Quasilinear PDEs Research Articles

Carleman weight functions for solving ill-posed Cauchy problems for quasilinear PDEs

This is the first publication in which a numerical method with rigorously guaranteed convergence is presented for some ill-posed Cauchy problems for semilinear PDEs of the second order. The key tool is the use of the Carleman weight functions in a Tikhonov-like functionals. The method works for elliptic, parabolic and hyperbolic semilinear equations of the second order.

Nonsolvability of the asymptotic Dirichlet problem for some quasilinear elliptic PDEs on Hadamard manifolds

We show, by modifying Borbély’s example, that there are 3-dimensional Cartan–Hadamard manifolds M , with sectional curvatures ≤ −1 , such that the asymptotic Dirichlet problem for a class of quasilinear elliptic PDEs, including the minimal graph equation, is not solvable.

Optimal Control of Quasistatic Plasticity with Linear Kinematic Hardening II: Regularization and Differentiability

We consider an optimal control problem governed by an evolution variational inequality arising in quasistatic plasticity with linear kinematic hardening. A regularization of the time-discrete problem is derived. The regularized forward problem can be interpreted as system of coupled quasilinear PDEs whose principal parts depend on the gradient of the state. We show the Fréchet differentiability of the solution map of this quasilinear system. As a consequence, we obtain a first order necessary optimality system. Moreover, we address certain convergence properties of the regularization.

The existence and uniqueness result for quasilinear stochastic PDEs with obstacle under weaker integrability conditions

We prove an existence and uniqueness result for quasilinear Stochastic PDEs with obstacle (in short OSPDE) under a weaker integrability condition on the coefficient and the barrier.

$G$-expectation weighted Sobolev spaces, backward SDE and path dependent PDE

We introduce a new notion of G-expectation-weighted Sobolev spaces, or in short, G-Sobolev spaces, and prove that a backward SDEs driven by G-Brownian motion are in fact path dependent PDEs in the corresponding Sobolev spaces under G-norms. For the linear case of G corresponding the classical Wiener probability space with Wiener measure P, we have established a 1-1 correspondence between BSDE and such new type of quasilinear PDE in the corresponding P-Sobolev space. When G is nonlinear, we also provide such 1-1 correspondence between a fully nonlinear PDE in the corresponding G-Sobolev space and BSDE driven by G-Brownian. Consequently, the existence and uniqueness of such type of fully nonlinear path-dependence PDE in G-Sobolev space have been obtained via a recent results of BSDE driven by G-Brownian motion.

A Fokker-Planck type approximation of parabolic PDEs with oblique boundary data

We consider solutions of quasi-linear parabolic PDEs with zero oblique boundary data in a bounded domain. Our main result states that the solutions can be approximated by solutions of a Fokker-Planck type PDE in the whole space with a penalizing drift term which also converges to zero outside the original domain. The convergence is locally uniform, and optimal error estimates are obtained.

KAM for autonomous quasi-linear perturbations of KdV

We prove the existence and the stability of Cantor families of quasi-periodic, small amplitude solutions of quasi-linear (i.e. strongly nonlinear) autonomous Hamiltonian differentiable perturbations of KdV. This is the first result that extends KAM theory to quasi-linear autonomous and parameter independent PDEs. The core of the proof is to find an approximate inverse of the linearized operators at each approximate solution and to prove that it satisfies tame estimates in Sobolev spaces. A symplectic decoupling procedure reduces the problem to the one of inverting the linearized operator restricted to the normal directions. For this aim we use pseudo-differential operator techniques to transform such linear PDE into an equation with constant coefficients up to smoothing remainders. Then a linear KAM reducibility technique completely diagonalizes such operator. We introduce the “initial conditions” as parameters by performing a “weak” Birkhoff normal form analysis, which is well adapted for quasi-linear perturbations.

A monotone scheme for high-dimensional fully nonlinear PDEs

In this paper we propose a feasible numerical scheme for high-dimensional, fully nonlinear parabolic PDEs, which includes the quasi-linear PDE associated with a coupled FBSDE as a special case. Our paper is strongly motivated by the remarkable work Fahim, Touzi and Warin [Ann. Appl. Probab. 21 (2011) 1322-1364] and stays in the paradigm of monotone schemes initiated by Barles and Souganidis [Asymptot. Anal. 4 (1991) 271-283]. Our scheme weakens a critical constraint imposed by Fahim, Touzi and Warin (2011), especially when the generator of the PDE depends only on the diagonal terms of the Hessian matrix. Several numerical examples, up to dimension 12, are reported.

Metastability for parabolic equations with drift: part II. The quasilinear case

This is the second part of our series of papers 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 quasi-linear parabolic equations with a drift and to obtain results similar to those in Freidlin and Koralov [Nonlinear stochastic perturbations of dynamical systems and quasi-linear parabolic PDE's with a small parameter, Probab. Theory Related Fields 147 (2010), ArXiv:0903.0428v2 (2012)].

Quasi-linear systems of PDE of first order with Cauchy data of higher codimensions

In this paper we discuss the local solvability of Cauchy problem for quasi-linear partial differential equations of first order. By using the classical method of characteristics we describe the non-uniqueness or the degree of freedom for solutions and also decide the conditions for the existence and the uniqueness of solutions for overdetermined systems of quasi-linear PDEs of first order.

The obstacle problem for quasilinear stochastic PDEs with non-homogeneous operator

We prove the existence and uniqueness of solution of the obstacle problem for quasilinear Stochastic PDEs with non-homogeneous second order operator. Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair $(u,\nu)$ where $u$ is a predictable continuous process which takes values in a proper Sobolev space and $\nu$ is a random regular measure satisfying minimal Skohorod condition. Moreover, we establish a maximum principle for local solutions of such class of stochastic PDEs. The proofs are based on a version of Ito's formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary.

Local well-posedness in Lovelock gravity

It has long been known that Lovelock gravity, being of Cauchy–Kowalevskaya type, admits a well defined initial value problem for analytic data. However, this does not address the physically important issues of continuous dependence of the solution on the data and the domain of dependence property. In this note, we fill this gap in our understanding of the (local) dynamics of the theory. We show that, by a known mathematical trick, the fully nonlinear harmonic-gauge-reduced Lovelock field equations can be made equivalent to a quasilinear PDE system. Due to this equivalence, an analysis of the principal symbol, as has appeared in recent works by other authors, is sufficient to decide the issue of local well-posedness of perturbations about a given background.

Local well-posedness of a class of hyperbolic PDE–ODE systems on a bounded interval

The well-posedness theory for hyperbolic systems of first-order quasilinear PDE's with ODE's boundary conditions (on a bounded interval) is discussed. Such systems occur in multi-scale blood flow models, as well as valveless pumping and fluid mechanics. The theory is presented in the setting of Sobolev spaces Hm (m ≥ 3 being an integer), which is an appropriate set-up when it comes to proving existence of smooth solutions using energy estimates. A blow-up criterion is also derived, stating that if the maximal time of existence is finite, then the state leaves every compact subset of the hyperbolicity region, or its first-order derivatives blow-up. Finally, we discuss physical examples which fit in the general framework presented.

Method of characteristics and first integrals for systems of quasi-linear partial differential equations of first order

Given a set of independent vector fields on a smooth manifold, we discuss how to find a function whose zero-level set is invariant under the flows of the vector fields. As an application, we study the solvability of overdetermined partial differential equations: Given a system of quasi-linear PDEs of first order for one unknown function we find a necessary and sufficient condition for the existence of solutions in terms of the second jet of the coefficients. This generalizes to certain quasi-linear systems of first order for several unknown functions.

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 obstacle problem for quasilinear stochastic PDEs: Analytical approach

We prove an existence and uniqueness result for quasilinear Stochastic PDEs with obstacle (OSPDE in short). Our method is based on analytical technics coming from the parabolic potential theory. The solution is expressed as a pair $(u,\nu)$ where $u$ is a predictable continuous process which takes values in a proper Sobolev space and $\nu$ is a random regular measure satisfying the minimal Skohorod condition.

Harnack estimates for degenerate parabolic equations modeled on the subelliptic p-Laplacian

We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype∂tu=−∑i=1mXi⁎(|Xu|p−2Xiu) where p⩾2, X=(X1,…,Xm) is a system of Lipschitz vector fields defined on a smooth manifold M endowed with a Borel measure μ, and Xi⁎ denotes the adjoint of Xi with respect to μ. Our estimates are derived assuming that (i) the control distance d generated by X induces the same topology on M; (ii) a doubling condition for the μ-measure of d-metric balls; and (iii) the validity of a Poincaré inequality involving X and μ. Our results extend the recent work in [16,36], to a more general setting including the model cases of (1) metrics generated by Hörmander vector fields and Lebesgue measure; (2) Riemannian manifolds with non-negative Ricci curvature and Riemannian volume forms; and (3) metrics generated by non-smooth Baouendi–Grushin type vector fields and Lebesgue measure. In all cases the Harnack inequality continues to hold when the Lebesgue measure is substituted by any smooth volume form or by measures with densities corresponding to Muckenhoupt type weights.

Maximum principle for quasilinear stochastic PDEs with obstacle

We prove a maximum principle for local solutions of quasi linear stochastic PDEs with obstacle (in short OSPDE). The proofs are based on a version of Ito's formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary.

Near Field Photometric Stereo with Point Light Sources

Shape recovery of an object based on shading variations resulting from different light sources has recently been reconsidered. Improvements have been made that allow for the photometric stereo approach to serve as a competitive alternative to other shape reconstruction methods. However, most photometric stereo methods tend to ignore factors that are relevant in practical applications. The setup considered in this paper tackles photometric stereo reconstruction in the case of a specific near-field imaging. This means that both the camera and the light sources are close to the imaged object, where close can be loosely considered as a setup having similar distances between lights, camera, and object. The known challenges that characterize the problem involve perspective viewing geometry, point light sources, and images that may include shadowed regions. Here, we pay special attention to the question of how to faithfully model these aspects and at the same time design an efficient and robust numerical solver. We present a mathematical formulation that integrates the above assumptions into a single coherent model based on quasi-linear PDEs. The well-posedness is proved showing uniqueness of a weak (i.e., Lipschitz continuous) solution. The surface reconstruction in our near-field scenario can then be executed efficiently in linear time. The merging strategy of the irradiance equations provided for each light source allows us to consider a characteristic expansion model which enables the direct computation of the surface. We evaluate several types of light attenuation models with a nonuniform albedo and noise on synthetic data. We also demonstrate the proposed method on surface reconstruction of real data using three images, each one taken with a different light source by a working prototype. We demonstrate the accuracy of the proposed method compared to other methods that ignore the near-field setup and assume distant, parallel beam light sources.

Backstepping Observers for Periodic Quasi-Linear Parabolic PDEs

The extension of the backstepping-based state observer design is considered for general periodic quasi-linear parabolic PDEs in one space dimension. Here, the extended linearization combined with a suitable coordinate and state transformation is employed. This allows for the application of the backstepping method for the determination of an extended Luenberger observer with observer gains entering the PDE and the boundary condition which ensures an exponential decay of the linearized observer error dynamics. This is confirmed analytically, while the convergence of the quasi-linear observer error system is studied in numerical simulations.