On Lp-viscosity solutions of parabolic bilateral obstacle problems with unbounded ingredients
Introduction
In this paper, we consider the following parabolic bilateral obstacle problem under the Cauchy-Dirichlet condition on . Here, for a bounded domain and , F is at least a measurable function on , and f, φ, ψ and g are given. We denote by the set of all real-valued symmetric matrices with the standard order, and set Moreover, we denote the parabolic boundary of by
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; 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 for 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 for , 1 and 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 -viscosity solutions of (1.1) in , the existence of -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 -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 -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 (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 and , we denote the quasi-norm: We note that satisfies Notice that we may choose when .
This paper is organized as follows: In Section 2, we recall the definition of -viscosity solutions, basic properties and exhibit main results. Section 3 is devoted to the weak Harnack inequality both in and near , which yields the global equi-continuity of -viscosity solutions. In Section 4, we establish the existence of -viscosity solutions of (1.1) when the obstacles are only continuous under appropriate hypotheses. We obtain Hölder estimates on the space derivative of -viscosity solutions in Section 5.
Section snippets
Preliminaries and main results
For and , we set For any measurable set , we denote by the -dimensional Lebesgue measure of A. The parabolic distance is defined by For U, , we define the distance between U and V by In what follows, means that is a compact set satisfying .
We denote by the space of functions such
Global equi-continuity estimates
In what follows, under (2.9), we denote by the modulus of continuity of φ and ψ in :
Existence results
In this section, we present an existence result of -viscosity solutions of (1.1) under suitable conditions when obstacles are merely continuous.
Using the parabolic mollifier by with in , in and , we introduce smooth approximations of f, μ and F by and for , where . 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 -viscosity solutions of (1.1) are also -viscosity solutions of
For any compact set , where is an -viscosity solution of (1.1), we first show that for some . Proposition 5.1 (cf. Theorem 7.3 in [9]) Assume (2.13), (2.3), (2.4), (2.5). Let be fixed, where is from Proposition 2.7, and . Then, there are and , depending on n, Λ, λ, p, q and β, such that if is an -viscosity solution of
Appendix: Local Hölder continuity of derivatives for elliptic problems
In this section, we consider the following elliptic bilateral obstacle problems where is a bounded domain. Hereafter, under the hypothesis we define by Suppose that The structure condition on F is that there exists such that for , ξ, , X, , For obstacles φ and ψ, we suppose that
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)
- et al.
Weak Harnack inequality for fully nonlinear uniformly parabolic equations with unbounded ingredients and applications
Nonlinear Anal.
(2019) - et al.
Hölder regularity for the gradient of the inhomogeneous parabolic normalized p-Laplacian
Commun. Contemp. Math.
(2018) - et al.
Applications of Variational Inequalities in Stochastic Control
(1982) - et al.
The principal eigenvalue and maximum principle for second-order elliptic operators in general domains
Commun. Pure Appl. Math.
(1994) Optimal Control of Variational Inequalities
(1984)Problémes unilatéraux
J. Math. Pures Appl. (9)
(1972)- et al.
On viscosity solutions of fully nonlinear equations with measurable ingredients
Commun. Pure Appl. Math.
(1996) - et al.
User's guide to viscosity solutions of second order partial differential equations
Bull. Am. Math. Soc. (N.S.)
(1992) - et al.
Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations
Electron. J. Differ. Equ.
(1999) - et al.
-theory for fully nonlinear uniformly parabolic equations
Commun. Partial Differ. Equ.
(2000)