An Application of Potential Estimates to A Priori Bounds for Elliptic Equations

Abstract

A potential estimate type approach is used in order to obtain some a priori bounds for the solutions of certain classes of Dirichlet problems associated with nondivergence structure elliptic equations.

1. Introduction

In this paper we are concerned with some aspects related to the study of the strong solvability of the classical Dirichlet problemin the space , , where is a sufficiently regular bounded open subset of , , and is the second-order linear differential operator in nondivergence form

As evidenced by a renewed example by C. Pucci (quoted by Talenti in [1]), for , the boundedness and ellipticity of the are not enough to derive the strong solvability of problem (1). There are two ways to overcome this difficulty: the first approach consists in imposing suitable regularity of the , while the second one assumes that the coefficients do not scatter too much, satisfying conditions stronger than the uniform ellipticity (see, e.g., [2] for a wider survey on this subject).

Concerning the first way, classical results (see, e.g., [3]) ensure that, for sufficiently regular sets and for coefficients , , one obtains the unique strong solvability of (1) together with the boundThe embedding theorems then givefor . Let us point out that the constant in (4) is independent of but depends on the required regularity of the coefficients.

Following the second approach, Talenti obtains existence and uniqueness results for problem (1), for , considering, in [1], coefficients satisfying the so-called Cordes condition:Condition (5) allows him to prove the estimateLater on, Campanato, in [4], extended these results to values of sufficiently close to 2.

Taking into account these results, in the recent paper [5], we assume that the coefficients , , are symmetric bounded measurable functions satisfying the following new hypothesis:Hypothesis (7) generalizes the classical Cordes condition, since (see Remark 3), for , the class of operators satisfying (7) is wider than the one verifying Cordes assumption.

In [5] we prove estimate (4), showing that, for the range of exponent , the constant is independent of the regularity of the coefficients. This can be done by assuming that has an extension, belonging to certain generalized Sobolev spaces (see Section 2) verifying quite restrictive conditions essentially due to the lack in literature of opportune extension results in the involved functional spaces. We point out that obtaining an estimate with a constant independent of the regularity of the coefficients is an important feature of the bound, since it can allow weakening of the hypotheses on the regularity of the .

The goal of the present work is to achieve an a priori bound for the solution of problem (1) in the same framework of [5], but dropping the restrictive conditions on the datum , required therein.

Our main result, obtained assuming that is the restriction to of a function belonging to (roughly speaking, is the restriction to of the Riesz potential of a function , with ), consists in the following estimate:for , where the constant is again independent of the regularity of the coefficients.

The main tools in achieving (8) are certain potential type estimates previously obtained for the solutions of problems of the same kind of (1), but with more regular datum. We refer to estimates (35) in Step and (52) in Step of Theorem 5. We remark that hypothesis (7) is the key point in the proof of Lemma 4, which allows showing the abovementioned potential type bounds.

2. A Class of Generalized Sobolev Spaces

This section is devoted to the definition and the main properties of a class of generalized Sobolev spaces where the datum will be taken (see also [5]).

To our aim, we recall the main features of Riesz potentials and hypersingular integrals focusing just on some specific aspects, required to our needs. For wider and deeper surveys we refer the reader, for instance, to [6–9].

Let and . It is known that if then the integralconverges absolutely for almost every .

The function in (9) defines, up to a positive multiplicative constant depending on , the Riesz potential of .

For and , it is therefore possible to consider the spacethat is, a Banach space, with respect to the normIf, in addition, , one has the following strict inclusion:and moreover

Now, let , , and . A characterization of hypersingular integrals in terms of Riesz potentials (see [7]) allows us to define the space of hypersingular integrals asThe function in (14) is called the Riesz derivative of ; it is denoted by and can be interpreted as an inverse of the Riesz potential.

For , , and , the space can be therefore rewritten asand it is a Banach space endowed with the normFrom definition (15) it appears to be clear that generalizes the notion of Sobolev space to the fractional case.

It is known that is dense in , where, as usual, stands for the class of all functions on with compact support.

Definition 1. Let be an open subset of , . For , , and , we define as the set of restrictions to of functions in . Namely, if there exists a function such that in .

The space is a Banach space endowed with the norm

3. Main Result

Let be an open subset of , . We want to prove some a priori estimates for strong solutions of the following Dirichlet problem:.

Here is a second-order linear differential operator in nondivergence formwith coefficients that are symmetric bounded measurable functions satisfying the uniform ellipticity condition; therefore, there exists a constant such thatfor almost every and for any .

In the sequel we assume thatand setand we suppose thatCondition (23) is equivalent towhere , , are the eigenvalues of the matrix .

Remark 2. By the definition of it easily follows that .

Remark 3. For the class of operators satisfying the assumption (23) is wider than the one verifying the Cordes condition. Indeed the Cordes assumption reads asor equivalentlyThus, if (26) is satisfied, by (26) and by the definition of , it follows that . This gives (24) for .

Let us now state the following lemma, proved in [5], that is an essential tool in the proof of our main result.

Lemma 4. Let be the operator defined in (19) satisfying (20) and (23). If , , and is a nonnegative function in , for some fixed , then for sufficiently small the potentialis such thatfor a.e. , and therefore is -superharmonic.

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

Theorem 5. Let be a bounded domain in , , with the -regularity property and let be the operator defined in (19) satisfying (20), (21), and (23). Let , . If is a solution of problem (18) with datum , then there exists a positive constant such thatwith as in Definition 1.

Proof. Theorem 5 will be proved in three steps. In the first two we show some potential type estimates for the solutions of certain auxiliary problems. In the last one, we exploit the estimate of Step to achieve (29).
Step  1. The datum ; thus by Definition 1 there exists a function such that in . Therefore, by (14) we can find such thatDenoted by and the positive and negative parts of the function , respectively, by classical results we get the fact that there exist two sequences of smooth and nonnegative functions and such thatLet take its values in a sequence of positive reals numbers converging to zero and setwith . Then, consider the classical solutions and of the problemsIn this step we want show thatwith   positive constant, whereTo this aim, putIn view of Lemma 4 there exists a positive constant such thatwith . ThenMoreover, by (33) and (37), one getsThus, as a consequence of our hypotheses, the maximum principle in its classical forms applies (see, e.g., [3], Theorems and ) and therefore by (32) and (37) we haveThis last inequality and the composition formula for fractional integrals (see, e.g., [6]) give thenwith positive constant. Following the same argument one obtains a similar estimate for the function :Combining inequalities (42) and (43) and definitions (36) we get (35).
Step  2. Let be the classical solution of the problemwhere .
In view of the uniqueness of the solutions of problems (33), (34), and (44) (see, e.g., [3], Theorem ) one has that and, therefore, since are nonnegative, as observed in (41), . Thus, (35) gives the following potential type estimate for :We want to pass to the limit as in (45).
By classical estimates (observe that the coefficients of problem (44) are smooth and, therefore [3], Lemma applies) we getFor and , for any fixed , by (36), (32), and Beppo Levi’s theorem, we obtainBy Lebesgue’s dominated convergence theorem, these last convergences take place in too:and thuswhere we setPutting together (46) and (49), we derive that is a Cauchy sequence in . Thus, by the completeness of , there exists such that in , as . Hence, up to a subsequence, still denoted by , we haveTaking into account (51), we can finally pass to the limit as in (45), obtaining the following potential type estimate for :Step  3. Combining (52) and (13) we havewith .
To prove our claim, we want to pass to the limit as in (53).
From (31), (36), and (13), it follows thatwhere we set.
Thus, since , by (54) we haveNow, observe that, by uniqueness, is the solution of the problemClassical results give thenand so, in view of convergence (56), we obtainwhere is the solution of problem (18).
The embedding then ensures thatFinally, by convergences (54) and (60) we can pass to the limit as in (53), obtainingWe are now able to conclude our proof. Indeed, by (61) and (13) we getwith .
Moreover,the functions and having disjoint supports.
Thus, by (62), (63), and (16), we obtain our estimatewith .

Disclosure

Sara Monsurrò and Maria Transirico are Members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.