initial values; for these, the above sets are shown to converge against corresponding surfaces of the IMCF as t → ∞ globally in Hausdorff distance and locally uniformly with respect to the C 1,α-norm.

Ergodic Type Problems and Large Time Behaviour of Unbounded Solutions of Hamilton–Jacobi Equations

We study the large time behavior of Lipschitz continuous, possibly unbounded, viscosity solutions of Hamilton–Jacobi Equations in the whole space ℝ N . The associated ergodic problem has Lipschitz continuous solutions if the analogue of the ergodic constant is larger than a minimal value λmin. We obtain various large-time convergence and Liouville type theorems, some of them being of completely new type. We also provide examples showing that, in this unbounded framework, the ergodic behavior may fail, and that the asymptotic behavior may also be unstable with respect to the initial data.

GLOBAL STABILITY OF SOLUTIONS WITH DISCONTINUOUS INITIAL DATA CONTAINING VACUUM STATES FOR THE RELATIVISTIC EULER EQUATIONS

The global stability of Lipschitz continuous solutions with discontinuous initial data for the relativistic Euler equations is established in a broad class of entropy solutions in L∞ containing vacuum states. As a corollary, the uniqueness of Lipschitz solutions with discontinuous initial data is obtained in the broad class of entropy solutions in L∞.