In this paper, we deal with obstacle problems involving p-Laplace-type operators in bad domains in . 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,  and references therein).
Let denote a conical domain (see Section 2 for definitions and properties) and let us consider the two obstacle problem:
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  and by Lieberman in  (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 there are no second-order 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  and by Mercuri, Riey, Sciunzi  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 (see ).
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 (see  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  and by Cianchi and Maz’ya . 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  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 is not convex and does not satisfy [16, condition (2.12)]. Actually, we use some ideas from  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 , ) is not required, but only the belonging to (see Theorems 3.6 and 3.7). To establish estimates near the conical point we follow the approach of Tolksdorf  and Dobrowolski .
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 , Cianchi and Maz’ya  and Barret and Liu  (for ).
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 . We remark that for Theorem 5.5 gives the sharp result of Grisvard (see [20, Corollary 184.108.40.206]). Moreover, Theorem 5.5 improves the results of : 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.
Let denote a plane domain with a polygonal boundary union of a finite number N of linear segments numbered according to the positive orientation. We denote by the angle between and and we assume that for any and . For simplicity, we assume that the corner point between and is the origin and that is included in the positive abscissa axis.
We consider the two obstacle problem:
Then there exists a unique function u that solves problem (2.1). Moreover,
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 of the functional
Now we introduce the Lewy–Stampacchia inequality that plays an important role in our approach to the regularity of the solution. We set
The Lewy–Stampacchia inequality was first proved in  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 , and for T-monotone operators like the p-Laplacian in . Actually, inequalities (2.5) hold under assumptions weaker than (2.4) according to [36, Remark 1 in Chapter 4.5].
where belongs to the space and
By using the Lewy–Stampacchia inequality and [32, Theorem 2], we stated in  for the following regularity result in terms of Besov spaces; the case can be treated analogously. We recall a characterization of Besov spaces
where , and is the real interpolation functor (see ).
Note that, putting in the previous theorem, we get 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 , and as , 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 ), 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 be the completion of the space with respect to the norm
where ρ denotes the distance function from the origin.
The weighted Sobolev space
is a Hilbert space with the norm
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.
Assume hypotheses (2.2) and
Then the solution u of obstacle problem (2.1) in belongs to the weighted Sobolev space
To our knowledge, for there are no second-order 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  and Mercuri, Riey, Sciunzi  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 (see ).
We note that for any fixed value of the function decreases as the variable χ increases, and it tends to the value as . Similarly, for any fixed value of the function increases as the variable p increases, and it tends to the value 1 as . If we choose , then the expression for γ becomes
Note that, putting in the previous formula, we get according to the by now classical results of Kondratiev for equations (see, e.g., ).
We point out that the regularity result of Theorem 3.1, also in the case of Dirichlet problems with datum , cannot be deduced from [4, Theorems 8.43, 8.44, 8.46] since we do not assume the differentiability of F, and, for any , the exponent of the weight in  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 (in our notation), it is increasing in p and its limit is equal to as , while μ in (3.2) is required to be greater than , it is decreasing in p and tends to 0 as .
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  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 , ) is not required, but only the belonging to (see Theorems 3.6 and 3.7).
The proof is obtained by combining some preliminary results that actually require weaker conditions than (3.1).
We assume hypothesis (2.2) and
Then the solution u of obstacle problem (2.1) in belongs to .
and, as ,
Then we use [16, (5.11) in the proof of Theorem 2.1] and we obtain
Now we derive estimates far away from the origin. Let and for . Let be such that is convex.
We assume hypothesis (2.2) and
Then the solution u of obstacle problem (2.1) in satisfies
for any and such that is convex.
We point out that far away from the origin, according to the terminology of , the weak second fundamental form on is non-positive. We choose the cut function with in .
We proceed as in [16, step 1 of the proof of Theorem 2.4]. We observe that on we have and on 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)])
The next theorem concerns estimates near the origin and it holds true for any .
If , we use [19, Theorem 3 and the subsequent remarks] and we deduce that u admits the singular expansion
with , and
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 ), we combine all the results and we deduce that the solution u of problem (2.1) belongs to the weighted Sobolev space for any as
4 Boundedness of the gradient far away from the origin
We assume hypotheses (2.2) and
Then the solution u of obstacle problem (2.1) belongs to the Sobolev space .
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 . Then the thesis follows from [35, Theorem 1] (see also [26, 18, 37]). ∎
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 with in and we obtain, for a smooth function v such that on ,
We have exploited the fact that the weak second fundamental form on is non-positive. ∎
We now state a further property for the gradient, useful for the application we have in mind when (see [2, Lemma 4.2]). Here, as before, for any we set and is chosen in such a way that is convex.
We suppose that the solution u of obstacle problem (2.1) belongs to the space , and for any the restriction of u to the set belongs to , . Then, for any , , we obtain
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 , and by (4.2) also . In particular, assumption (3.11) of Theorem 3.8 is satisfied with . We deduce from (3.13) that behaves like in a neighborhood of O, and hence near O. Far away from the origin, we apply Theorem 4.1 to obtain that .
Let G be a domain with . Then and . It follows that and . Moreover, we have that
and if , then .
5 Error estimates
Obstacle problems in fractal domains have been studied in  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 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 of the obstacle problem in converge to the fractal solution of the obstacle problem in the Koch Island .
For any (fixed) n, the number of reentrant angles is fixed and hence we can prove, for the solution of the obstacle problem in , all the results of previous sections with , where
We recall that by we denote the opening of the rotation angle of the similarities involved in the construction of the Koch curve, that is,
Then in the case of outward curves or in the case of inward curves.
In this framework, the involved weighted Sobolev space is
which is a Hilbert space with the norm
Here is the completion of the space with respect to the norm
and denotes the distance function from the set of vertices of the reentrant corners of . In this setting, we state the following theorems.
Then the solution of obstacle problem (2.1) in belongs to the weighted Sobolev space
with and ω in (5.1).
If , then an analog of Theorem 4.3 holds.
We assume (5.2) and
If the solution of obstacle problem (2.1) in belongs to the space , then for any and we obtain
We introduce the triangulation of the domain in order to define the approximate solutions according to the Galerkin method. Let be a partitioning of the domain into disjoint, open regular triangles τ, each side being bounded by h so that . Associated with , we consider the finite-dimensional spaces
By we denote the interpolation operator such that for any vertex of the partitioning .
The family of triangulations is adapted to the -regularity if the following conditions hold:
The vertices of the polygonal curves are nodes of the triangulations.
The meshes are conformal and regular.
There exists such that, as ,
Here is the size of the triangulation and denotes the distance of the point x from the set of the vertices of the reentrant corners of .
The construction of triangulations adapted to the -regularity was introduced by Grisvard in . 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 :
with and .
As previously, the solution to problem (5.9) realizes the minimum on the convex of the functional , i.e.,
Let be a triangulation of adapted to the -regularity of the solution . Then
For any we put
Repeating the proof of [12, Lemma 5.2] (given for ), we prove for any and
where , 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 and in an appropriate way. According to Theorem 5.1, the function belongs to the weighted Sobolev space for any (see (5.4) and (5.5)).
We choose , and by using approximation estimates of Grisvard (see [20, Section 8.4.1]), we derive
Then we choose and, as in [12, Lemma 4.4], we have
Again using Grisvard estimates and assumption (5.10), we derive
We compare the seminorm with (defined in (5.12)) and we obtain
We now evaluate the term , where . By the embedding of weighted Sobolev spaces in the fractional Sobolev spaces (see, for instance, ), belongs to the space for any . Taking into account the Sobolev embedding (see, for instance, ), we have
By the Hölder inequality, we obtain
Now we use [20, Theorem 220.127.116.11] and we obtain
We note that in Theorem 5.5 we assume ; if the following result holds.
Let us denote by and the solutions of problems (2.1) in and (5.9), respectively. Let us assume hypotheses (5.2), (5.7), (5.10) and that the solution belongs to the space . Let be a triangulation of adapted to the -regularity of the solution . Then
, , and for we require .
Here we have used the Hölder inequality and estimate (5.8). ∎
From the previous proofs we deduce that, for the linear case , Theorem 5.5 gives the sharp result of Grisvard (see [20, Corollary 18.104.22.168]): in fact, we have and, in particular, formula (5.18) holds true for .
We note that Theorem 5.5 improves the results of : in particular, estimate (5.11) gives a faster convergence than the convergence in [12, estimate (5.63)]. In fact, the solution belongs to the weighted Sobolev space for any . This space is continuously embedded in the fractional Sobolev space for any (see, e.g., ). Hence, by the Sobolev embedding (see, e.g., ), for any , , the fractional Sobolev space properly contains the weighted Sobolev space for some . Actually, for every the exponent r in (5.11) is strictly greater than . Namely by writing the expression of γ in (5.5) in terms of the parameters and , we obtain that if and only if
Of course, inequality (5.20) holds for any choice of the parameters.
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About the article
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.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0