Double Obstacle Problem Research Articles

Doubly reflected generalized BSDEs with jumps and an obstacle problem of parabolic IPDEs with nonlinear Neumann boundary conditions

Abstract A one-dimensional generalized backward stochastic differential equation with jumps and two barriers is the main objective of this paper. When the generators are monotone and the barriers are right continuous with left limits and completely separated, we prove the existence and uniqueness of a solution. As in application, we provide a probabilistic interpretation of a solution of a double obstacle problem of second-order parabolic integral-partial differential equations with nonlinear Neumann boundary conditions.

Nonlocal Double Phase Implicit Obstacle Problems with Multivalued Boundary Conditions

Nonlocal Double Phase Implicit Obstacle Problems with Multivalued Boundary Conditions

Regularity for non-uniformly elliptic double obstacle problems with fractional maximal operators

Regularity for non-uniformly elliptic double obstacle problems with fractional maximal operators

Generalized solution of the double obstacle problem for Musielak-Orlicz Dirichlet energy integral on metric measure spaces

Generalized solution of the double obstacle problem for Musielak-Orlicz Dirichlet energy integral on metric measure spaces

Regularity for Asymptotically Regular Elliptic Double Obstacle Problems of Multi-phase

Regularity for Asymptotically Regular Elliptic Double Obstacle Problems of Multi-phase

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.

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.

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.

A double obstacle problem in an optimal investment problem

In this paper, we study a double obstacle problem in partial differential equation that arises in an optimization problem in finance. Precisely, we consider the double obstacle problem which is related to the optimal investment problem with proportional transaction costs of an investor with the logarithmic utility in finite time under the constant elasticity of variance (CEV) model.First, we construct a solution of the double obstacle problem and prove the monotonicity of its free boundaries. From the solution to the double obstacle problem, we construct the solution of the optimization problem. Hence, our result regarding monotonicity indicates the optimal strategy for the optimization problem.

Double phase implicit obstacle problems with convection term and multivalued operator

This paper is devoted to studying a complicated implicit obstacle problem involving a nonhomogenous differential operator, called double phase operator, a nonlinear convection term (i.e. a reaction term depending on the gradient), and a multivalued term which is described by Clarke’s generalized gradient. We develop a general framework to deliver an existence result for the double phase implicit obstacle problem under consideration. Our proof is based on the Kakutani–Ky Fan fixed point theorem together with the theory of nonsmooth analysis and a surjectivity theorem for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded pseudomonotone mapping.

The virtual element method for a nonhomogeneous double obstacle problem of Kirchhoff plate

This paper is intended to propose and analyze the lowest order conforming and nonconforming virtual element methods (VEMs) for a nonhomogeneous double obstacle problem in fourth-order variational inequality. We first present an abstract framework for the error analysis of an abstract discrete method. Then, we develop the conforming and fully nonconforming VEMs for the previous problem, with the optimal error estimates obtained by using the previous abstract framework combined with two modified interpolation operators and the related estimates. The discrete problem is solved by the primal–dual active set algorithm and the numerical results are in good agreement with theoretical findings.

An inverse problem for a double phase implicit obstacle problem with multivalued terms

In this paper, we study an inverse problem of estimating three discontinuous parameters in a double phase implicit obstacle problem with multivalued terms and mixed boundary conditions which is formulated by a regularized optimal control problem. Under very general assumptions, we introduce a multivalued function called a parameter-to-solution map which admits weakly compact values. Then, by employing the Aubin-Cellina convergence theorem and the theory of nonsmooth analysis, we prove that the parameter-to-solution map is bounded and continuous in the sense of Kuratowski. Finally, a generalized regularization framework for the inverse problem is developed and a new existence theorem is provided.

Double phase obstacle problems with multivalued convection and mixed boundary value conditions

&lt;p style='text-indent:20px;'&gt;In this paper, we consider a mixed boundary value problem with a double phase partial differential operator, an obstacle effect and a multivalued reaction convection term. Under very general assumptions, an existence theorem for the mixed boundary value problem under consideration is proved by using a surjectivity theorem for multivalued pseudomonotone operators together with the approximation method of Moreau-Yosida. Then, we introduce a family of the approximating problems without constraints corresponding to the mixed boundary value problem. Denoting by &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathcal S $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; the solution set of the mixed boundary value problem and by &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \mathcal S_n $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; the solution sets of the approximating problems, we establish the following convergence relation&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{align*} \emptyset\neq w-\limsup\limits_{n\to\infty}{\mathcal S}_n = s-\limsup\limits_{n\to\infty}{\mathcal S}_n\subset \mathcal S, \end{align*} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;where &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ w $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-&lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ \limsup_{n\to\infty}\mathcal S_n $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ s $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-&lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ \limsup_{n\to\infty}\mathcal S_n $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; stand for the weak and the strong Kuratowski upper limit of &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ \mathcal S_n $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, respectively.&lt;/p&gt;

Lagrange multipliers and nonlinear variational inequalities with gradient constraints.

The paper deals with nonlinear monotone variational inequalities with gradient constraints. In particular, using a new strong duality principle, the equivalence between the problem under consideration and a suitable double obstacle problem is proved. Moreover, the existence of [Formula: see text] Lagrange multipliers is achieved. This article is part of the theme issue 'Non-smooth variational problems and applications'.

Double phase obstacle problems with variable exponent

This paper is devoted to the study of a quasilinear elliptic inclusion problem driven by a double phase differential operator with variable exponents, an obstacle effect and a multivalued reaction term with gradient dependence. By using an existence result for mixed variational inequalities with multivalued pseudomonotone operators and the theory of nonsmooth analysis, we examine the nonemptiness, boundedness and closedness of the solution set to the problem under consideration. In the second part of the paper, we present some convergence analysis for approximated problems. To be more precise, when the obstacle function is approximated by a suitable sequence, applying a generalized penalty technique, we introduce a family of perturbed problems without constraints associated to our problem and prove that the solution set of the original problem can be approached by the solution sets of the perturbed problems in the sense of Kuratowski.

Identification of discontinuous parameters in double phase obstacle problems

Abstract In this article, we investigate the inverse problem of identification of a discontinuous parameter and a discontinuous boundary datum to an elliptic inclusion problem involving a double phase differential operator, a multivalued convection term (a multivalued reaction term depending on the gradient), a multivalued boundary condition and an obstacle constraint. First, we apply a surjectivity theorem for multivalued mappings, which is formulated by the sum of a maximal monotone multivalued operator and a multivalued pseudomonotone mapping to examine the existence of a nontrivial solution to the double phase obstacle problem, which exactly relies on the first eigenvalue of the Steklov eigenvalue problem for the p p -Laplacian. Then, a nonlinear inverse problem driven by the double phase obstacle equation is considered. Finally, by introducing the parameter-to-solution-map, we establish a continuous result of Kuratowski type and prove the solvability of the inverse problem.

Double Phase Implicit Obstacle Problems with Convection and Multivalued Mixed Boundary Value Conditions

In this paper we consider a mixed boundary value problem with a nonhomogeneous, nonlinear differential operator (called double phase operator), a nonlinear convection term (a reaction term depending on the gradient), three multivalued terms and an implicit obstacle constraint. Under very general assumptions on the data, we prove that the solution set of such implicit obstacle problem is nonempty (so there is at least one solution) and weakly compact. The proof of our main result uses the Kakutani-Ky Fan fixed point theorem for multivalued operators along with the theory of nonsmooth analysis and variational methods for pseudomonotone operators.

Upper-Bound Error Estimates for Double Phase Obstacle Problems with Clarke’s Subdifferential

The main goal of this article is to investigate upper-bound error estimates (also called error bounds) for a class of double phase obstacle problems. We first recall double phase implicit obstacle problems involving Clarke’s subdifferential given by Zeng et al. [Calc. Var. Partial Differential Equations 59 (2020): 176]. Then, based on regularized gap functions introduced by Yamashita and Fukushima, we establish some regularized gap functions for the double phase obstacle problem. Finally, using the properties of Clarke’s subdifferential and double phase operators, the upper-bound error estimates for such double phase obstacle problems in terms of regularized gap functions are provided.

Weighted distribution approach to gradient estimates for quasilinear elliptic double-obstacle problems in Orlicz spaces

We construct an efficient approach to deal with the global regularity estimates for a class of elliptic double-obstacle problems in Lorentz and Orlicz spaces. Moreover, in the setting of the paper, we are led to the study of problems with nonlinearity is supposed to be partially weak BMO condition (is measurable in one fixed variable and only satisfies locally small-BMO seminorms in the remaining variables). This work also extends some abstract results proved in Nguyen and Tran (2021) [43] by using the weighted fractional maximal distributions (WFMDs). We further investigate a pointwise estimate of the gradient of weak solutions via fractional maximal operators and Riesz potential of data.