Qualitative Analysis of Differential, Difference Equations, and Dynamic Equations on Time ScalesView this Special Issue
-Solvability of the Dirichlet Problem for Elliptic Equations with Singular Data
We give an overview on some results concerning the unique solvability of the Dirichlet problem in , , for second-order linear elliptic partial differential equations in nondivergence form and with singular data in weighted Sobolev spaces. We also extend such results to the planar case.
Let be an open subset of , . Consider in the Dirichlet problem where and is the second-order linear elliptic differential operator in nondivergence form defined by
If and is a bounded set with a suitable regularity property, the well-posedness of the Dirichlet problem (1) has been largely studied by several authors under various hypotheses of discontinuity on the coefficients. It must be mentioned the classical contribution by Miranda , where the author assumed that the belong to and considered the case . This result was later on generalized in [2, 3] by considering the coefficients belonging to the wider class of spaces VMO and .
In the framework of open sets, nonnecessarily bounded, whose boundary has various singularities, for example, corners or edges, in accordance with the linear theory, it is natural to assume that the lower order coefficients and the right-hand side of the equation of problem (1) belong to some weighted Sobolev spaces, where the weight is usually a power of the distance function from the “singular set” of the boundary of domain. In these cases, near to the “singular set,” the solution of the boundary value problem may have a singularity which can be often characterized by a weight of mentioned type. For instance, if is a bounded weight function related to the distance function from nonempty subset of the boundary of an arbitrary domain , not necessarily bounded and regular (see Section 2 for the definition of such weight function), a problem similar to (1) has been studied, in the weighted case, by several authors under suitable hypotheses on the weight function and when the coefficients of lower order terms are singular near to . This kind of Dirichlet problem has been dealt, for example, in [4–6] under hypotheses as those in [2, 3] with . To be more precise, in [4–6], the authors consider the problem where , and are some weighted Sobolev spaces whose weight functions are suitable powers of . The existence and uniqueness of problem (3) have been firstly proved in [4, 5] when . Such results have been later on used in  to get the unique solvability also for . We point out that in this last case we needed to employ some variational results (see  and its bibliography).
The aim of this work is to give an overview on the above-mentioned results concerning the solvability of (3) when as well as to extend such results to the planar case. Here we note also that, for , if the required summability on can be equal to while if we have to take a summability greater than (see hypothesis ).
At last, we observe that the existence and uniqueness result for the problem (3) are based on the unique solvability of the problem (3) for and of a problem similar to (3) whose associated operator differs from that in (2) for a compact operator (see Section 4).
2. Notation and Function Spaces
In this section we recall the definitions and the main properties of the class of weights we are interested in and of certain classes of function spaces where the coefficients of our operator belong. Thus, from now on, let be an unbounded open subset of , . By we denote the -algebra of all Lebesgue measurable subsets of . For , is its characteristic function, is its Lebesgue measure and , where is the open ball with center in and radius . The class of restrictions to of functions (resp., ) with is (resp., ). For , is the class of all functions , defined on , such that for all .
We denote by the class of all measurable weight functions such that where is independent on and . Given , we put
For and , the related weighted Sobolev space is made up of all the distributions on such that for . We observe that is a Banach space with the norm defined by
Moreover, it is separable if , reflexive if , and, in particular, is an Hilbert space. We also denote by the closure of in and we put .
Clearly the following embeddings hold:
For and , let be the set of all the functions such that equipped with the norm defined in (10). Obviously, the space is a Banach space containing and as well (see ). Therefore, we denote by (resp., ) the closure of (resp., ) in .
Now, let us define the moduli of continuity of functions belonging to or .
Given a function in , the following characterization holds: (see ).
Thus, if is a function in , a modulus of continuity of in is a map such that
Next, we introduce a class of mappings needed to define a modulus of continuity of in . Fix in satisfying the conditions and equivalent to dist (for more details on the existence of such an see, for instance, Theorem 2, Chapter VI in  and Lemma 3.6.1 in ). Hence, for we define the functions
It is easy to prove that each belongs to and where .
Given in , it is known (see ) that
Thus, a modulus of continuity of in is a map such that
Further properties of the spaces , , and can be found in .
In the end, we recall the definitions of two other classes of function spaces where the leading coefficients of our operator belong.
If has the property where is a positive constant independent of and , it is possible to consider the space () of functions such that where
If , where we say that if for . A function is called a modulus of continuity of in if
3. Preliminary Results
In this section we recall some embedding and compactness estimates for a multiplication operator as well as a regularity result which will be also extended to the planar case.
Let us assume that has the segment property (for all definitions of regularity properties of open subsets of we will refer to ) and fix such that .
For our purposes, we suppose that the following condition on holds. () There exists an open subset of with the uniform -regularity property such that
We note that since the required segment property gives that lies on one side of the “singular” part of its boundary the hypothesis , roughly speaking, means that it is possible to “widen” suitably on the other side of .
Remark 1. We observe that, as a result of condition , there exists such that where is an open infinite cone with vertex in and opening and (see Remark 5.1 in ). As a consequence, there exists a function which is equivalent to and such that where is independent of (see ).
Let us start collecting two results of  which provide the boundedness and compactness of the multiplication operator where the function belongs to suitable spaces .
Theorem 2. Let be numbers such that
If condition holds, then for all and for any we have and where the constant is independent of and .
Furthermore, if then for any there exist a constant and a bounded open set with the cone property such that and we have that the multiplication operator is compact.
We go on collecting a density result, an a priori estimate, and a regularity result, which will be some of the crucial analytic tools of our main results.
Let , . At first we recall a density result (see Lemma 3.2 in ).
Lemma 3. If verifies condition , then for every there exists a sequence such that
Now consider in the second-order linear differential operator in nondivergence form
Assume that the leading coefficients satisfy the hypothesis. () There exist extensions of to such that
For the lower order terms coefficients suppose that ()
We observe that, in view of Theorem 2, under the assumptions , , and , the operator is bounded.
The following a priori estimate holds.
Lemma 4. Let be defined in (32). If hypotheses , , and are satisfied, then there exists such that where depends on , , , , , , , , , , , , , and .
In the next Lemma we recall a regularity result of  (see Theorem 3.4) and we extend it to the case .
Lemma 5. Let be defined in (32). Suppose that conditions , , and hold with and . Then any solution of the problem with and , belongs to the space and verifies the bound where depends only on , , , , , , , , , , , , , , and .
Let us give an overview of the proof of Lemma 5. The idea is to use a local regularity result which is based on an analogous result for solutions of boundary value problems in classical Sobolev spaces defined on regular domains. The mentioned regularity result has been proved in  when (see, Theorem 5.1 of ) and it can be easily extended to the planar case, taking into account to apply, at right time, Lemma 4.2 in  in place of Lemma 4.2 in .
4. Main Results
Moreover, suppose that the following condition on holds: ()
We are interested in the study of the Dirichlet problem: with and .
We point out that the unique solvability of (41) has been firstly proved in [4, 5] for and . Later on, the results of [4, 5] have been used in  to get the existence and uniqueness of solution also for . Here, we collect the main results of the above-mentioned papers and we also extend them to the case .
For the case where the assumptions are taken into account, with , and , and for , the unique solvability of (41) has been proved in Lemma 4.1 of , which can be easily extended to the planar case, applying our Lemmas 4 and 5 at the right time. If , the idea is to exploit the previous case and the unique solvability of a problem similar to (41), whose associated differential operator differs from that in (32) for a compact operator.
Let , and consider in the differential operator
Lemma 6. Suppose that conditions hold. Then for any and there exists such that
where depends on , , , , , , , , , , , , , and .
Furthermore, if and then the problem is uniquely solvable.
Here, we only give an overview of the proof, pointing out to the crucial aspects. Since the operator verifies the same assumptions of the operator , we can write for it the estimate (35). Now a crucial point is to provide a bound for in terms of , when the function is more regular; that is, . To this aim, we build the following bilinear form related, in appropriate way, to the operator : where .
By simple computations, we get
Now we study separately the cases and . If , applying Lemma 4.1 of  in (46), with a suitable choice of function , we easily deduce the claimed bound on . Thus, from Lemma 3 we get the estimate (43). Furthermore, in this case, the uniqueness of the solution of problem (44) easily follows from (43). For the existence of solution, we refer to Theorem 4.2 in , whose analytic technique exploits the bound (43), which also gives the closure of the range of operator , the unique solvability of our problem when , and the regularity result of Lemma 5, which allow us to go up in summability.
Suppose now . In this case, in order to get a bound on , when is more regular, we use a variational result (see the proof of Lemma 3.3 in ) whose main analytic tools exploit the existence and uniqueness result of previous case and Lemma 5. The mentioned variational result gives us a bound for , where is the unique solution of the Dirichlet problem associated with and is the conjugate exponent of . From such estimate and (46), with a suitable choice of function , we obtain our claim. Using again Lemma 3, we get (43) also in this case. Finally, for the existence and uniqueness of problem (44) we can refer to Lemma 3.4 of , whose proof can be easily extended also to the planar case, in view of previous considerations.
Theorem 7. Suppose that conditions hold. Then for any and there exist and a bounded open set , with the cone property, such that where and depend on , , , , , , , , , , , , , , , , , and .
Proof. Fix . We observe that
Thus, from estimate (43) we deduce that there exists , depending on , , , , , , , , , , , , , and , such that
On the other hand, by Theorem 2 it follows that for any there exist and a bounded open set , with the cone property, such that where and depend on , , , , , , , , , , , , , and . The result easily follows from relations (50) and (51).
Proof. First we prove that the problem (41) is an index problem with index equal to zero. In fact, from Lemma 6 we deduce that the operator is a Fredholm operator with index zero. On the other hand, by Theorem 2 it follows that the operator
is a compact operator. Thus, from (48), (49), and well known results of the classical Fredholm index theory, we deduce that the problem (41) is an index problem with index equal to zero.
Assume now . We prove only the uniqueness of solution of problem (41); the existence will easily follow from uniqueness and from what we have proved in the case . At first, suppose . Then, by the unique solvability of problem (41) for there exists a unique such that . Moreover, by our Lemma 5 and by Lemma 3.2 of  it follows that belongs to and thus we deduce that . Now let and such that . Using again our Lemma 5 and Lemma 3.2 of  we get with and . Exploiting again the existence and uniqueness of (41) for , we obtain that . This concludes the proof.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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