Parabolic Obstacle Problem Research Articles

Uniform estimates near the initial state for solutions of the two-phase parabolic problem

This paper is devoted to a proof of optimal regularity, near the initial state, for weak solutions to the two-phase parabolic obstacle problem. The approach used here is general enough to allow us to consider the initial data belonging to the class $C^{1,1}$.

Calderón–Zygmund theory for parabolic obstacle problems with nonstandard growth

Abstract. We establish local Calderón–Zygmund estimates for solutions to certain parabolic problems with irregular obstacles and nonstandard p ( x , t ) ${p(x,t)}$ -growth. More precisely, we will show that the spatial gradient D u ${Du}$ of the solution to the obstacle problem is as integrable as the obstacle ψ ${\psi }$ , i.e. | D ψ | p ( · ) , | ∂ t ψ | γ 1 ' ∈ L loc q ⇒ | D u | p ( · ) ∈ L loc q , for any q > 1 , $ |D\psi |^{p(\,\cdot \,)},|\partial _t\psi |^{\gamma _1^{\prime }}\in L^q_\mathrm {loc}\Rightarrow |Du|^{p(\,\cdot \,)}\in L^q_\mathrm {loc},\quad \text{for any}~q>1, $ where γ 1 ' = γ 1 γ 1 - 1 ${\gamma _1^{\prime }=\frac{\gamma _1}{\gamma _1-1}}$ and γ 1 ${\gamma _1}$ is the lower bound for p ( · ) ${p(\,\cdot \,)}$ .

Hölder estimates for parabolic obstacle problems

In this paper, we establish the local Hölder continuity of the spatial gradient of the solution \(u\) to the parabolic obstacle problem with superquadratic growth. More precisely, we prove that $$\begin{aligned} Du\in C^{0;\alpha ,\frac{\alpha }{2}}_\mathrm {loc}\qquad \text {for some}~\alpha \in (0,1), \end{aligned}$$provided the coefficients and the obstacle are regular enough.

Lorentz estimates for obstacle parabolic problems

We prove that the spatial gradient of (variational) solutions to parabolic obstacle problems of p-Laplacian type enjoys the same regularity of the data and of the derivatives of the obstacle in the scale of Lorentz spaces.

Quasilinear parabolic variational inequalities with multi-valued lower-order terms

In this paper, we provide an analytical frame work for the following multi-valued parabolic variational inequality in a cylindrical domain \({Q = \Omega \times (0, \tau)}\) : Find \({{u \in K}}\) and an \({{\eta \in L^{p'}(Q)}}\) such that $$\eta \in f(\cdot,\cdot,u), \quad \langle u_t + Au, v - u\rangle + \int_Q \eta (v - u)\,{\rm d}x{\rm d}t \ge 0, \quad \forall \, v \in K,$$ where \({{K \subset X_0 = L^p(0,\tau;W_0^{1,p}(\Omega))}}\) is some closed and convex subset, A is a time-dependent quasilinear elliptic operator, and the multi-valued function \({{s \mapsto f(\cdot,\cdot,s)}}\) is assumed to be upper semicontinuous only, so that Clarke’s generalized gradient is included as a special case. Thus, parabolic variational–hemivariational inequalities are special cases of the problem considered here. The extension of parabolic variational–hemivariational inequalities to the general class of multi-valued problems considered in this paper is not only of disciplinary interest, but is motivated by the need in applications. The main goals are as follows. First, we provide an existence theory for the above-stated problem under coercivity assumptions. Second, in the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence, comparison, and enclosure results. Third, the order structure of the solution set enclosed by sub-supersolutions is revealed. In particular, it is shown that the solution set within the sector of sub-supersolutions is a directed set. As an application, a multi-valued parabolic obstacle problem is treated.

Formula omitted]-adaptive IPDG/TDG-FEM for parabolic obstacle problems

For a parabolic obstacle problem two equivalent hp-FEM discretization methods based on interior penalty discontinuous Galerkin in space and discontinuous Galerkin in time are presented. The first approach is based on a variational inequality (VI) formulation and the second approach on a mixed method in which the non-penetration condition is resolved by a Lagrange multiplier. The discrete Lagrange multiplier is a linear combination of biorthogonal basis functions, allowing to write the discrete VI-constraints as a set of complementarity problems. Employing a penalized Fischer–Burmeister non-linear complementarity function, the discrete mixed problem can be solved by a locally Q-quadratic converging semi-smooth Newton (SSN) method. The hierarchical a posteriori error estimator for the VI-formulation, which under the saturation assumption is both efficient and reliable, allows hp-adaptivity. The numerical experiments show improved convergence compared to uniform and h-adaptive meshes. Furthermore, an a priori error estimate is given for the VI-formulation.

Increasing powers in a degenerate parabolic logistic equation

The purpose of this paper is to study the asymptotic behavior of the positive solutions of the problem $$\partial _t u - \Delta u = au - b\left( x \right)u^p in \Omega \times \mathbb{R}^ + , u(0) = u_0 , \left. {u(t)} \right|_{\partial \Omega } = 0,$$ as p → + ∞, where Ω is a bounded domain, and b(x) is a nonnegative function. The authors deduce that the limiting configuration solves a parabolic obstacle problem, and afterwards fully describe its long time behavior.

Area Estimation between the Early Exercise Boundaries for the American Put Option with Different Local Volatilities

In this paper, an estimate of the area between the early exercise boundaries is established for the American put option with different local volatility functions through the uniform distance between the volatilities. The constant factor on the right-hand side of the estimate is given in explicit form. The derivation of the estimate is based on the detailed analysis of the related parabolic obstacle problem with different principal parts in the corresponding differential operator.

Pointwise estimates for the heat equation. Application to the free boundary of the obstacle problem with Dini coefficients

We study the pointwise regularity of solutions to parabolic equations. As a first result, we prove that if the modulus of mean oscillation of $\Delta u -u_t$ at the origin is Dini (in $L^p$ average), then the origin is a Lebesgue point of continuity (still in $L^p$ average) for $D^2 u$ and $\dd_t u$. We extend this pointwise regularity result to the parabolic obstacle problem with Dini right hand side. In particular, we prove that the solution to the obstacle problem has, at regular points of the free boundary, a Taylor expansion up to order two in space and one in time (in the $L^p$ average). Moreover, we get a quantitative estimate of the error in this Taylor expansion. Our method is based on decay estimates obtained by contradiction, using blow-up arguments and Liouville type theorems. As a by-product of our approach, we deduce that the regular points of the free boundary are locally contained in a $C^1$ hypersurface for the parabolic distance $\sqrt{x^2 +|t|}$.

Potential estimates in parabolic obstacle problems

For parabolic obstacle problems with quadratic growth, we give pointwise estimates both for the solutions and their gradients in terms of potentials of the given data. As applications, we derive Lorentz space estimates if the data satisfies the corresponding Lorentz space regularity. Moreover, we discuss a borderline case in the regularity theory, the question of boundedness and continuity of the gradients as well as of the solutions itself. In the present work, we establish pointwise estimates by potentials for solutions to parabolic obstacle problems. Obstacle problems play a prominent role in various applications, for example in mechanics or control theory, cf. (4, 22), but also in other fields of mathematics such as potential theory, where solutions to obstacle problems prove useful as approximations of super-solutions (16, 18, 23). Here, we treat obstacle problems that are related to equations of the type

Higher integrability in parabolic obstacle problems

In this paper we establish the self-improving property of integrability for parabolic variational inequalities satisfying an obstacle constraint and involving possibly degenerate respectively singular operators in divergence form. In particular, our results apply to the model case of the variational inequality associated to the parabolic p-Laplacean operator. Thereby we do not impose any monotonicity assumption in time on the obstacle function.

Optimal investment with transaction costs based on exponential utility function: a parabolic double obstacle problem

This paper concerns optimal investment problem with proportional transaction costs and finite time horizon based on exponential utility function. Using a partial differential equation approach, we reveal that the problem is equivalent to a parabolic double obstacle problem involving two free boundaries that correspond to the optimal buying and selling policies. Numerical examples are obtained by the binomial method.

Hp‐TDG/IPDG for Parabolic Obstacle Problems

AbstractWe present a mixed hp‐FE time DG method combined with an interior penalty DG method for parabolic obstacle problems. The discrete Lagragian multiplier set is spanned by basis functions which are biorthogonal to the basis functions of the primal variable. This allows us to write the discrete mixed problem as a problem of finding the root of a strongly semi‐smooth function. In turn, this problem is solved by a locally Q‐quadratic converging semi‐smooth Newton method. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Degenerate problems with irregular obstacles

We establish the natural Calderón and Zygmund theory for solutions of elliptic and parabolic obstacle problems involving possibly degenerate operators in divergence form of p-Laplacian type, and proving that the (spatial) gradient of solutions is as integrable as that of the assigned obstacles. We also include an existence and regularity theorem where obstacles are not necessarily considered to be non-increasing in time.

Obstacle problem for nonlinear parabolic equations

We show the existence of a continuous solution to a nonlinear parabolic obstacle problem with a continuous time-dependent obstacle. The solution is constructed by an adaptation of the Schwarz alternating method. Moreover, if the obstacle is Hölder continuous, we prove that the solution inherits the same property.

Boundary estimates for solutions to the two-phase parabolic obstacle problem

Estimates for the second-order derivatives of a solution to the two-phase parabolic obstacle problem are established. Similar results in the elliptic case were obtained by the authors in 2006. Bibliography: 4 titles.

A Posteriori Error Estimates for Parabolic Variational Inequalities

We study a posteriori error estimates in the energy norm for some parabolic obstacle problems discretized with a Euler implicit time scheme combined with a finite element spatial approximation. We discuss the reliability and efficiency of the error indicators, as well as their localization properties. Apart from the obstacle resolution, the error indicators vanish in the so-called full contact set. The case when the obstacle is piecewise affine is studied before the general case. Numerical examples are given.

Free boundary regularity close to initial state for parabolic obstacle problem

In this paper we study the behavior of the free boundary ∂ { u > ψ } \partial \{u>\psi \} , arising in the following complementary problem: ( H u ) ( u − ψ ) = 0 , u ≥ ψ ( x , t ) i n Q + , H u ≤ 0 , u ( x , t ) ≥ ψ ( x , t ) o n ∂ p Q + . \begin{gather*} (Hu)(u-\psi )=0,\qquad u\geq \psi (x,t) \quad \mathrm {in}\ Q^+, Hu \leq 0, u(x,t) \geq \psi (x,t) \quad \mathrm {on}\ \partial _p Q^+. \end{gather*} Here ∂ p \partial _p denotes the parabolic boundary, H H is a parabolic operator with certain properties, Q + Q^+ is the upper half of the unit cylinder in R n + 1 \textbf {R}^{n+1} , and the equation is satisfied in the viscosity sense. The obstacle ψ \psi is assumed to be continuous (with a certain smoothness at { x 1 = 0 \{x_1=0 , t = 0 } t=0\} ), and coincides with the boundary data u ( x , 0 ) = ψ ( x , 0 ) u(x,0)=\psi (x,0) at time zero. We also discuss applications in financial markets.

On the size of the non-coincidence set of parabolic obstacle problems with applications to American option pricing

The following paper is devoted to the study of the positivity set $U=\{\mathcal{L}\phi>0\}$ arising in parabolic obstacle problems. It is shown that $U$ is contained in the non-coincidence set with a positive distance between the boundaries uniformly in the spatial variable if the boundary of $U$ satisfies an interior $C^1$-Dini condition in the space variable and a Lipschitz condition in the time variable. We apply our results to American option pricing and we thus show that the positivity set is strictly contained in the continuation region, which means that the option should not be exercised in $U$ or on the boundary of $U$.

Residual-type a posteriori error estimate for parabolic obstacle problems

In this paper, a posteriori error estimates were derived for piecewise linear finite element approximations to parabolic obstacle problems. The instrumental ingredient was introduced as a new interpolation operator which has optimal approximation properties and preserves positivity. With the help of the interpolation operator the upper and lower bounds were obtained.