Elsevier

Journal of Differential Equations

Volume 296, 25 September 2021, Pages 724-758
Journal of Differential Equations

On Lp-viscosity solutions of parabolic bilateral obstacle problems with unbounded ingredients

https://doi.org/10.1016/j.jde.2021.06.015Get rights and content

Abstract

The global equi-continuity estimate on Lp-viscosity solutions of parabolic bilateral obstacle problems with unbounded ingredients is established when obstacles are merely continuous. The existence of Lp-viscosity solutions is established via an approximation of given data. The local Hölder continuity estimate on the space derivative of Lp-viscosity solutions is shown when the obstacles belong to C1,β, and p>n+2.

Introduction

In this paper, we consider the following parabolic bilateral obstacle problemmin{max{ut+F(x,t,Du,D2u)f,uψ},uφ}=0in ΩT under the Cauchy-Dirichlet condition u=g on pΩT. Here, ΩT:=Ω×(0,T] for a bounded domain ΩRn and T>0, F is at least a measurable function on ΩT×Rn×Sn, and f, φ, ψ and g are given. We denote Sn by the set of all n×n real-valued symmetric matrices with the standard order, and setSλ,Λn:={XSn:λIXΛI}for 0<λΛ. Moreover, we denote the parabolic boundary of ΩT bypΩT:=Ω×{0}Ω×[0,T).

To begin with, the theory of obstacle problems is motivated by numerous applications, e.g. in stochastic control theory, in economics, in mechanics, in mathematical physics or in mathematical biology.

An existence theory for parabolic unilateral obstacle problems was first introduced by J.-L. Lions and G. Stampacchia in [21]. In [5], regularity of solutions of parabolic unilateral obstacle problems was studied by H. Brézis. Then, A. Friedman in [12], [13] considered stochastic games and studied regularity of solutions of bilateral obstacle problems. Afterwards, there appeared numerous researches on parabolic obstacle problems when F are partial differential operators of divergence form. We only refer to [14], [17], [2], [15], [4], [25] and references therein for the existence and regularity of solutions of parabolic obstacle problems and applications.

In [23], [26], we considered unilateral obstacle problems for fully nonlinear uniformly parabolic operators under appropriate assumptions for applying the regularity theory of viscosity solutions in [27], [28], [29]. It is natural to ask whether the results can be extended to bilateral obstacle problems. We refer to [24] for an accomplished overview of bilateral obstacle problems.

Although bilateral obstacle problems have been studied since the 1960s, some results on bilateral obstacle problems for non-divergence form operators only have been obtained very recently. In particular, L.F. Duque in [10] showed interior Hölder estimates on viscosity solutions of bilateral obstacle problems for fully nonlinear uniformly parabolic operators with no variable coefficients, no first derivative terms and constant inhomogeneous terms when the obstacles are independent of time and Hölder continuous;{F(x,t,ξ,X)=F(X) for (x,t,ξ,X)ΩT×Rn×Sn,fC,φ(x,t)=φ(x),ψ(x,t)=ψ(x) for (x,t)ΩT,φ,ψCα(ΩT) for α(0,1). Under the above hypotheses, in [10], we obtain the existence of viscosity solutions of (1.1) under the Cauchy-Dirichlet condition, and interior Hölder estimates on the space derivative when the obstacles are in C1,β for β(0,1) and separated. The corresponding results for elliptic problems are also established in [10]. It was later extended for fully nonlinear uniformly elliptic equations with unbounded coefficients and inhomogeneous terms in [20]. We will give the definition of Ck,α for k=0, 1 and α(0,1) in Section 2.

This paper is the parabolic counterpart of [20] on fully nonlinear elliptic bilateral obstacle problems. Our aim in this paper is to extend results in [10] when F is a fully nonlinear uniformly parabolic operator. More precisely, under more general hypotheses than those in [10], we show the equi-continuity of Lp-viscosity solutions of (1.1) in ΩT, the existence of Lp-viscosity solutions of (1.1), and their local Hölder continuity of space derivatives under additional assumptions. In [10], it is assumed that the obstacles are separated in order to obtain interior Hölder estimates on the space derivative of viscosity solutions of bilateral obstacle problems. In this paper, we remove this hypothesis (for elliptic case see the Appendix).

Because most of results on the equi-continuity and existence of Lp-viscosity solutions of (1.1) follow the same line of arguments as that of its elliptic counterpart used in [20], we shall give the outline of proofs. As for the local Hölder continuity of space derivatives of Lp-viscosity solutions of (1.1), we cannot use our argument used in [20] because the domain, where the infimum is taken, differs from that of the Lε0 (quasi-) norm in the weak Harnack inequality, which arises in Proposition 2.4. Instead, we use a compactness-based technique developed in [26].

For any p>0 and u:ΩTR, we denote the quasi-norm:uLp(ΩT)=(0TΩ|u(x,t)|pdxdt)1p. We note that Lp(ΩT) satisfiesu+vLp(ΩT)Cp(uLp(ΩT)+vLp(ΩT))for some Cp1. Notice that we may choose Cp=1 when p1.

This paper is organized as follows: In Section 2, we recall the definition of Lp-viscosity solutions, basic properties and exhibit main results. Section 3 is devoted to the weak Harnack inequality both in KΩT and near pΩT, which yields the global equi-continuity of Lp-viscosity solutions. In Section 4, we establish the existence of Lp-viscosity solutions of (1.1) when the obstacles are only continuous under appropriate hypotheses. We obtain Hölder estimates on the space derivative of Lp-viscosity solutions in Section 5.

Section snippets

Preliminaries and main results

For (x,t)Rn+1 and r>0, we setBr:={yRn:|y|<r},Br(x):=x+Br,Qr:=Br×(r2,0],andQr(x,t):=(x,t)+Qr. For any measurable set ARn+1, we denote by |A| the (n+1)-dimensional Lebesgue measure of A. The parabolic distance is defined byd((x,t),(y,s)):=|xy|2+|ts|. For U, VRn+1, we define the distance between U and V bydist(U,V):=inf{d((x,t),(y,s)):(x,t)U,(y,s)V}. In what follows, KΩT means that KΩT is a compact set satisfying dist(K,pΩT)>0.

We denote by C2,1(ΩT) the space of functions uC(ΩT) such

Global equi-continuity estimates

In what follows, under (2.9), we denote by σ0 the modulus of continuity of φ and ψ in ΩT:σ0(r):=sup{|φ(x,t)φ(y,s)||ψ(x,t)ψ(y,s)|:(x,t),(y,s)ΩTd((x,t),(y,s))<r}.

Existence results

In this section, we present an existence result of Lp-viscosity solutions of (1.1) under suitable conditions when obstacles are merely continuous.

Using the parabolic mollifier by ρC0(Rn+1) with ρ0 in Rn+1, ρ0 in Rn+1Q1 and Rn+1ρdxdt=1, we introduce smooth approximations of f, μ and F byfε:=fρε,με:=μρε andFε(x,t,ξ,X):=Rn+1ρε(xy,ts)F(y,s,ξ,X)dyds for (x,t,ξ,X)Rn+1×Rn×Sn, where ρε(x,t):=εn2ρ(xε,tε2). Here and later, we use the same notion f, μ and F for their zero extension outside

Estimates in the non-coincidence set

We first note that Lp-viscosity solutions uC(ΩT) of (1.1) are also Lp-viscosity solutions ofut+F(x,t,Du,D2u)f=0in N[u].

For any compact set KN[u], where uC(ΩT) is an Lp-viscosity solution of (1.1), we first show that DuCβ(K) for some β(0,1).

Proposition 5.1

(cf. Theorem 7.3 in [9]) Assume (2.13), (2.3), (2.4), (2.5). Let β(0,βˆβ0) be fixed, where βˆ is from Proposition 2.7, and β0=1n+2p. Then, there are δ0>0 and r1>0, depending on n, Λ, λ, p, q and β, such that if uC(ΩT) is an Lp-viscosity solution of

Appendix: Local Hölder continuity of derivatives for elliptic problems

In this section, we consider the following elliptic bilateral obstacle problemsmin{max{F(x,Du,D2u)f,uψ},uφ}=0in Ω, where ΩRn is a bounded domain. Hereafter, under the hypothesisqp>n, we define β0(0,1) byβ0=1np. Suppose thatfLp(Ω). The structure condition on F is that there existsμLq(Ω),μ0in Ω such that for xΩ, ξ, ζRn, X, YSn,{F(x,0,O)=0, andP(XY)μ(x)|ξζ|F(x,ξ,X)F(x,ζ,Y)P+(XY)+μ(x)|ξζ|. For obstacles φ and ψ, we suppose thatφψin Ω,andφ,ψC1,β1(Ω)for β1(0,1).

Under the

Acknowledgements

The author would like to thank Professor S. Koike for his helpful comments. The author is supported by Grant-in-Aid for Japan Society for Promotion Science Research Fellow 16J02399, in part by Grant-in-Aid for Scientific Research (No. 16H06339) of Japan Society for Promotion Science, and by Foundation of Research Fellows, The Mathematical Society of Japan.

References (29)

  • S. Koike et al.

    Weak Harnack inequality for fully nonlinear uniformly parabolic equations with unbounded ingredients and applications

    Nonlinear Anal.

    (2019)
  • A. Attouchi et al.

    Hölder regularity for the gradient of the inhomogeneous parabolic normalized p-Laplacian

    Commun. Contemp. Math.

    (2018)
  • A. Bensoussan et al.

    Applications of Variational Inequalities in Stochastic Control

    (1982)
  • H. Berestycki et al.

    The principal eigenvalue and maximum principle for second-order elliptic operators in general domains

    Commun. Pure Appl. Math.

    (1994)
  • V. Barbu

    Optimal Control of Variational Inequalities

    (1984)
  • H. Brézis

    Problémes unilatéraux

    J. Math. Pures Appl. (9)

    (1972)
  • L.A. Caffarelli et al.

    On viscosity solutions of fully nonlinear equations with measurable ingredients

    Commun. Pure Appl. Math.

    (1996)
  • M.G. Crandall et al.

    User's guide to viscosity solutions of second order partial differential equations

    Bull. Am. Math. Soc. (N.S.)

    (1992)
  • M.G. Crandall et al.

    Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations

    Electron. J. Differ. Equ.

    (1999)
  • M.G. Crandall et al.

    Lp-theory for fully nonlinear uniformly parabolic equations

    Commun. Partial Differ. Equ.

    (2000)
  • L.F. Duque

    The double obstacle problem on non divergence form

  • L. Escauriaza

    W2,n a priori estimates for solutions to fully nonlinear equations

    Indiana Univ. Math. J.

    (1993)
  • A. Friedman

    Stochastic games and variational inequalities

    Arch. Ration. Mech. Anal.

    (1973)
  • A. Friedman

    Regularity theorems for variational inequalities in unbounded domains and applications to stopping time problems

    Arch. Ration. Mech. Anal.

    (1973)
  • Cited by (0)

    View full text