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Axioms 2018, 7(3), 65; doi:10.3390/axioms7030065
Interior Regularity Estimates for a Degenerate Elliptic Equation with Mixed Boundary Conditions
Departamento de Análise Matemática, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
Department of Applied Mathematics & Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK
Author to whom correspondence should be addressed.
Received: 9 August 2018 / Accepted: 27 August 2018 / Published: 1 September 2018
The Marchaud fractional derivative can be obtained as a Dirichlet-to–Neumann map via an extension problem to the upper half space. In this paper we prove interior Schauder regularity estimates for a degenerate elliptic equation with mixed Dirichlet–Neumann boundary conditions. The degenerate elliptic equation arises from the Bernardis–Reyes–Stinga–Torrea extension of the Dirichlet problem for the Marchaud fractional derivative.
Keywords:marchaud fractional derivative; interior regularity; schauder estimate; extension problem; fractional order weighted Sobolev spaces
In the last years there has been a growing interest in the study of fractional elliptic equations involving the right fractional Marchaud derivative , such as equations of the formwhere without loss of generality , with and .
Fractional diffusion problems of type (1) arise for example in the modelling of neuronal transmission in Purkinje cells, whose malfunctioning is known to be related to the lack of voluntary coordination and the appearance of tremors . Further motivation comes from various experimental results which showed anomalous diffusion of fractional type, see for example [2,3] and references therein.
The right fractional Marchaud derivative of a function is defined via Fourier transforms asand it can also be expressed by the pointwise formulawhere is a positive normalization constant. We observe from (3) that the right fractional Marchaud derivative is a nonlocal operator. Nonlocal operators have the peculiarity of taking memory effects into account and capturing long-range interactions, i.e., events that happen far away in time or space. Further discussion of the difference between local integro-differential operators and nonlocal or fractional ones can be found in  and references therein. In this context, the nonlocality of the fractional Marchaud derivative prevents us from applying local PDE techniques to treat nonlinear problems for . To overcome this difficulty, Bernardis, Reyes, Stinga and Torrea showed in  that the right fractional Marchaud derivative can be determined as an operator that maps a Dirichlet boundary condition to a Neumann-type condition via an extension problem. Similar extension properties have been found for the fractional Laplacian by Caffarelli and Silvestre in .
To be more precise, consider the function that solves the boundary value problem
Then we have :where is a positive multiplicative constant depending only on . Here the differential operators and are given respectively by:
We use the notation for the derivative from the right at the point , that is:for good enough functions v. Observe that equals the negative of the lateral derivative as usually defined in calculus .
This characterization of via the local (degenerate) PDE (5) was used for the first time in  to get maximum principles. To solve (4), Stinga and Torrea noted that (5) can be thought of as the harmonic extension of v into extra dimensions (see ). From there, they established the fundamental solution and, using a conjugate equation, a Poisson formula for . Furthermore, taking advantage of the general theory of degenerate elliptic equations developed by Fabes, Jerison, Kenig and Serapioni in 1982–83, they proved comparison principles for (and thus for v).
The aim of this paper is to prove an interior Schauder estimate for the problem (4), involving any fractional power of the derivative as an operator that maps a Dirichlet condition to a Neumann-type condition via an extension problem as in .
A significant contribution of the above extension problem is to provide a way of applying classical analysis methods to partial differential equations containing one-sided Marchaud derivative operators. By means of such extension techniques, a series of important results, such as comparison principles, Harnack inequalities, and regularity estimates for solutions to degenerate elliptic equations involving the fractional Laplacian, have been studied by many authors, for example [6,7,8,9,10,11,12,13,14,15,16,17]. The same analysis was done for the one-sided fractional derivative operator in the sense of Marchaud ( Theorem 1.1 and Corollary 1.2).
In view of these results, we immediately observe that interior regularity and boundary regularity for the degenerate elliptic equation with mixed boundary conditions involving the one-sided Marchaud derivative is missing in the literature. Indeed, from the pioneering work of [5,7,18] on the analogue extension problem for nonlocal operators that map Dirichlet to Neumann, one can reduce a nonlocal problem involving fractional derivatives to a local one by keeping their qualitative properties. Using this technique, one can study interior and boundary regularity. Hence the raison d’être for this work.
In order to underline what makes the difference between the extension problems introduced by Bernardis, Reyes, Stinga and Torrea , and the one introduced by Bucur and Ferrari , we point out that the extension problem established in  is based on a time-dependent initial condition, which leads to a heat conduction problem. Indeed, considering the function of one variable, formally representing the time variable, their approach relies on constructing a parabolic local operator by adding an extra variable, say the space variable, on the positive half-line, and working on the following problem in the half-plane :
The problem (8) is not the usual Cauchy problem for the heat operator, but a heat conduction problem.
In view of the type of problem we are interested in here, we choose to deal with the Bernardis–Reyes–Stinga–Torrea extension problem . Our main result, which will be proved in Section 3 below, is as follows. We note that this result can be proved only using extension techniques.
Let and let be a weak solution to
- For , if and is such that , then . Moreover,
- If and , then . Moreover,
The paper is organised as follows. In Section 2, we give some notations and definitions of function spaces and their associated norms which will be needed in this work. We also provide some preliminary results and finally state our main result. In Section 3, we prove an intermediate result and provide the proof of the regularity estimate up to the boundary for the degenerate Equation (4) with the Neumann boundary condition stated in Theorem 1. Finally we end with the conclusion in Section 4.
2. Notations and Preliminary Results
In this section we introduce some notations, definitions, and preliminary results used throughout the paper.
Here and in the following, we consider , and a bounded Lipschitz domain. For an open set , an integer , and a real number , the Hölder spaces are defined as the subspaces of consisting of functions whose k-th order derivatives are uniformly Hölder continuous with exponent in .
Furthermore, we introduce the following notation for intervals, boxes, and balls:
We consider the function space
For an open set, we say is in , i.e., Hölder continuous with exponent , if
We recall the following definition of Sobolev spaces.
Definition 1 (Sobolev spaces).
For any real number α, the αth Sobolev space on is defined to bewhere the Sobolev norm is defined by
For a general domain , the αth Sobolev space on X is defined to be
Let and be two arbitrary parameters. We define the functional space
We denote here by the space of absolutely continuous functions on I.
Definition 2 (Caputo derivative).
The Caputo derivative of with initial point at the point is given by
The right Marchaud derivative of a well defined function v is given bywith a positive normalisation constant.
Notice that the one-sided nonlocal derivative in the sense of Marchaud can also be obtained by extending the Caputo derivative. Indeed, by making an integration by parts of Equation (11), we obtain an equivalent definition [19,20] as follows:for all , so that , where is a constant depending on α. Indeed, for sufficiently regular functions v, we have:
Hence, we take the convention that for any . With this extension, one has that, for any ,
So one can write (13) as
Note that the integral in (12) is absolutely convergent for functions in the Schwartz class . Furthermore one should notice that the nonlocal operators and depend on the values of v on the whole half line .
We recall that the inverse of the right fractional Marchaud derivative is defined aswhere the Riesz potential (see [7,21]) is defined aswith the constant .
From , we have that for , , where
The topology in is given by the family of seminorms , for . Let be the dual space of ; then defines a continuous operator from into .
2.1. Weighted Spaces
Weighted spaces of smooth functions play an important role in the context of partial differential equations (PDEs). They are widely used, for instance, to treat PDEs with degenerate coefficients or domains with a nonsmooth geometry (see e.g., [22,23,24,25]), as is the case here. For evolution equations, power weights in time play an important role in order to obtain results for rough initial data (see [26,27]). This subsection dedicated to weighted spaces is motivated by the appearance of the Muckenhoupt weight which appears in (5) and (6). For general literature on weighted function spaces we refer to [23,24,25,28,29,30,31] and references therein.
In a general framework, a function , for an integer , is called a weight if w is locally integrable and the zero set has Lebesgue measure zero. For we denote by the Muckenhoupt class of weights. In the case , we say that if
In the case , we say that belongs to if there exists some constant C such thatfor all and all balls . In the case , we define . Note that, for functions with support contained in or , the class of weights is denoted by or respectively. We refer to [27,30,32] for the general properties of these classes.
Problem (4) is a weighted—singular or degenerate, depending on the value of —elliptic equation on with mixed boundary conditions. The weight belongs to the Muckenhoupt class , i.e., there exists a constant C such that for any ,
For this reason, when working with one-sided weights, we can assume without loss of generality that (see e.g.,  for more details).
Next, for a strongly measurable function f and a number , we define the weighted norm byand we define the weighted space to be the following Banach space:
(see ). Given , , and an open set , we denoteendowed with the norm
We also denotewith the induced norm
Using the variable , the space coincides with the trace on of
So by a density argument, every has a well defined trace . Conversely, any is the trace (restriction to ) of a function .
2.2. The Extension Problem
In the next statement we recall the results obtained from  which show that the fractional derivatives on the line are Dirichlet-to–Neumann operators for an extension degenerate PDE problem in , where the data f have been taken in the more general setting: more precisely a weighted space, where w satisfies the one-sided version (see ) of the familiar condition of Muckenhoupt.
Fix . Given a semigroup acting on real functions, the generalized Poisson integral of f is given bysee ( (1.9)) for more details.
By considering the semigroup of translations , , we findwhere
Since the kernel is increasing and integrable in , it is well known that the functionis pointwise controlled by the usual Hardy–Littlewood maximal operator. However, since the support of is , a sharper control can be obtained by using the one-sided Hardy–Littlewood maximal operator. This control and the behavior of in weighted -spaces will be used in the results of this paper. We revise briefly recall the two fundamental theorems from .
(). Consider the semigroup of translations , , initially acting on functions . Let , , be as in (17). Then:
- For , is a bounded linear operator from into itself and .
- When , the Fourier transform of is given by
- The maximal operator defined by is bounded from into itself, for , and from into weak-, for
- Let , for , . The function is a classical solution to the extension problem (4).
Theorem 3 (Extension problem).
Let , . Then the functionis a classical solution to the extension problem
Moreover, for , we have
3. Regularity Estimate up to the Boundary for the Degenerate Equation with the Neumann Boundary Condition
In this section, we prove the interior regularity estimate up to the boundary for the degenerate equation with the Neumann boundary condition associated to problem (4). Namely we provide the proof of Theorem 1. But before we get into that, it is necessary to explain the main ideas in the proof of interior regularity provided by Theorem 1. The proof of Theorem 1 is inspired by [5,7,8,34]. The method for this proof differs substantially from interior regularity methods for second-order equations, but is similar to the proof for the fractional Laplacian. Recall that for second-order equations, one first shows that is bounded, and then the estimate for equations with bounded measurable coefficients implies a estimate for . This is also true for the boundary regularity for solutions to fully nonlinear equations .
We shall start by the regularity property of the problem (1). We show in Proposition 1 that the solution of the problem (1) is of class . To the best of the authors’ knowledge, the proofs available in the literature are those dealing with the case of the fractional Laplacian (see for instance [7,36] (Proposition 2.1.9)). With this result in hand, and by making an appropriate change of variables, we will use this result and estimate to prove our main theorem.
We start by recalling the following lemma from , which gives a Liouville-type theorem for (1) in the case .
Let be a function satisfying in , in , and for some . Then .
The proof of this lemma relies on similar reasoning as the proof of ( (Theorem 2.2.3)) for the Caputo density function.
In the case where we have a non-vanishing right hand side () as in (1), we state the following Liouville-type theorem for the one-sided Marchaud derivative.
Let and let be the solution to
- For , if and and is such that , then and there exists a constant such that
- If and and , then and there exists a constant such that
We will show that u has the corresponding regularity in a neighbourhood of the origin. We split the proof into two parts, as follows.
Proof of (a): . Let be a smooth cutoff function such that on , on , and on . Consider the Riesz potential as defined in (15). Then the functionsatisfies
We first estimate the norm of v for . Since the kernel is positive and is a smooth function with compact support in , we write . We note that, by using a similar argument as for the Poisson equation for the fractional Laplacian, we find that is an element of with norm depending only on .
Since is compactly supported, we getFor and and , we have
Next we consider the following inequalities , valid for and with and for every :
For , and for , we can writesince . Using the fact that the support of is always contained in the ball of radius 2 centred at , we have thatup to relabelling of the positive constant that depends on and . Replacing by its value and using the polar coordinates , , we get that
Hence, we conclude thatfor every .
Next, by change of variables, the function satisfies in by (21). Then, thanks to the derivative estimate, for every ,
The difference function is smooth in and is bounded. From this observation, together with (23), we have thatfor every with , as required.
Proof of (b): . The proof in this case is similar to the previous one. We consider as above a smooth cutoff function such that on , on , and on . Then we consider the Riesz potential as defined in (15), so that we can estimate the norm of v for . Since the kernel is positive and is a smooth function with compact support in , we get
Next, by using the inequality stated in (22), we get that for and ,
Hence, we conclude thatfor every .
Next, by change of variables, the function satisfies in by (21). Therefore, thanks to ( (Corollary 1.13)), we have the derivative estimate for every :
The difference function is smooth in and is bounded. From this, together with (24), we conclude thatfor every . ☐
Now we are in a position to state and prove our main result on the interior Schauder estimate for the solution function on the set .
Proof of the Main Result: Theorem 1
Again, we present the two parts of the proof separately.
Proof of (a): . We choose a cut-off function such that on and on . Let be the unique solution to the equationwhere . Making use of the previous result Proposition 1, we know that andwhere is a constant that depends only on , , and p.
The next step is to consider the Bernardis–Reyes–Stinga–Torrea extension of , i.e., the functionwhich satisfies the equations
By a change of variables, we havewhere
Then, if we set , we have the estimate
By direct computation from (25), and using Theorem 2, we have:
Therefore,for a positive constant depending only on , p and .
Next we put , so that satisfies
Considering the even reflection of in the variable t, as described in ( (Lemma 4.1)), we have that
From the definition (7) of , and using ( (Corollary 1.13)) or ( (Corollary 1.5)), we have that for and fixed,
Next, from the fact thatand from the inequality (28), we obtain
For any point , we have, by (28), thatwhich implies that
Thus, we have that such that
We finally obtainsince . This ends the proof of the first case.
Proof of (b): . The proof here is similar to the first case above. By considering the same cut-off function with on and on , we let be the unique solution to the equationwhere . Making use of the previous result Proposition 1, we have that andwhere is a constant that depends only on and .
The next step is to consider the Bernardis–Reyes–Stinga–Torrea extension of , i.e.,which satisfies the equation
Proceeding as in the previous case, it follows thatfor a positive constant depending only on and .
Next we put , so that satisfies
We finally obtainwhich ends the proof. ☐
Regularity theorems are an important result in the theory of PDEs, and their fractional counterparts also play a significance role in the study of problems involving nonlocal behaviour. As already observed in various papers [6,8,9,34,40,41,42] in the theory of fractional nonlocal PDEs, it is possible to find the qualitative behaviour of a solution. In this paper we have shown that the degenerate elliptic equation with mixed boundary conditions for a problem with fractional Marchaud derivative admits an interior regularity estimate. The current work fits in with some results obtained in the case of fractional Laplacians with Caffareli–Silvestre extensions. We stress that the types of regularity results proved herein form only a small subset of many possible versions of regularity theorems which can only be obtained using extension techniques.
J.-D.D. and A.F. contributed equally to this work.
The second author’s research was supported by a student grant from the Engineering and Physical Sciences Research Council, UK.
Conflicts of Interest
The authors declare no conflict of interest.
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