Elliptic problems with nonhomogeneous boundary condition and derivatives of nonlinear terms
© Motreanu and Motreanu; licensee Springer. 2014
Received: 22 October 2013
Accepted: 6 December 2013
Published: 7 January 2014
The paper presents existence results for nonlinear elliptic problems under a nonhomogeneous Dirichlet boundary condition. The considered elliptic equations exhibit nonlinearities containing derivatives of the solution.
The aim of the paper is two-fold: first, to study nonlinear elliptic problems under nonhomogeneous Dirichlet boundary condition; second, to incorporate in the problem statement nonlinearities exhibiting derivatives of the solution. These requirements need to develop a nonstandard approach, in particular prevent the use of variational methods.
where , , are Carathéodory functions (that is, they are measurable in and continuous in the other variables), , and denotes the space of sized symmetric matrices. In the following definition we make clear what we understand by solution to problem (1).
Here , and g are as in problem (1), while and () are Carathéodory functions. The meaning of solution of problem (2) is as follows.
for all .
Problems of type (1) and (2) have been investigated in settings that are different from ours (see, e.g., [1–6]). For instance, problem (1) is studied in  when and with functions and corresponding to certain physical models, as described by Reynolds equation where with , , , and . Whereas many of the previous results on problems (1) and (2) involve technical and somewhat restrictive assumptions on the data, the purpose of the present paper is to provide an elementary resolution of problems (1) and (2) in geometrically relevant situation. As an example of such a geometrically relevant situation, we mention the assumption on the term in problem (1) to vanish at two points.
Our results are stated as Theorems 1 and 2. They are existence and location theorems on problems (1) and (2), respectively, guaranteeing solutions in the sense of Definitions 1 and 2 that fulfill an estimate with given constants . This a priori estimate of the solution is derived through natural geometric hypotheses that can be directly checked. It is also worthwhile to remark that we cannot drop by translation the nonhomogeneous boundary conditions to become homogeneous because our hypotheses would be no longer verified. The arguments used in the proof are based on truncation techniques and Schauder’s fixed point theorem. We emphasize that, due to the type of assumptions we impose, it is essential in our approach to keep separate the two terms in divergence form appearing in the statement of (1) and (2). A careful inspection of our proofs shows that we rely on the linearity with respect to the gradient ∇u in the first divergence term and on the vanishing at suitable points in the second divergence term.
The rest of the paper is organized as follows. Section 2 is devoted to problem (1). Section 3 studies problem (2).
2 Result on problem (1)
Throughout the paper the notation and stands for the usual norms on (or ) and , respectively. By we denote the Euclidean norm of .
We suppose the following hypotheses on the data a, b, f, and g in problem (1):
Remark 1 The constants and are not solutions of problem (1), unless or on ∂ Ω. Thus, in general, problem (1) has no evident solution.
Remark 2 Due to their different structure and requirements, the two terms in (1) that are in divergence form cannot be combined.
and it is more general than both of them.
The result that we set forth in this section is the following theorem ensuring existence and location of solution for problem (1).
with and given in (H2).
which is a nonempty, bounded, closed, convex subset in .
for all . Notice that the operators A and B are well defined due to our hypotheses.
Moreover, relations (5)-(6) ensure that the operator A is maximal monotone, so pseudomonotone (see, e.g., [, §2.3.1]). Since A is bounded, coercive, and pseudomonotone, it is surjective (see, e.g., [, Theorem 2.99]), whence the existence of in Claim 1. The uniqueness of is a direct consequence of (6) (notice that ). This establishes Claim 1.
Now, taking advantage of Claim 1, we define the operator by for all , where is the unique element corresponding to as proved in Claim 1.
Claim 2: The mapping is continuous.
To this end, we show that any subsequence of possesses a subsequence converging to 0 in . Since in , we have that, along a relabeled subsequence, for a.a. . Invoking (H3), we have that , with some constant . Through Lebesgue’s dominated convergence theorem, we conclude that as , so (8) holds true.
Then, in view of (7), we infer that . Since the domain Ω is bounded and , we can make use of the Poincaré inequality for , which yields , whence in . This establishes Claim 2.
Note that S takes values in .
Claim 3: The mapping has a fixed point.
Since is continuous by Claim 2 (thus a fortiori is continuous) and τ is a bounded continuous function, we infer that is continuous. We claim that is a compact operator. To this end, it suffices to check that is relatively compact in . Because of the compact embedding of in , it is sufficient to prove that is bounded in .
with a constant independent of u.
with constants independent of u. It follows that the set is bounded in , so according to what was said before, the map is compact. Consequently, Schauder’s fixed point theorem can be applied (see, e.g., [, p.452]), through which it follows that S admits a fixed point in C. This shows Claim 3.
Claim 4: Let be a fixed point of S. Then there holds .
whence a.e. in . On the other hand, we have in . Altogether, we obtain that in Ω. Since , we conclude that a.e. in Ω, thus a.e. in Ω. This proves Claim 4.
By Claims 3 and 4, the operator T admits a fixed point . Then the definition of T implies that , so u is a solution of problem (1). In addition, the fact that guarantees that a.e. in Ω. The proof of Theorem 1 is complete. □
3 Result on problem (2)
The hypotheses on the data a, (), f, and g in problem (2) that we suppose are as follows: (H1) in Section 2,
Remark 4 As in the case of problem (1), we note that the constant functions and are not solutions of problem (2), unless or on ∂ Ω.
Now we state our result of existence and location of solutions for problem (2).
with and as in ().
which is a nonempty, bounded, closed, convex subset of . We proceed by proving four claims regarding problem (2) that correspond to those in the proof of Theorem 1 for problem (1). We provide the proof since there are some differences with respect to the proof of Theorem 1.
Since (as postulated in ()), we infer that is also coercive.
where stands for the Lebesgue measure of Ω. Therefore is linear and continuous. The properties of the mappings A and B permit to apply the Lax-Milgram theorem, through which we conclude that problem (13) admits a unique solution. This establishes Claim 1.
As in the proof of Theorem 1, we introduce the operator defined by for all , with given in Claim 1.
Claim 2: The mapping is continuous.
with μ and in ().
Now it suffices to combine (18), (19), (20) and recall that (see ()) to conclude that . Then, because and Ω is bounded, by the Poincaré inequality, we also deduce that . This amounts to saying that in , which proves Claim 2.
Following the approach developed in the proof of Theorem 1, we introduce the operator given by (10), with the truncation function defined in (9) corresponding to the constants and in ().
Claim 3: The mapping has a fixed point.
with independent of u. We conclude that the set is bounded in , so relatively compact in . Therefore the map is compact. This enables us to apply Schauder’s fixed point theorem (see, e.g., [, p.452]), which implies that S possesses a fixed point in C. Claim 3 is thus shown.
Claim 4: If is a fixed point of S, then .
Let be a fixed point of S and set . In order to show that u is a fixed point of T, it is needed to be fulfilled a.e. in Ω. The proof is done following the pattern of the corresponding part in the proof of Theorem 1. We outline the proof of a.e. in Ω (the proof of the other inequality is similar).
It turns out that a.e. in . Also, it is clear that in . Consequently, the equality in Ω is valid, which results in a.e. in Ω because . This reads as a.e. in Ω, so Claim 4 is fulfilled.
Now we can conclude the proof. Claims 3 and 4 ensure that there exists a fixed point of the operator T. This means that and u is a solution of problem (2). Moreover, since , we also have a.e. in Ω. The desired conclusion is achieved. □
The second author is supported by the Marie Curie Intra-European Fellowship for Career Development within the European Community’s 7th Framework Program (Grant Agreement No. PIEF-GA-2010-274519).
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