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International Journal of Stochastic Analysis
Volume 2011 (2011), Article ID 469806, 35 pages
The Cauchy-Dirichlet Problem for a Class of Linear Parabolic Differential Equations with Unbounded Coefficients in an Unbounded Domain
Comisión Nacional de Seguros y Fianzas, Distrito Federal, Mexico
Received 20 December 2010; Revised 18 March 2011; Accepted 19 April 2011
Academic Editor: Lukasz Stettner
Copyright © 2011 Gerardo Rubio. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider the Cauchy-Dirichlet problem in for a class of linear parabolic partial differential equations. We assume that is an unbounded, open, connected set with regular boundary. Our hypotheses are unbounded and locally Lipschitz coefficients, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution to the nonhomogeneous Cauchy-Dirichlet problem using stochastic differential equations and parabolic differential equations in bounded domains.
In this paper, we study the existence and uniqueness of a classical solution to the Cauchy-Dirichlet problem for a linear parabolic differential equation in a general unbounded domain. Let be the differential operator where , , and . The Cauchy-Dirichlet problem is where is an unbounded, open, connected set with regular boundary.
In the case of bounded domains, the Cauchy-Dirichlet problem is well understood (see [1, 2] for a detailed description of this problem). Moreover, when the domain is unbounded and the coefficients are bounded, the existence of a classical solution to (1.2) is well known. For a survey of this theory see [3, 4] where the problem is studied with analytical methods and  for a probabilistic approach.
In the last years, parabolic equations with unbounded coefficients in unbounded domains have been studied in great detail. For the particular case when , there exist many papers in which the existence, uniqueness, and regularity of the solution is studied under different hypotheses on the coefficients; see for example, [6–17].
In the case of general unbounded domains, Fornaro et al. in  studied the homogeneous, autonomous Cauchy-Dirichlet problem. They proved, using analytical methods in semigroups, the existence and uniqueness of a solution to the Cauchy-Dirichlet problem when the coefficients are locally , with bounded, and functions with a Lyapunov type growth; that is, there exists a function such that and for some , It is also assumed that has a boundary. Schauder-type estimates were obtained for the gradient of the solution in terms of the data. Bertoldi and Fornaro in  obtained analogous results for the Cauchy-Neumann problem for an unbounded convex domain. Later, in  Bertoldi et al. generalized the method to nonconvex sets with boundary. They studied the existence, uniqueness, and gradient estimates for the Cauchy-Neumann problem. For a survey of this results, see .
Using the theory of semigroups, Da Prato and Lunardi studied, in [22, 23], the realization of the elliptic operator , in the functions spaces , and , when is an unbounded convex function defined in a convex set . They proved existence and uniqueness for the elliptic and parabolic equations associated with and studied the regularity of the semigroup generated by . Geissert et al. in , made a similar approach for the Ornstein-Uhlenbeck operator.
In the paper of Hieber et al. , the existence and uniqueness of a classical solution for the autonomous, nonhomogeneous Cauchy-Dirichlet and Cauchy-Neumann problems is proved. The domain is considered to be an exterior domain with boundary. The coefficients are assumed to be continuous functions with Lyapunov type growth. The continuity properties of the semigroup generated by the solution of the parabolic problem are studied in the spaces and .
In all the papers cited above, the uniformly elliptic condition is assumed; that is, there exists such that for all we have that for all .
In this paper, we prove the existence and uniqueness of a classical solution to (1.2), when the coefficients are locally Lipschitz continuous in and locally Hölder continuous in , has a quadratic growth, has linear growth, and is bounded from above. We allow , , and to have a polynomial growth of any order. We also consider the elliptic condition to be local; that is, for any , there exists such that for all , and . We assume that is an unbounded, connected, open set with regular boundary (see  Chapter , Section 4, for a definition of regular boundary). Furthermore, we prove that the solution is locally Hölder continuous up to the second space derivative and the first time derivative.
Our approach is using stochastic differential equations and parabolic differential equations in bounded domains. For proving existence, many analytical methods construct the solution by solving the problem in nested bounded domains that approximate the domain . In these cases, the convergence of the approximating solutions is always a very difficult task. Unlike these methods, first we propose, as a solution to (1.2), a functional of the solution to a SDE, where Using the continuity of the paths of the SDE, we prove that this function is continuous in . Then, using the theory of parabolic equations in bounded domains, we study locally the regularity of the function and prove that it is a function. Finally, with some standard arguments, we prove that it solves the Cauchy-Dirichlet problem. This kind of idea has been used for several partial differential problems (see [5, 26, 27]).
In Section 2, we introduce the notation and the hypotheses used throughout the paper. Section 3 presents the main result. In this section we prove that if the function is smooth, then it has to be the solution to the Cauchy-Dirichlet problem. Section 4 is devoted to prove the required differentiability for the candidate function. Finally, in Section 5, the reader will find some of the results used in the proof of our main theorem.
2. Preliminaries and Notation
In this section, we present the hypotheses and the notation used in this paper.
Let be an unbounded, open, connected set with boundary and closure . We assume that has a regular boundary; that is, for any , is a regular point (see  Chapter , Section 4 or  Chapter 2, Section 4, for a detailed discussion of regular points). We denote the hypotheses on by H0.
2.2. Stochastic Differential Equation
Let be a complete filtered probability space and let be a -dimensional brownian motion defined in it. For and , consider the stochastic differential equation where and . Although this process is the natural one for solving equation (1.2), it does not posses many good properties. The continuity of the flow process does not imply the continuity with respect to . Furthermore, although this process is a strong Markov process, it is not homogeneous in time, a very useful property for proving the results in this paper.
To overcome these difficulties, we augment the dimension considering the following process Then the process is solution to with . Throughout this paper we will use both processes, and , in order to simplify the exposition. For the expectation, we use the notation when considering the process defined as in (2.1) and the notation when working with the joint process defined in (2.3).
We need to define the following stopping times
Remark 2.1. Observe that is the exit time of the process from the set , that is, We cannot guarantee that the process leaves the set in a finite time however, the process reaches the boundary at time . Thus, the joint process leaves the set in a bounded time.
We assume the following hypotheses on the coefficients and . We denote them by H1. The matrix norm considered is .H1.
Let be continuous functions such that Continuity. Let . For all , , there exists such that for all , , . Linear-Growth. For each , there exists a constant such that for all , . Local Ellipticity. Let be any bounded, open, connected set and . There exists such that, for all and , where .
Remark 2.2. Observe that the local ellipticity is only assumed on . This condition is used to prove the existence of a classical solution to (1.2) and so is only needed in that set. The local Lipschitz condition and the linear growth are assumed on to ensure the existence of a strong solution to (2.3) for .
Remark 2.3. If we assume, in the more natural case, that are continuous functions satisfying the hypotheses in H1 restricted to the set , then we can extend them to be defined for negative values of as follows: let and be defined as It is easy to see that these functions satisfy H1 with the same constants and .
The next proposition presents some of the properties of the process required in this work.
Proposition 2.5. As a consequence of , has the following properties. (i)For all , there exists a unique strong solution to (2.3). (ii)The process is a strong homogeneous Markov process. (iii)The process does not explode in finite time a.s.. (iv)For all , , and ,
Proof. See(i)Theorem 1.1 Chapter in  or Theorem 3.2, Chapter 6 in . (ii)Theorem 4.6, Chapter 5 in  or Proposition 3.15, Chapter 6 in . (iii)Theorem in . (iv)Theorem 2.3, Chapter 5 in  or Corollary 3.3, Chapter 6 in .
2.3. The Cauchy-Dirichlet Problem
Consider the following differential operator: where , , and . For the rest of the paper, we assume that the coefficients of satisfy H1.
The Cauchy-Dirichlet problem for a linear parabolic equation is We assume the following hypotheses for the functions , , , and . We denote them by H2.H2. (1)Let be continuous functions such that (i)Continuity. Let . For all , , there exists a constant such that for all , , with , . (ii)Growth. There exists such that There exists , such that for all , a constant exists such that for all , (2)Let be continuous functions such that (i)Growth. There exists , such that, for all , there exists a constant such that for all . (ii)Consistency. There exists consistency in the intersection of the space and the time boundaries, that is, for .
2.4. Additional Notation
If is a locally Lipschitz function defined in some set , then, for any bounded open set for which , we denote, by and , the constants If , then, for all , The space is the space of all functions such that they and all their derivatives up to the second order in and first order in , are locally Hölder continuous of order .
3. Main Result
In this section, we present the main result of this work and some parts of the proof.
Theorem 3.1. Assume , , and . Then, there exists a unique solution to (2.17). The solution has the representation where is the solution to the stochastic differential equation Furthermore, for all , where , , , , and are the constants defined in and .
The proof of this theorem is given by several lemmas. The method we will use has the following steps: first we define a functional of the process as a candidate solution. Let be defined as If , then there exist some standard arguments (see [27, chapter 4]) to prove that is the unique solution to (2.17). The rest of this section is devoted to proving Theorem 3.1 in the case when is a “regular” function. The proof is divided into two lemmas: the first one proves that if , then is a solution to equation. The second one proves that in case of existence of a classical solution, , to problem (2.17), then it is unique and has the form given by in (3.4). The regularity of is proved in Section 4 below.
Proposition 3.2. Assume . Then, there exists a continuous function such that
Proof. Thanks to the consistency condition in and the continuity of and , we can extend by Tietze's extension theorem (see [32, Section 2.6]) the functions , from the closed set to a continuous function defined in .
Lemma 3.3. Assume , , and . Let be defined as in (3.6) and assume that . Then, fulfils the following equation: Furthermore, for all , there exists such that where , , , , and are the constants defined in H1 and H2.
Proof. Let , then, following the same argument used to prove (4.80) in the proof of Theorem 4.4 in Section 4 we have that
Because of and , we have that the random variable inside the conditional expectation is integrable and so the left-hand side of (3.9) is a -martingale, for . Since , we can apply Ito's formula to to get
It follows from the continuity of , and that
is a.s. finite and then
is a local martingale for . So, combining (3.9) and (3.10), we get that
is a continuous local martingale for . Since is locally of bounded variation, then . This implies that for all .
For the boundary condition, it follows from the regularity of the set and the local ellipticity (see Remark 2.4) that From this, it is clear that the first addend of the right-hand side of (3.6) is zero. For the second addend, we get that the exponential term is equal to one and that , and so we conclude that satisfies the boundary condition.
The second statement of the theorem is proved with the same argument used to prove (4.9) and (4.42) in the proofs of Lemmas 4.2 and 4.3 in Section 4.
The next Lemma proves the uniqueness of the solution.
Lemma 3.4. Assume , , and . Assume that there exists a classical solution to equations such that, for all , there exists for which for some . Then, has the following representation: and hence the solution is unique.
Proof. Consider, for , the process Applying Ito's rule, we get A similar argument as the one used in the proof of Lemma 3.3 shows that is a local martingale. Due to (3.15), we conclude that is a local martingale for . Let be a sequence of localization times for ; that is, a.s. as and is a martingale for all . Then, for all , Since , using estimate (3.16), we get By (2.15) and the dominated convergence theorem, letting , we get Letting , a similar argument and the boundary condition proof that and the proof is complete.
4. Regularity of
In this section, we prove that . First, we prove, using the continuity of the flow process , that is a continuous function in . Since we are only assuming the continuity of the coefficients, then the flow is not necessarily differentiable and so we can not prove the regularity of in terms of the regularity of the flow. To prove that , we show that is the solution to a parabolic differential equation in a bounded domain, for which we have the existence of a classical solution and hence .
4.1. Continuity of
Theorem 4.1. Assume , , and . Let be defined as in (4.1). Then, is continuous on .
The proof of this theorem is divided into two lemmas.
Lemma 4.2. Assume , , and . Let be defined as in (4.2). Then, is continuous on .
Proof. First, we prove the continuity on . For that, let
in and . We need to prove that there exists such that, for all
Denote by and the solutions to (2.3) with initial conditions and , respectively. To simplify the notation in the proof, let and denote their corresponding exit times from .
Let , then there exists such that, for all Observe that, for all , we get Define the random variables as The sequence is uniformly integrable. Thanks to Theorem 4.2, in Chapter 5 of , it is sufficient to prove that . So, where and . We use (4.6), (4.7), (2.15) and the polynomial growth of .
Let , , and and define the set Then, Since the sequence is uniformly integrable, there exists such that for any that satisfies , we have It follows, from Remark 2.4, Proposition 2.5, and Theorems 5.3 and 5.4 in Section 5, the existence of and such that for all . Then for , we get that For simplicity of notation, we write the set as and define and let be an open set such that .
On the set , for all and , it is satisfied that We have that We first analyse (4.19): For (4.18), we get Now, For (4.22), we need the following bound: since . If we choose such that for all and , then we get by the mean value theorem that Then, To summarize, we get with the above estimations that Hence, to prove continuity, we proceed as follows, (i)Let and . (ii)Let such that, for all , (iii)Let fulfil the uniformly integrability condition (4.12). (iv)Take such that . (v)Define and . (vi)Let (vii)Choose such that for all (viii)Let be such that for all .(ix)Let and choose such that, for all , .
Thus, if , then, for all , Therefore, is continuous in .
For the continuity at the boundary we make a similar argument. Let , where and , that is, either or . In both cases we get that a.s. and so . Then, we need to prove that Let and be such that We get for all . For the continuity, we have The convergence follows from the uniform integrability of and the fact that (see Theorem 5.2 in Chapter 5 of  and Theorem 5.3 in Section 5). This completes the proof.
Lemma 4.3. Assume , , and . Let be defined as in (4.3). Then, is continuous on .
Proof. We use an analogous argument to the one in the proof of Lemma 4.2. First, we prove the continuity in . Let
with . Denote by and the solutions to (2.3) with initial conditions and , respectively, and let and be their corresponding exit times from . Let and be such that, for all ,
This implies that
First, we prove that the sequence of random variables
is uniformly integrable for all . As in (4.9),
where and . We use (4.39), (4.40), (2.15), and the polynomial growth of in . As in Lemma 4.2, let , then there exists such that
for all , with .
Let be defined as in (4.10), and choose and such that for all .
For simplicity of notation, denote as . Then, Let , , and be defined as in Lemma 4.2 (see (4.15) and (4.16)). Then, on the set , we get that, for all and , So, We study each addend of the right-hand side separately: First, we get a bound for (4.49). Since is continuous, then it is uniformly continuous on . Then, for , there exists such that for all with . On the set , we have and So, if we choose , then we get Next, we study (4.50). Thanks to Theorem 5.3, we know that