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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Regularity results for p-Laplacians in pre-fractal domains

Raffaela CapitanelliORCID iD: https://orcid.org/0000-0003-1771-1522 / Salvatore Fragapane
  • Dipartimento di Scienze di Base e Applicate per l’Ingegneria, “Sapienza” Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy
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  • De Gruyter OnlineGoogle Scholar
/ Maria Agostina VivaldiORCID iD: https://orcid.org/0000-0003-1931-2402
Published Online: 2018-06-14 | DOI: https://doi.org/10.1515/anona-2017-0248

Abstract

We study obstacle problems involving p-Laplace-type operators in non-convex polygons. We establish regularity results in terms of weighted Sobolev spaces. As applications, we obtain estimates for the FEM approximation for obstacle problems in pre-fractal Koch Islands.

Keywords: Degenerate elliptic equations; smoothness and regularity of solutions; FEM; fractals

MSC 2010: 35J70; 35B65; 65N30; 28A80

1 Introduction

In this paper, we deal with obstacle problems involving p-Laplace-type operators in bad domains in 2. This kind of problems occurs in many mathematical models of physical processes: nonlinear diffusion and filtration, power-law materials and quasi-Newtonian flows (see, for example, [17] and references therein).

Let Ωω denote a conical domain (see Section 2 for definitions and properties) and let us consider the two obstacle problem:

find u𝒦 such thatap(u,v-u)-Ωωf(v-u)𝑑x𝑑y0for all v𝒦,(1.1)

where

ap(u,v)=Ωω(k2+|u|2)p-22uvdxdy

and

𝒦={vW01,p(Ωω):φ1vφ2 in Ωω}.

Then, under natural assumptions (see (2.2)), there exists a unique function u that solves problem (1.1). Properties of first-order derivatives have been established by Li and Martio in [25] and by Lieberman in [27] (see also the references quoted there). In this paper, we face the study of the regularity of the second-order derivatives. To our knowledge, for p>2 there are no second-order L2 regularity results concerning obstacle problems even if the differentiability of the data and the smoothness of the boundary are assumed; in particular, recent results by Brasco, Santambrogio [5] and by Mercuri, Riey, Sciunzi [29] do not seem to work for obstacle problems. Global regularity results in terms of Sobolev (or Besov) spaces with smoothness index greater than 1 are up to now only established for solutions of obstacle problems for p=2 (see [11]).

In this paper, we establish a regularity result for the solution of obstacle problem (1.1) in terms of the weighted Sobolev spaces, where the weight is the distance from the conical point (see Theorem 3.1). In our approach, the Lewy–Stampacchia inequality (see Proposition 2.2) plays a crucial role. We note that this result is new not only for obstacle problems but also in the case of Dirichlet problems. In fact, there is a huge literature about the regularity in the Hölder classes for both the solution u and the gradient u (see [22] and the references quoted there), while the smoothness of the second derivatives is little investigated in such type of irregular domains. Actually, on this topic we have only the contributions by Borsuk and Kondratiev [4] and by Cianchi and Maz’ya [16]. More precisely, Borsuk and Kondratiev (see [4, Theorems 8.43, 8.44, 8.46]) deal with Dirichlet problems in conical domains, but they require a stronger assumption of the datum and prove a weaker regularity. In particular, the exponent of the weight in [4] is greater than the one in our paper (see (3.2)). On the other side, Cianchi and Maz’ya (see [16, Theorem 2.4]) deal with Dirichlet problems in domains that either satisfy [16, condition (2.12)]) or are convex: here the domain Ωω is not convex and Ωω does not satisfy [16, condition (2.12)]. Actually, we use some ideas from [16] in order to obtain local estimates and estimates far away from the conical point. We note that in this part the boundedness of the data (f and Ap(φi), i=1,2) is not required, but only the belonging to L2(Ωω) (see Theorems 3.6 and 3.7). To establish estimates near the conical point we follow the approach of Tolksdorf [34] and Dobrowolski [19].

In the present paper, we prove also the boundedness of the gradient far away from the conical point (see Theorem 4.3). Essential tools are some results by Tolksdorf [34], Cianchi and Maz’ya [15] and Barret and Liu [2] (for k=0).

We think that the established results are interesting in themselves and also from the point of view of numerical analysis. In fact, as is well known, the regularity results are crucial tools to establish error estimates for the FEM approximation (see, for instance, [6, 8]). To face the numerical approach of the solutions of obstacle problems in fractal domains, it is natural to consider the solutions of obstacle problems in pre-fractal approximating domains and the corresponding FEM-solutions and to evaluate the approximation error. In this spirit, we apply Theorems 3.1 and 4.3 in the study of the obstacle problems in pre-fractal Koch Islands. More precisely, in Theorem 5.5 we prove a sharp error estimate for the FEM approximations using the sharp approach of Grisvard [20]. We remark that for p=2 Theorem 5.5 gives the sharp result of Grisvard (see [20, Corollary 8.4.1.7]). Moreover, Theorem 5.5 improves the results of [12]: in particular, estimate (5.11) gives a faster convergence than the convergence in [12, estimate (5.63)].

The plan of the paper is the following. In Section 2, we describe the geometry of our domain, we introduce the obstacle problems and we state existence, uniqueness, energy estimates, the Lewy–Stampacchia inequality and a first regularity result for the solutions in terms of the Besov spaces. In Section 3, we establish our main result in terms of the weighted Sobolev spaces. In Section 4, we establish some further results concerning the boundedness of the gradient. In the last section, we show an application of these estimates.

2 Preliminary

Let Ωω denote a plane domain with a polygonal boundary Ωω union of a finite number N of linear segments Γj numbered according to the positive orientation. We denote by ωj the angle between Γj and Γj+1 and we assume that ωj<π for any j<N and ωN=ω>π. For simplicity, we assume that the corner point between ΓN and Γ1 is the origin and that Γ1 is included in the positive abscissa axis.

We consider the two obstacle problem:

find u𝒦 such thatap(u,v-u)-Ωωf(v-u)𝑑x𝑑y0for all v𝒦,(2.1)

where

ap(u,v)=Ωω(k2+|u|2)p-22uvdxdy,k,

and

𝒦={vW01,p(Ωω):φ1vφ2 in Ωω}.

By using the Poincaré inequality (see, e.g., [28]), the monotonicity properties of the p-Laplacian and choosing v=φ2(φ10) as test function in (2.1), we can prove the following result.

Proposition 2.1.

Let

{fW-1,p(Ωω),1p+1p=1,φiW1,p(Ωω),i=1,2,φ1φ2 in Ωω,φ10φ2 in Ωω.(2.2)

Then there exists a unique function u that solves problem (2.1). Moreover,

uW1,p(Ωω)C{fW-1,p(Ωω)+φ1W1,p(Ωω)+φ2W1,p(Ωω)+|k|}.(2.3)

From now on, we denote by C possibly different constants.

We recall that the solution u to problem (2.1) realizes the minimum on the convex 𝒦 of the functional

Jp(u)=minv𝒦Jp(v),whereJp(v)=1pΩω(k2+|v|2)p2𝑑x𝑑y-Ωωfv𝑑x𝑑y.

Now we introduce the Lewy–Stampacchia inequality that plays an important role in our approach to the regularity of the solution. We set

Ap(u)=-div((k2+|u|2)p-22u).

Proposition 2.2.

We assume hypothesis (2.2) and

f,Ap(φi)Lp(Ωω),i=1,2,1p+1p=1.(2.4)

Let u be the solution of (2.1). Then

Ap(φ2)fAp(u)Ap(φ1)fin Ωω.(2.5)

The Lewy–Stampacchia inequality was first proved in [24] for superharmonic functions which solve a minimum problem, the proof being deeply based on the properties of the Green function. This result has been extended to more general (linear) operators and more general obstacles by Mosco and Troianiello in [31], and for T-monotone operators like the p-Laplacian in [30]. Actually, inequalities (2.5) hold under assumptions weaker than (2.4) according to [36, Remark 1 in Chapter 4.5].

Proposition 2.3.

We assume hypotheses (2.2) and (2.4). Then the solution u of problem (2.1) is the solution of the Dirichlet problem

{Ap(u)=f*in Ωω,u=0in Ωω,(2.6)

where f* belongs to the space Lp(Ωω) and

f*Lp(Ωω)C{fLp(Ωω)+Ap(φ1)Lp(Ωω)+Ap(φ2)Lp(Ωω)}.

By using the Lewy–Stampacchia inequality and [32, Theorem 2], we stated in [12] for k=0 the following regularity result in terms of Besov spaces; the case k0 can be treated analogously. We recall a characterization of Besov spaces

Bp,q1-λ(Ωω):=(W1,p(Ωω),Lp(Ωω))λ,q,Bp,q2-λ(Ωω):=(W2,p(Ωω),W1,p(Ωω))λ,q={uW1,p(Ωω):uBp,q1-λ(Ωω:2)},

where λ[0,1], p,q[1,+] and (,)λ,q is the real interpolation functor (see [3]).

Theorem 2.4.

We assume hypotheses (2.2) and (2.4). Let u be the solution of (2.1). Then u belongs to the Besov space Bp,+1+1/p(Ωω). Moreover,

uBp,+1+1/p(Ωω)C{1+fLp(Ωω)p/p+Ap(φ1)Lp(Ωω)p/p+Ap(φ2)Lp(Ωω)p/p}.

Note that, putting p=2 in the previous theorem, we get uH3/2-ϵ(Ωω) in the Sobolev scale. We point out that the previous result is, in some sense, the best possible as it holds for any value of ω(π,2π), and as ω2π, the domain becomes very bad.

A natural question is then if we can expect sharper regularity results if we consider a fixed value of ω. Having in mind the by now classical results of Kondratiev (see [21]), we think that the natural spaces to study regularity properties in non-convex polygons are the weighted Sobolev spaces of which we now recall the definition.

Let L2,μ(Ωω) be the completion of the space C(Ω¯ω) with respect to the norm

vL2,μ(Ωω)={Ωω|v|2ρ2μ𝑑x}1/2,

where ρ denotes the distance function from the origin.

The weighted Sobolev space

H2,μ(Ωω)={vW1,2(Ωω):DβvL2,μ(Ωω) for all |β|=2},β=(β1,β2),β1,β2{0},

is a Hilbert space with the norm

vH2,μ(Ωω)={|β|=2DβvL2,μ(Ωω)2+vW1,2(Ωω)2}1/2.

In the next section, we state our regularity result in terms of weighted Sobolev spaces.

3 Main result

In this section, we state our regularity result in terms of weighted Sobolev spaces.

Theorem 3.1.

Assume hypotheses (2.2) and

{k0,f,Ap(φi)L(Ωω),i=1,2,Ap(φ2)f0.(3.1)

Then the solution u of obstacle problem (2.1) in Ωω belongs to the weighted Sobolev space

H2,μ(Ωω),μ>1-γ,(3.2)

where

γ=γ(p,χ)=1+p(1-χ)2+(1-χ)p2-χ(2-χ)(p-2)22χ(2-χ)(p-1)(3.3)

with χ=ωπ.

Moreover,

uH2,μ(Ωω)C{1+fL(Ωω)+Ap(φ1)L(Ωω)+Ap(φ2)L(Ωω)}.(3.4)

We note that γ is the least positive eigenvalue and ϕ(θ) is the corresponding eigenfunction of the problem (see [34] and [4, Theorem 8.12 and Remark 8.13])

{θ{(λ2ϕ2+|θϕ|2)p-22θϕ}+λ(λ(p-1)+2-p)(λ2ϕ2+|θϕ|2)p-22ϕ=0in 0<θ<ω,ϕ(0)=ϕ(ω)=0.(3.5)

Remark 3.2.

To our knowledge, for p>2 there are no second-order L2 regularity results concerning obstacle problems even if the differentiability of the data and the smoothness of the boundary are assumed; in particular, recent results of Brasco, Santambrogio [5] and Mercuri, Riey, Sciunzi [29] do not seem to work for obstacle problems. For properties of first-order derivatives we refer to [25, 27] and to the references quoted there. Global regularity results in terms of Sobolev (or Besov) spaces with smoothness index greater than 1 for solutions of obstacle problems are up to now only established for p=2 (see [11]).

Remark 3.3.

We note that for any fixed value of p>2 the function γ(p,) decreases as the variable χ increases, and it tends to the value p-1p as χ2. Similarly, for any fixed value of χ<2 the function γ(,χ) increases as the variable p increases, and it tends to the value 1 as p+. If we choose ω=4π3, then the expression for γ becomes

γ(p,43)=1+p-p2+32p-3216(p-1).

Note that, putting p=2 in the previous formula, we get γ=34 according to the by now classical results of Kondratiev for equations (see, e.g., [4]).

The behavior of γ(p,43) is shown in Figure 1 for 2<p<10, and in Figure 2 for 2<p<10.000.

The function γ for 2<p<10{2<p<10}.
Figure 1

The function γ for 2<p<10.

The function γ for 2<p<10.000{2<p<10.000}.
Figure 2

The function γ for 2<p<10.000.

Remark 3.4.

We point out that the regularity result of Theorem 3.1, also in the case of Dirichlet problems with datum FL, cannot be deduced from [4, Theorems 8.43, 8.44, 8.46] since we do not assume the differentiability of F, and, for any p>2, the exponent of the weight in [4] is greater than the one in (3.2). In fact, the exponent of the weight in formula [4, (8.4.35)] is required to be greater than p2(1-γ) (in our notation), it is increasing in p and its limit is equal to 12 as p, while μ in (3.2) is required to be greater than (1-γ), it is decreasing in p and tends to 0 as p.

Remark 3.5.

We point out that this regularity result cannot be deduced from [16, Theorem 2.4] as our boundaries do not satisfy [16, condition (2.12)]. Actually, we use some ideas from [16] in order to obtain local estimates and estimates far away from the origin. We note that in this part the boundedness of the data (f and Ap(φi), i=1,2) is not required, but only the belonging to L2(Ωω) (see Theorems 3.6 and 3.7).

The proof is obtained by combining some preliminary results that actually require weaker conditions than (3.1).

Theorem 3.6.

We assume hypothesis (2.2) and

{k0,f,Ap(φi)Lloc2(Ωω)i=1,2.(3.6)

Then the solution u of obstacle problem (2.1) in Ωω belongs to Hloc2(Ωω).

Proof.

From the Lewy–Stampacchia inequality (2.5) and assumption (3.6), we derive that the solution u of problem (2.1) is the solution of the equation Ap(u)=f*, where f* belongs to the space Lloc2(Ωω) and

f*Lloc2(Ωω)C{fLloc2(Ωω)+Ap(φ1)Lloc2(Ωω)+Ap(φ2)Lloc2(Ωω)}.

Moreover,

supt>0(p-2)t2(k2+t2)p-42(k2+t2)p-22=p-2

and, as k0,

inft>0(p-2)t2(k2+t2)p-42(k2+t2)p-22=0.

Then we use [16, (5.11) in the proof of Theorem 2.1] and we obtain

BR(k2+|u|2)p-2|β|=2|Dβu|2dxC(fL2(B2R)2+Ap(φ1)L2(B2R)2+Ap(φ2)L2(B2R)2+1R4(B2R(k2+|u|2)p-22|u|𝑑x)2)(3.7)

for any ball B2RΩω with C independent of k. Then we repeat [16, steps 2 and 3 of the proof of Theorem 2.1], and by (2.3) we obtain that uHloc2(Ωω). ∎

Now we derive estimates far away from the origin. Let xΩωO and Ωs(x):=Bs(x)Ω¯ω for s>0. Let 0<R<dist(x,O)4 be such that Ω2R(x):=B2R(x)Ω¯ω is convex.

Theorem 3.7.

We assume hypothesis (2.2) and

f,Ap(φi)L2(Ωω)i=1,2.(3.8)

Then the solution u of obstacle problem (2.1) in Ωω satisfies

ΩR(x)(k2+|u|2)p-2|β|=2|Dβu|2dx  C(fL2(Ω2R(x))2+Ap(φ1)L2(Ω2R(x))2+Ap(φ2)L2(Ω2R(x))2+1R2Ω2R(x)(k2+|u|2)p-2|u|2𝑑x)(3.9)

for any xΩωO and R(0,dist(x,O)4) such that Ω2R(x)=B2R(x)Ω¯ω is convex.

Proof.

From the Lewy–Stampacchia inequality (2.5) and assumption (3.8) we derive that the solution u of the problem is the solution of the Dirichlet problem (2.6), where f* belongs to the space L2(Ωω) and

f*L2(Ωω)C{fL2(Ωω)+Ap(φ1)L2(Ωω)+Ap(φ2)L2(Ωω)}.(3.10)

We point out that far away from the origin, according to the terminology of [16], the weak second fundamental form on Ωω is non-positive. We choose the cut function ξC0(B2R(x)) with ξ=1 in BR(x).

We proceed as in [16, step 1 of the proof of Theorem 2.4]. We observe that on ΩωB2R(x) we have ξ=0 and on ΩωB2R(x) the Dirichlet condition holds, so the boundary integrals (see [16, (4.18)]) can be neglected. By using estimate (3.10), we obtain (see [16, (4.74)])

Ωωξ2(k2+|u|2)p-2|β|=2|Dβu|2dxC(ξ2fL2(Ωω)2+ξ2Ap(φ1)L2(Ωω)2+ξ2Ap(φ2)L2(Ωω)2+Ωω|ξ|2(k2+|u|2)p-2|u|2𝑑x).

Then we repeat [16, steps 2, 3 and 4 of the proof of Theorem 2.3] and we achieve estimate (3.9), where the constant C is independent of k. ∎

The next theorem concerns estimates near the origin and it holds true for any k.

Theorem 3.8.

Assume hypotheses (2.2), (2.4) and

Ap(φ2)f0,Ap(φ1)fC1rλ0with λ0>γ(p-1)-p in Ωω,(3.11)

where γ is defined in (3.3). Then the following estimates hold for the solution u of obstacle problem (2.1):

|u(x)|Crγ,|u(x)|Crγ-1,|Dβu|Crγ-2,|β|=2.(3.12)

Proof.

From the Lewy–Stampacchia inequality (2.5) and assumption (3.11), we derive that the solution u of problem (2.1) is the solution of the Dirichlet problem (2.6) with a datum f* having the property

0f*C1rλ0with λ0>γ(p-1)-p.

Moreover, we can suppose that f*0. In fact, if f*=0, then the unique solution u of problem (2.1) is identically zero and estimates (3.12) are trivial.

If f*0, we use [19, Theorem 3 and the subsequent remarks] and we deduce that u admits the singular expansion

u(r,θ)=C2rγϕ(θ)+v(x)(3.13)

with C2>0, and

|v(x)|C3rγ+δ,|v(x)|C3rγ+δ-1,|Dβu|C3rγ+δ-2,|β|=2.(3.14)

Here γ is defined in (3.3), ϕ(θ) is the corresponding eigenfunction in problem (3.5) and the maximum δ>0 depends on γ and λ0. We deduce estimates (3.12) from (3.13) and (3.14). ∎

We are now in a position to prove our main result.

Proof of Theorem 3.1.

Since assumptions (2.2) and (3.1) imply the assumptions of Theorem 3.6, Theorem 3.7 and Theorem 3.8 (with λ0=0), we combine all the results and we deduce that the solution u of problem (2.1) belongs to the weighted Sobolev space H2,μ(Ωω) for any μ>1-γ as

rμ|Dβu|L2(Ωω),|β|=2.

Finally, estimate (3.4) follows from (2.3), (3.7), (3.9) and (3.12). ∎

4 Boundedness of the gradient far away from the origin

We now investigate boundedness of the gradient in L far away from the origin. We stress the fact that the results of Theorems 4.1 and 4.2 hold for any k.

Theorem 4.1.

We assume hypotheses (2.2) and

f,Ap(φi)L(Ωω),i=1,2.(4.1)

Then the solution u of obstacle problem (2.1) belongs to the Sobolev space Wloc1,(Ωω).

Proof.

From the Lewy–Stampacchia inequality (2.5) and assumption (4.1) we derive that the solution u of problem (2.1) is the solution of the Dirichlet problem (2.6) with datum f*L(Ωω). Then the thesis follows from [35, Theorem 1] (see also [26, 18, 37]). ∎

Theorem 4.2.

We assume hypotheses (2.2) and (4.1). Then the solution u of obstacle problem (2.1) belongs to the Sobolev space W1,(ΩR(x)) for any xΩωO and R(0,dist(x,O)4) such that Ω2R(x)=B2R(x)Ω¯ω is convex.

Proof.

From the Lewy–Stampacchia inequality (2.5) and assumption (4.1) we derive that the solution u of problem (2.1) is the solution of the Dirichlet problem (2.6) with datum f*L(Ωω).

Then we can proceed as in [15, Theorem 2.2 and Remark 2.7]: more precisely, we replace [15, Lemma 5.4] by a localized version involving a cut-off function ξC0(B2R(x)) with ξ=1 in BR(x) and we obtain, for a smooth function v such that v=0 on Ωω,

C(k2+t2)p-22t{|v|=t}ξ2||v||𝑑1(x)t{|v|=t}ξ2|div((k2+|v|2)p-22v)|𝑑1(x)+{|v|>t}ξ21(k2+|v|2)p-22|div((k2+|v|2)p-22v)|2𝑑x   +C{|v|>t}ξ2|v|p𝑑x.

We have exploited the fact that the weak second fundamental form on ΩωB2R(x) is non-positive. ∎

We now state a further property for the gradient, useful for the application we have in mind when k=0 (see [2, Lemma 4.2]). Here, as before, for any xΩωO we set Ω2R(x)=B2R(x)Ω¯ω and R(0,dist(x,O)4) is chosen in such a way that Ω2R(x)=B2R(x)Ω¯ω is convex.

Theorem 4.3.

We assume (2.2), (4.1) and

k=0,Ap(φ2)fc*>0.(4.2)

We suppose that the solution u of obstacle problem (2.1) belongs to the space Wloc2,s(Ωω), and for any xΩωO the restriction of u to the set Ω2R(x) belongs to W2,s(ΩR(x)), s[1,2]. Then, for any q1, p>2, we obtain

|u|-(p-t)qt-qL1(Ωω)

with

tq(p+(p-2)s)q+(p-2)s.

Proof.

From the Lewy–Stampacchia inequality (2.5) and assumption (4.1) we derive that the solution u of problem (2.1) is the solution of the Dirichlet problem (2.6) with datum f*L(Ωω), and by (4.2) also f*c*>0. In particular, assumption (3.11) of Theorem 3.8 is satisfied with λ0=0. We deduce from (3.13) that |u| behaves like rγ-1 in a neighborhood of O, and hence |u|-1L near O. Far away from the origin, we apply Theorem 4.1 to obtain that uWloc1,(Ωω).

Let G be a domain with GΩω. Then (v1,v2)u(W1,s(G))2 and v|u|L(G). It follows that vW1,s(G) and v=(v1v1+v2v2)/v. Moreover, we have that

f*=-div(vp-2u)=-{vp-2(v1x1+v2x2)+(p-2)vp-2v1vx1+v2vx2v}.

Then

c*f*M(x)|u|p-2a.e. in G,

where M(x)Ls(G).

We obtain

G|u|-(p-t)qt-q𝑑xCG(M(x))(p-t)q(p-2)(t-q)𝑑x,

and if tq(p+(p-2)s)q+(p-2)s, then (p-t)q(p-2)(t-q)s.

We repeat the previous proof by replacing G by ΩR(x) and Theorem 4.1 by 4.2 to complete the proof. ∎

5 Error estimates

Obstacle problems in fractal domains have been studied in [10] in the framework of reinforcement problems. To face the numerical approach to the solutions of obstacle problems in fractal domains, it is natural to consider the solutions of obstacle problems in pre-fractal approximating domains and the corresponding FEM-solutions and to evaluate the approximation error. We consider the pre-fractal Koch Islands Ωαn that are polygonal domains having as sides pre-fractal Koch curves. We start by a regular polygon and we replace each side by a pre-fractal Koch curve (see Figures 3 and 4); we refer to [12, Section 2] for the definition and details.

In [12, Section 3], we showed that, assuming some natural conditions, the solutions un of the obstacle problem in Ωαn converge to the fractal solution of the obstacle problem in the Koch Island Ωα.

For any (fixed) n, the number of reentrant angles is fixed and hence we can prove, for the solution un of the obstacle problem in Ωαn, all the results of previous sections with χ=ωπ, where

ω={π+θ(α)if the sides of the polygons are obtained by outward curves,π+2θ(α)if the sides of the polygons are obtained by inward curves.(5.1)

We recall that by θ(α) we denote the opening of the rotation angle of the similarities involved in the construction of the Koch curve, that is,

θ(α)=arcsin(α(4-α)2).

Then χ(1,32) in the case of outward curves or χ(1,2) in the case of inward curves.

Pre-fractal Koch Islands Ωα2{\Omega^{2}_{\alpha}} with α=2.1{\alpha=2.1}, α=3{\alpha=3} and α=3.75{\alpha=3.75}, respectively.
Figure 3

Pre-fractal Koch Islands Ωα2 with α=2.1, α=3 and α=3.75, respectively.

Pre-fractal Koch Islands Ωα2{\Omega^{2}_{\alpha}} with α=2.1{\alpha=2.1}, α=3{\alpha=3} and α=3.75{\alpha=3.75}, respectively.
Figure 4

Pre-fractal Koch Islands Ωα2 with α=2.1, α=3 and α=3.75, respectively.

In this framework, the involved weighted Sobolev space is

H2,μ(Ωαn)={vW1,2(Ωαn):DβvL2,μ(Ωαn) for all |β|=2},β=(β1,β2),β1,β2,,

which is a Hilbert space with the norm

vH2,μ(Ωαn)={|β|=2DβvL2,μ(Ωαn)2+vW1,2(Ωαn)2}1/2.

Here L2,μ(Ωαn) is the completion of the space C(Ω¯αn) with respect to the norm

vL2,μ(Ωαn)={Ωαn|v|2ρ2μ𝑑x}1/2

and ρ=ρn(x) denotes the distance function from the set of vertices of the reentrant corners of Ωαn. In this setting, we state the following theorems.

Theorem 5.1.

We assume

{φiW1,p(Ωαn),i=1,2,φ1φ2 in Ωαn,φ10φ2 in Ωαn,(5.2)

and

{k0f,Ap(φi)L(Ωαn),i=1,2,Ap(φ2)f0.(5.3)

Then the solution un of obstacle problem (2.1) in Ωαn belongs to the weighted Sobolev space

H2,μ(Ωαn),μ>1-γ,(5.4)

where

γ=γ(p,χ)=1+p(1-χ)2+(1-χ)p2-χ(2-χ)(p-2)22χ(2-χ)(p-1)(5.5)

with χ=ωπ and ω in (5.1).

Moreover,

unH2,μ(Ωαn)C{1+fL(Ωαn)+Ap(φ1)L(Ωαn)+Ap(φ2)L(Ωαn)}.(5.6)

If k=0, then an analog of Theorem 4.3 holds.

Theorem 5.2.

We assume (5.2) and

{k=0f,Ap(φi)L(Ωαn),i=1,2,Ap(φ2)fc*>0.(5.7)

If the solution un of obstacle problem (2.1) in Ωαn belongs to the space H2,μ(Ωαn), then for any q1 and p>2 we obtain

|un|-(p-t)qt-qL1(Ωαn),(5.8)

with

tq(p+(p-2)2)q+(p-2)2.

We introduce the triangulation of the domain Ωαn in order to define the approximate solutions uh according to the Galerkin method. Let Th be a partitioning of the domain Ωαn into disjoint, open regular triangles τ, each side being bounded by h so that Ω¯αn=τThτ¯. Associated with Th, we consider the finite-dimensional spaces

Sh={vC(Ω¯αn):v|τ is affine for all τTh}andSh,0={vSh:v=0 on Ωαn}.

By πh we denote the interpolation operator πh:C(Ω¯αn)Sh such that πhv(Pi)=v(Pi) for any vertex Pi of the partitioning Th.

Definition 5.3.

The family of triangulations Th is adapted to the H2,μ(Ωαn)-regularity if the following conditions hold:

  • The vertices of the polygonal curves Ωαn are nodes of the triangulations.

  • The meshes are conformal and regular.

  • There exists σ*>0 such that, as h0,

    hτσ*h11-μfor all τTh such that one of the vertices of τ belongs to n,hτσ*hinfτρμfor all τTh with no vertix in n.

Here h=sup{hτ=diam(τ):τTh} is the size of the triangulation and ρ=ρn(x) denotes the distance of the point x from the set n of the vertices of the reentrant corners of Ωαn.

The construction of triangulations Th adapted to the H2,μ-regularity was introduced by Grisvard in [20]. This tool has been fruitfully used for the FEM approximation of linear problems in pre-fractal domains by [38, 39, 23, 1, 13, 14].

Consider the two obstacle problem in the finite-dimensional space Sh,0:

find u𝒦h such thatap(u,v-u)-Ωαnf(v-u)𝑑x𝑑y0for all v𝒦h,(5.9)

where

ap(u,v)=Ωαn(k2+|u|2)p-22uvdxdyand𝒦h={vSh,0:φ1,hvφ2,h in Ωαn},

with φ1,h=πhφ1 and φ2,h=πhφ2.

Proposition 5.4.

Let us assume hypothesis (5.2). Then, for any fLp(Ωαn), there exists a unique function uh that solves problem (5.9). Moreover,

uhW1,p(Ωαn)C{|k|+fLp(Ωαn)p/p+φ1W1,p(Ωαn)+φ2W1,p(Ωαn)}.

As previously, the solution uh to problem (5.9) realizes the minimum on the convex 𝒦h of the functional Jp(), i.e.,

Jp(u)=minv𝒦hJp(v),whereJp(v)=1pΩαn(k2+|v|2)p2𝑑x𝑑y-Ωαnfv𝑑x𝑑y.

Theorem 5.5.

Let us denote by un and uh the solutions of problems (2.1) in Ωαn and (5.9), respectively. Let us assume hypotheses (5.2), (5.3) and

φiH2,μ(Ωαn),i=1,2.(5.10)

Let Th be a triangulation of Ωαn adapted to the H2,μ(Ωαn)-regularity of the solution un. Then

un-uhW1,t(Ωαn)ChrtunH2,μ(Ωαn)(5.11)

for any

r[1,2p2-χ(2-χ)(p-2)2p2-χ(2-χ)(p-2)2+(χ-1)(p-2)),t[2,p].

Proof.

For any σ[0,p] we put

|v|(p,σ)=(Ωαn(|k|+|un|+|v|)p-σ|v|σ𝑑x𝑑y)1p.(5.12)

Repeating the proof of [12, Lemma 5.2] (given for k=0), we prove for any vh𝒦h and

v𝒦n:={vW01,p(Ωαn):φ1vφ2 in Ωαn}

that

|un-uh|(p,t)pC{|un-vh|(p,r)p+f+Ap(un)L2(Ωαn)(un-vhL2(Ωαn)+v-uhL2(Ωαn))},(5.13)

where r[1,2], t[2,p] and the constant C does not depend on h. Now we evaluate the terms on the right-hand side in estimate (5.13) by choosing the test functions vh𝒦h and v𝒦n in an appropriate way. According to Theorem 5.1, the function un belongs to the weighted Sobolev space H2,μ(Ωαn) for any μ>1-γ (see (5.4) and (5.5)).

We choose vh=πhun, and by using approximation estimates of Grisvard (see [20, Section 8.4.1]), we derive

un-πhunL2(Ωαn)Ch2unH2,μ(Ωαn).(5.14)

Then we choose v=φ2(uhφ1) and, as in [12, Lemma 4.4], we have

v-uhL2(Ωαn)2πhφ2-φ2L2(Ωαn)2+πhφ1-φ1L2(Ωαn)2.

Again using Grisvard estimates and assumption (5.10), we derive

v-uhL2(Ωαn)Ch2.(5.15)

We compare the seminorm |un-uh|W1,t(Ωαn) with |un-uh|(p,t)p (defined in (5.12)) and we obtain

|un-uh|W1,t(Ωαn)tC|k|p-t|un-uh|(p,t)p.(5.16)

We now evaluate the term |un-vh|(p,r)p, where vh=πhun. By the embedding of weighted Sobolev spaces in the fractional Sobolev spaces (see, for instance, [33]), un belongs to the space Wσ2,2(Ωαn) for any σ2<1+γ. Taking into account the Sobolev embedding (see, for instance, [7]), we have

|un|Lr*(Ωαn)with r*=22-σ2.(5.17)

By the Hölder inequality, we obtain

|un-vh|(p,r)pC(r)|un-πhun|W1,2(Ωαn)r,(5.18)

where we have used estimate (5.6) with r=2(r*-p)r*-2. Hence, as σ2<1+γ, r* is given in (5.17) and γ in (5.5), we have to choose r<p+2-pγ and we obtain by calculations that

r<2p2-χ(2-χ)(p-2)2p2-χ(2-χ)(p-2)2+(χ-1)(p-2).

Now we use [20, Theorem 8.4.1.6] and we obtain

|un-πhun|W1,2(Ωαn)Ch.(5.19)

By taking into account estimates (5.13)–(5.16), (5.18) and (5.19), we conclude the proof using once again the Poincaré inequality. ∎

We note that in Theorem 5.5 we assume k0; if k=0 the following result holds.

Theorem 5.6.

Let us denote by un and uh the solutions of problems (2.1) in Ωαn and (5.9), respectively. Let us assume hypotheses (5.2), (5.7), (5.10) and that the solution un belongs to the space H2,μ(Ωαn). Let Th be a triangulation of Ωαn adapted to the H2,μ(Ωαn)-regularity of the solution un. Then

un-uhW1,q(Ωαn)ChrtunH2,μ(Ωαn)

for any

r[1,2p2-χ(2-χ)(p-2)2p2-χ(2-χ)(p-2)2+(χ-1)(p-2))

t[2,p] , q[1,t], and for q<p we require tq(p+(p-2)2)q+(p-2)2.

Proof.

We proceed as in the proof of Theorem 5.5: we replace estimate (5.16) by

|un-uh|W1,q(Ωαn)t|un|-(p-t)qt-qL1(Ωαn)(t-q)/qΩαn|(un-uh)|t|un|p-t𝑑x𝑑yC|un-uh|(p,t)p.

Here we have used the Hölder inequality and estimate (5.8). ∎

Remark 5.7.

From the previous proofs we deduce that, for the linear case p=2, Theorem 5.5 gives the sharp result of Grisvard (see [20, Corollary 8.4.1.7]): in fact, we have p=t=2 and, in particular, formula (5.18) holds true for r=2=p.

Remark 5.8.

We note that Theorem 5.5 improves the results of [12]: in particular, estimate (5.11) gives a faster convergence than the convergence in [12, estimate (5.63)]. In fact, the solution un belongs to the weighted Sobolev space H2,μ(Ωαn) for any μ=μ(p)>1-γ. This space is continuously embedded in the fractional Sobolev space Wσ2,2(Ωαn) for any σ2<2-μ (see, e.g., [33]). Hence, by the Sobolev embedding (see, e.g., [7]), for any σ<γ+2p, p2, the fractional Sobolev space Wσ,p(Ωαn) properly contains the weighted Sobolev space H2,μ(Ωαn) for some μ=μ(p)>1-γ. Actually, for every p2 the exponent r in (5.11) is strictly greater than γ+2p. Namely by writing the expression of γ in (5.5) in terms of the parameters p[2,+) and χ(1,2), we obtain that γ+2p<r if and only if

χ(2-χ)(p-1)(p-2)+(χ-1)p2(χ-1)2+4χ(2-χ)(p-1)>0.(5.20)

Of course, inequality (5.20) holds for any choice of the parameters.

Remark 5.9.

We note that the constant C in estimate (5.11) does not depend on n. However, to deduce from (5.11) error estimates for the fractal solution we have to bound the norms unH2,μ(Ωαn) uniformly in n. Up to now, this type of results is only established for p=2 (see [9, 11]).

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About the article

Received: 2017-10-31

Revised: 2017-12-21

Accepted: 2018-01-28

Published Online: 2018-06-14


The authors are members of GNAMPA (INdAM) and are partially supported by Grant Ateneo “Sapienza” 2017. Capitanelli and Vivaldi are partially supported by INdAM GNAMPA Project 2017.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 1043–1056, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2017-0248.

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