Regularity results for solutions to obstacle problems with Sobolev coefficients

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Abstract

We establish the higher differentiability of solutions to a class of obstacle problems of the typemin{Ωf(x,Dv(x))dx:vKψ(Ω)}, where ψ is a fixed function called obstacle, Kψ(Ω)={vWloc1,p(Ω,R):vψ a.e. in Ω} and the convex integrand f satisfies p-growth conditions with respect to the gradient variable. We derive that the higher differentiability property of the weak solution v is related to the regularity of the assigned ψ, under a suitable Sobolev assumption on the partial map xDξf(x,ξ). The main novelty is that such assumption is independent of the dimension n and this, in the case pn2, allows us to manage coefficients in a Sobolev class below the critical one W1,n.

Introduction

We are interested in the study of the regularity of the gradient of the solutions to variational obstacle problems of the formmin{Ωf(x,Dv(x)):vKψ(Ω)}, where ΩRn is a bounded open set, ψ:Ω[,+) belonging to the Sobolev class Wloc1,p+22 is the obstacle, andKψ(Ω)={vWloc1,p(Ω,R):vψ a.e. in Ω} is the class of the admissible functions.

Let us observe that uWloc1,p(Ω) is a solution to the obstacle problem (1.1) in Kψ(Ω) if and only if uKψ(Ω) and u is a solution to the variational inequalityΩA(x,Du(x)),D(φ(x)u(x))dx0φKψ(Ω), where the operator A(x,ξ):Ω×RnRn is defined as followsAi(x,ξ)=Dξif(x,ξ)i=1,...,n.

We assume that A is a p-harmonic operator, that satisfies the following p-ellipticity and p-growth conditions with respect to the ξ-variable. There exist positive constants ν,L, and an exponent p2 and a parameter 0μ1 such thatA(x,ξ)A(x,η),ξην|ξη|2(μ2+|ξ|2+|η|2)p22|A(x,ξ)A(x,η)|L|ξη|(μ2+|ξ|2+|η|2)p22|A(x,ξ)|(μ2+|ξ|2)p12, for all ξ,ηRn and for almost every xΩ.

The regularity for solutions of obstacle problems has been object of intense study not only in the case of variational inequalities modeled upon the p-Laplacian energy [11], [12], [20], [40] but also in the case of more general structures [6], [7], [16], [23], [24].

It is usually observed that the regularity of solutions to the obstacle problems depends on the regularity of the obstacle itself: for linear problems the solutions are as regular as the obstacle; this is no longer the case in the nonlinear setting for general integrands without any specific structure. Hence along the years, in this situation there has been an intense research activity in which extra regularity has been imposed on the obstacle to balance the nonlinearity (see [4], [5], [12], [22], [23], [38]).

In some very recent papers the authors analyzed how an extra differentiability of integer or fractional order of the gradient of the obstacle transfers to the gradient of the solutions (see [20], [21]).

The analysis comes from the fact that the regularity of the solutions to the obstacle problem (1.1) is strictly connected to the analysis of the regularity of the solutions to partial differential equation of the formdivA(x,Du)=divA(x,Dψ). It is well known that no extra differentiability properties for the solutions can be expected even if the obstacle ψ is smooth, unless some assumption is given on the x-dependence of the operator A. Therefore, inspired by recent results concerning the higher differentiability of integer ([8], [14], [17], [18], [19], [26], [27], [30], [31], [33], [41], [42]) and fractional ([2], [13], [32]) order for the solutions to elliptic equations or systems, in a number of papers the higher differentiability of the solution of an obstacle problem is proved under a suitable Sobolev assumption on the partial map xA(x,ξ). More precisely, in [20], the higher differentiability of the solution of an homogeneous obstacle problem with the energy density satisfying p-growth conditions is proved; in [21] the integrand f depends also on the v variable; in [24], [25] the energy density satisfies (p,q)-growth conditions. The nonhomogeneous obstacle problem is considered in [39] when the energy density satisfies p-growth conditions and in [10] when the energy density satisfies (p,q)-growth conditions. All previous quoted higher differentiability results have been obtained under a W1,r with rn Sobolev assumption on the dependence on x of the operator A; some of them reveal also crucial in order to prove local Lipschitz continuity results for the obstacle problem, see for istance in [3], [10], [15].

It is well known that the local boundedness of the solutions to a variational problem is a turning point in the regularity theory. Actually, in [33] it has been proved that, when dealing with bounded solutions to (1.3), the higher differentiability holds true under weaker assumptions on the partial map xA(x,ξ) with respect to W1,n. Recently, in [10] it has been proved that a local bound assumption on the obstacle ψ implies a local bound for the solutions to the obstacle problem (1.1), and this allows us to prove that the higher differentiability of solutions to (1.1) persists assuming that the partial map xA(x,ξ) belongs to a Sobolev class that is not related to the dimension n but to the growth exponent of the functional.

More precisely, we assume that there exists a non-negative function κLlocp+2(Ω) such that|A(x,ξ)A(y,ξ)|(κ(x)+κ(y))|xy|(μ2+|ξ|2)p12 for almost every x,yΩ and for all ξRn. The condition (A4) is equivalent to assume that the operator A has a Sobolev-type dependence on the x-variable (see [35]). Such assumption has been use for non constrained minimizers in [36], [37].

We will prove a higher differentiability result assuming that DψWloc1,p+22(Ω). More precisely, we shall prove the following

Theorem 1.1

Let A(x,ξ) satisfy the conditions (A1)(A4) for an exponent p2 and let uKψ(Ω) be a solution to the obstacle problem (1.2). Then, if ψLloc(Ω) the following implication holdsDψWloc1,p+22(Ω)(μ2+|Du|2)p24DuWloc1,2(Ω), with the following estimateBR4|D[(μ2+|Du(x)|2)p24Du(x)]|2dxc(ψL2+uLp(BR)2)Rp+22BR[1+|D2ψ(x)|p+22+|Dψ(x)|p+22+κp+2(x)+|Du(x)|p]dx.

Note that, in the case p<n2, Theorem 1.1 improves the results in [20] and [21]. The proof of Theorem 1.1 is achieved combining a suitable a priori estimate for the second derivative of the local solutions, obtained using the difference quotient method, with a suitable approximation argument. The local boundedness allows us to use an interpolation inequality that gives the higher local integrability Lp+2 of the gradient of the solutions. Such higher integrability is the key tool in order to weaken the assumption on κ that in previous results has been assumed at least in Ln.

Moreover, our result is obtained under a weaker assumption also on the gradient of the obstacle. Indeed, previous results assumed DψW1,p while our assumption is DψW1,p+22 with p>2.

Finally, we observe that the assumption of boundedness of the obstacle ψ is needed to get the boundedness of the solution (see Theorem 2.2). Therefore if we deal with a priori bounded minimizers, then the result holds without the hypothesis ψL.

Section snippets

Notations and preliminary results

In this section we list the notations that we use in this paper and recall some tools that will be useful to prove our results.

We shall follow the usual convention and denote by C or c a general constant that may vary on different occasions, even within the same line of estimates. Relevant dependencies on parameters and special constants will be suitably emphasized using parentheses or subscripts. The norm we use on Rn, will be the standard Euclidean one.

For a C2 function f:Ω×RnR, we writeDξf(x

Proof of the Theorem 1.1

The proof of the theorem will be divided in two steps: in the first one, we will establish the a priori estimate, while in the second one we will conclude through an approximation argument.

Proof

Step 1: The a priori estimate. Suppose that u is a local solution to the obstacle problem in Kψ(Ω) such thatDuWloc1,2(Ω)and(μ2+|Du|2)p24DuWloc1,2(Ω). By estimate (2.4) and Lemma 2.1, we also have |Du|Llocp+2(Ω). Note that the a priori assumption |Du|Llocp+2(Ω) implies that the variational inequality (1.2)

Acknowledgements

The third author has been partially supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and by Università degli Studi di Napoli Parthenope through the projects “sostegno alla Ricerca individuale” (triennio 2015 - 2017) and “Sostenibilità, esternalità e uso efficiente delle risorse ambientali” (triennio 2017-2019).

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