Obstacle Problem Research Articles

On global solutions of the obstacle problem

On global solutions of the obstacle problem

Noncoercive parabolic obstacle problems

Abstract We prove an existence result for obstacle problems related to convection-diffusion parabolic equations with singular coefficients in the convective term. Our operator is not coercive, the obstacle function is time-dependent irregular, and the coefficients in the lower-order term belong to a borderline mixed Lebesgue-Marcinkiewicz space.

Regularity theory for fully nonlinear parabolic obstacle problems

We study the free boundary of solutions to the parabolic obstacle problem with fully nonlinear diffusion. We show that the free boundary splits into a regular and a singular part: near regular points the free boundary is C∞ in space and time. Furthermore, we prove that the set of singular points is locally covered by a Lipschitz manifold of dimension n−1 which is also ε-flat in space, for any ε>0.

Collision avoidance fault-tolerant control for dynamic positioning vessels under thruster faults

This paper explores a collision avoidance fault-tolerant control scheme for dynamic positioning vessels in the presence of velocity constraints and thruster faults. First, the collision avoidance problem for multiple static obstacles is described as security zone constraint and integrated into the controller design via an artificial potential function. Second, an indirect error vector including the gradient of potential function and vessel velocity information is constructed, and a time-varying asymmetric barrier Lyapunov function is adopted to concurrently guarantee the satisfaction of velocity constraints. With the help of robust adaptive techniques, the derived controller not only provides robustness against unknown parameters affections, ocean disturbances, and thruster faults, but also has a simple structure and inexpensive online computations for real-time implementation. Finally, the asymptotic convergence of the gradient of potential function and vessel velocity is demonstrated, while a positioning control task for a dynamic positioning vessel is presented to evaluate the effectiveness of the proposed method.

Existence and regularity for nonlinear parabolic double obstacle problems

We study nonlinear parabolic equations of p-Laplacian type with irregular double obstacles to establish the existence and an optimal global Calderón–Zygmund theory.

Contact points with integer frequencies in the thin obstacle problem

AbstractFor the thin obstacle problem, we develop a unified approach that leads to rates of convergence to blow‐up profiles at contact points with integer frequencies. For these points, we also obtain a stratification result.

Fractional differentiability for elliptic double obstacle problems with measure data

We consider elliptic double obstacle problems involving p -Laplacian type operators and measure data. We prove maximal fractional differentiability results in a completely linearized form under suitable differentiability assumptions on the obstacles. We also investigate fractional Sobolev–Morrey regularity for such problems under a density condition on the measures.

A neural network warm-start approach for the inverse acoustic obstacle scattering problem

In this paper, we consider the inverse acoustic obstacle problem for sound-soft star-shaped obstacles in two dimensions wherein the boundary of the obstacle is determined from measurements of the scattered field at a collection of receivers outside the object. One of the standard approaches for solving this problem is to reformulate it as an optimization problem: finding the boundary of the domain that minimizes the L2 distance between computed values of the scattered field and the given measurement data. The optimization problem is computationally challenging since the local set of convexity shrinks with increasing frequency and results in an increasing number of local minima in the vicinity of the true solution. In many practical experimental settings, low frequency measurements are unavailable due to limitations of the experimental setup or the sensors used for measurement. Thus, obtaining a good initial guess for the optimization problem plays a vital role in this environment.We present a neural network warm-start approach for solving the inverse scattering problem, where an initial guess for the optimization problem is obtained using a trained neural network. We demonstrate the effectiveness of our method with several numerical examples. For high frequency problems, this approach outperforms traditional iterative methods such as Gauss-Newton initialized without any prior (i.e., initialized using a unit circle), or initialized using the solution of a direct method such as the linear sampling method. The algorithm remains robust to noise in the scattered field measurements and also converges to the true solution for limited aperture data. However, the number of training samples required to train the neural network scales exponentially in frequency and the complexity of the obstacles considered. We conclude with a discussion of this phenomenon and potential directions for future research.

Domain variations of the first eigenvalue via a strict Faber-Krahn type inequality

For $d\geq 2$ and $\frac{2d+2}{d+2} < p < \infty $, we prove a strict Faber-Krahn type inequality for the first eigenvalue $\lambda _1(\Omega )$ of the $p$-Laplace operator on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^d$ (with mixed boundary conditions) under the polarizations. We apply this inequality to the obstacle problems on the domains of the form $\Omega \setminus \mathscr{O}$, where $\mathscr{O}\subset \subset \Omega $ is an obstacle. Under some geometric assumptions on $\Omega $ and $\mathscr{O}$, we prove the strict monotonicity of $\lambda _1 (\Omega \setminus \mathscr{O})$ with respect to certain translations and rotations of $\mathscr{O}$ in $\Omega $.

Implicit Finite Element Scheme with a Penalty for a Nonlocal Parabolic Obstacle Problem of Kirchhoff Type

Implicit Finite Element Scheme with a Penalty for a Nonlocal Parabolic Obstacle Problem of Kirchhoff Type

The regularity theory for the double obstacle problem for fully nonlinear operator

In this paper, we prove the existence and uniqueness of W2,p (n<p<∞) solutions of a double obstacle problem. Moreover, we show the optimal regularity of the solution and the local C1 regularity of the free boundary under a thickness assumption at the free boundary point on the intersection of two free boundaries. In the study of the regularity of the free boundary, we deal with a general problem, the no-sign reduced double obstacle problem with an upper obstacle ψ, F(D2u,x)=fχΩ(u)∩{u<ψ}+F(D2ψ,x)χΩ(u)∩{u=ψ},u≤ψinB1, where Ω(u)=B1∖{u=0}∩{∇u=0}.

A Nonconstant Gradient Constrained Problem for Nonlinear Monotone Operators

The purpose of the research is the study of a nonconstant gradient constrained problem for nonlinear monotone operators. In particular, we study a stationary variational inequality, defined by a strongly monotone operator, in a convex set of gradient-type constraints. We investigate the relationship between the nonconstant gradient constrained problem and a suitable double obstacle problem, where the obstacles are the viscosity solutions to a Hamilton–Jacobi equation, and we show the equivalence between the two variational problems. To obtain the equivalence, we prove that a suitable constraint qualification condition, Assumption S, is fulfilled at the solution of the double obstacle problem. It allows us to apply a strong duality theory, holding under Assumption S. Then, we also provide the proof of existence of Lagrange multipliers. The elements in question can be not only functions in L2, but also measures.

Analysis of a new kind of elastic-rigid bilateral obstacle problem

This paper is devoted to a study of a new kind of bilateral obstacle problem. The obstacles are made of rigid bodies covered by soft layers which are deformable and allow penetration. A model of an elastic rigid bilateral obstacle problem is established, and three equivalent descriptions are derived: the energy form, the variational inequality form, and the differential equation form. We prove the solution existence and uniqueness of this model and provide an error estimate of the numerical solutions. The optimal order error estimate for the linear finite element method is derived under the proper regularity assumption. A penalty method is introduced to solve the finite element approximation problem, and the convergence results are obtained when the penalty parameter tends to infinity. Several numerical examples are reported, and the results are in good agreement with the theoretical analysis.

A dynamical method for optimal control of the obstacle problem

Abstract In this paper, we consider the numerical method for an optimal control problem governed by an obstacle problem. An approximate optimization problem is proposed by regularizing the original non-differentiable constrained problem with a simple method. The connection between the two formulations is established through some convergence results. A sufficient condition is derived to decide whether a solution of the first-order optimality system is a global minimum. The method with a second-order in time dissipative system is developed to solve the optimality system numerically. Several numerical examples are reported to show the effectiveness of the proposed method.

Systems of fully nonlinear degenerate elliptic obstacle problems with Dirichlet boundary conditions

In this paper, we prove existence and uniqueness of viscosity solutions to the following system: For $$ i\in \left\{ 1,2,\dots ,m\right\} $$ $$\begin{aligned}{} & {} \min \biggl \{ F\bigl ( y,x,u_{i}(y,x),D u_{i}(y,x),D^2 u_{i}(y,x)\bigl ), u_{i}(y,x)-\max _{j\ne i}\bigl ( u_{j}(y,x)-c_{ij}(y,x)\bigl )\biggl \}\\{} & {} =0, \left( y,x \right) \in \Omega _{L}\\{} & {} u_{i}(0,x)=g_{i}(x), x\in \bar{\Omega },\ u_i(y,x)=f_i(y,x), (y,x)\in (0,L)\times \partial {\Omega } \end{aligned}$$ where $$ \Omega \subset \mathbb {R}^n $$ is a bounded domain, $$ \Omega _{L}:=(0,L)\times \Omega $$ and $$ F:\left[ 0,L\right] \times \mathbb {R}^n\times \mathbb {R}\times \mathbb {R}^n\times \mathcal {S}^n\rightarrow \mathbb {R}$$ is a general second-order partial differential operator which covers even the fully nonlinear case. (We will call a second-order partial differential operator $$F:\left[ 0,L\right] \times \mathbb {R}^n\times \mathbb {R}\times \mathbb {R}^n\times \mathcal {S}^n\rightarrow \mathbb {R}$$ fully nonlinear if and only if, it has the following form $$\begin{aligned} F \left( y,x,u,D_x u,D_{xx}^2 u\right) :=\sum _{|\alpha |=2}\alpha _{\alpha }\left( y,x,u,D_x u,D_{xx}^2 u \right) D^{\alpha }u(y,x)+\alpha _{0}\left( y,x,u,D_x u \right) \end{aligned}$$ with the restriction that at least one of the functional coefficients $$ \alpha _{\alpha },\ |\alpha |=2, $$ contains a partial derivative term of second order.) Moreover, F belongs to an appropriate subclass of degenerate elliptic operators. Regarding uniqueness, we establish a comparison principle for viscosity sub and supersolutions of the Dirichlet problem. This system appears among others in the theory of the so-called optimal switching problems on bounded domains.

On inverse problems arising in fractional elasticity

We first formulate an inverse problem for a linear fractional Lamé system. We determine the Lamé parameters from exterior partial measurements of the Dirichlet-to-Neumann map. We further study an inverse obstacle problem as well as an inverse problem for a nonlinear fractional Lamé system. Our arguments are based on the unique continuation property for the fractional operator as well as the associated Runge approximation property.

Zero-sum stochastic differential games of impulse versus continuous control by FBSDEs

We consider a stochastic differential game in the context of forward-backward stochastic differential equations, where one player implements an impulse control while the opponent controls the system continuously. Utilizing the notion of “backward semigroups” we first prove the dynamic programming principle (DPP) for a truncated version of the problem in a straightforward manner. Relying on a uniform convergence argument then enables us to show the DPP for the general setting. Our approach avoids technical constraints imposed in previous works dealing with the same problem and, more importantly, allows us to consider impulse costs that depend on the present value of the state process in addition to unbounded coefficients. Using the dynamic programming principle we deduce that the upper and lower value functions are both solutions (in viscosity sense) to the same Hamilton-Jacobi-Bellman-Isaacs obstacle problem. By showing uniqueness of solutions to this partial differential inequality we conclude that the game has a value.

Supremum-norm a posteriori error control of quadratic discontinuous Galerkin methods for the obstacle problem

Supremum-norm a posteriori error control of quadratic discontinuous Galerkin methods for the obstacle problem

Obstacle problem for a stochastic conservation law and Lewy Stampacchia inequality

Our interest in this paper evolves around rigorous mathematical formalisation of obstacle problem for stochastic conservation laws. We formulate a stochastic entropy solution framework for a bilateral obstacle problem and establish well-posedness. Our method primarily relies on a combination of techniques involving penalization and regularisation by viscous/parabolic perturbation. In tandem with Kruzkhov's doubling of the variables method, this combination manifests itself via Lagrange multipliers and we are able to obtain existence and uniqueness of entropy solution.

On a class of nonlocal obstacle type problems related to the distributional Riesz fractional derivative

In this work, we consider the nonlocal obstacle problem with a given obstacle \psi in a bounded Lipschitz domain \Omega in \mathbb{R}^{d} , such that \mathbb{K}_\psi^s=\{v\in H^s_0(\Omega):v\geq\psi \text{ a.e. in }\Omega\}\neq\emptyset , given by u\in\mathbb{K}_\psi^s:\quad\langle\mathcal{L}_au,v-u\rangle\geq\langle F,v-u\rangle\quad\forall v\in\mathbb{K}^s_\psi, for F in H^{-s}(\Omega) , the dual space of the fractional Sobolev space H^s_0(\Omega) , 0&lt;s&lt;1 . The nonlocal operator \mathcal{L}_a:H^s_0(\Omega)\to H^{-s}(\Omega) is defined with a measurable, bounded, strictly positive singular kernel a(x,y):\mathbb{R}^d\times\mathbb{R}^d\to[0,\infty) , by the bilinear form \langle\mathcal{L}_au,v\rangle=\mathrm{P.V.}\int_{\mathbb{R}^d}\int_{\mathbb{R}^d} \tilde{v}(x)(\tilde{u}(x)-\tilde{u}(y))a(x,y) \,dy\,dx=\mathcal{E}_a(u,v), which is a (not necessarily symmetric) Dirichlet form, where \tilde{u},\tilde{v} are the zero extensions of u and v outside \Omega respectively. Furthermore, we show that the fractional operator \tilde{\mathcal{L}}_A=-D^s\cdot AD^s:H^s_0(\Omega)\to H^{-s}(\Omega) defined with the distributional Riesz fractional D^s and with a measurable, bounded matrix A(x) corresponds to a nonlocal integral operator \mathcal{L}_{k_A} with a well-defined integral singular kernel a=k_A . The corresponding s -fractional obstacle problem for \tilde{\mathcal{L}}_A is shown to converge as s\nearrow1 to the obstacle problem in H^1_0(\Omega) with the operator -D\cdot AD given with the classical gradient D . We mainly consider obstacle type problems involving the bilinear form \mathcal{E}_a with one or two obstacles, as well as the N -membranes problem, thereby deriving several results, such as the weak maximum principle, comparison properties, approximation by bounded penalization, and also the Lewy–Stampacchia inequalities. This provides regularity of the solutions, including a global estimate in L^\infty(\Omega) , local Hölder regularity of the solutions when a is symmetric, and local regularity in fractional Sobolev spaces W^{2s,p}_{\mathrm{loc}}(\Omega) and in C^1(\Omega) when \mathcal{L}_a=(-\Delta)^s corresponds to fractional s -Laplacian obstacle type problems u\in\mathbb{K}^s_\psi(\Omega):\quad\int_{\mathbb{R}^d}(D^su-{f})\cdot D^s(v-u)\,dx\geq0\quad\forall v\in\mathbb{K}^s_\psi\text{ for }{f}\in [L^2(\mathbb{R}^d)]^d. These novel results are complemented with the extension of the Lewy–Stampacchia inequalities to the order dual of H^s_0(\Omega) and some remarks on the associated s -capacity and the s -nonlocal obstacle problem for a general \mathcal{L}_a .