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Trace operator and a nonlinear boundary value problem in a new space
Boundary Value Problems volume 2014, Article number: 153 (2014)
We develop a new function space and discuss trace operator on the same genealogical spaces. We also prove that the nonlinear boundary value problem with Dirichlet condition: in the given domain, on the boundary, possesses only a trivial solution if obeys the slope condition: , where is the anti-derivative of with .
We are interested in the nonlinear boundary value problem with Dirichlet condition:
where is an open subset of the -dimensional Euclidean space . Questions of this kind occur in many problems of mathematical physics, in the theory of traveling waves, homogenization, stationary states, boundary layer theory, biology, flame propagation, probability theory, and so on. This problem has been one of the most important and most discussed topics in the theory of partial differential equations during the past several decades.
Many physicists and mathematicians have studied the simplest form
which is a special case of (1) with . A critical reason for studying this special case is that the function spaces that have been used to deal with these problems are just the Lebesgue spaces (especially for the existence theory). However, it is too good to be true in reality!
In this paper we build up a new function space which has been designed to handle solutions of the general nonlinear boundary value problem (1) without imposing too much assumption on the function . As a matter of fact, in , one can find series of attempts to construct new function spaces which generalize classical Lebesgue spaces. We extend those ideas to obtain a better space. The motivation of these research comes from taking a close look at the -norm: of the classical Lebesgue spaces , . It can be rewritten as
Even though the positive real-variable function has very beautiful and convenient algebraic and geometric properties, it also has some practical limitations for the theory of differential equations, as pointed out above. The new space is devised to overcome these limitations without hurting the beauty of -norm too much.
There are two different attempts to generalize the classical Lebesgue spaces - Orlicz spaces and Lorentz spaces . The theory of Orlicz spaces has been well developed which is similar to our new spaces. The Orlicz space requires the convexity of the -function for the triangle inequality of the norm, whereas the norm for the space does not require the convexity of the Hölder function and it has indeed inherited the beautiful and convenient properties from the classical Lebesgue norm.
Based on the new function spaces , we present two main results. One of them is the trace theorem in the space . The trace theorem is one of the basic requirements to deal with the boundary value problems. We state and prove it in Section 3. We also discuss the non-existence of non-trivial solutions for the problem (1) if the given function is of fast growth. To be more precise, we prove that the boundary value problem (1) possesses only a trivial solution if obeys the following slope condition:
where is the anti-derivative of with and (see Section 4 for details).
Throughout this paper, represents an open subset of and is an abstract measure space (Section 2). Also, denotes various real positive constants.
We introduce some terminologies to define the Lebesgue type function spaces which improve the original version introduced in . In this section, .
A pre-Hölder function is an absolutely continuous bijective function satisfying . If there exists a pre-Hölder function satisfying
for all , then is called the conjugate (pre-Hölder) function of . In the relation (5), the notations , are meant to be the inverse functions of , , respectively. Examples of pre-Hölder pairs are for , and for , . In fact, for any Orlicz -function together with complementary -function , is a pre-Hölder pair with .
Some basic identities for a pre-Hölder pair are listed:
In the following discussion, a function represents the two-variable function on defined by
provided that a pre-Hölder pair exists.
Let . A pre-Hölder function with the conjugate function is said to be a Hölder function if for any positive constants , there exist constants , (depending on , ) such that
and that a comparable condition
holds for all .
The following proposition and the proof may illustrate that the comparable condition (13) is not far-fetched.
Letbe a convex pre-Hölder function together with the convex conjugate function. Suppose that for any, there are constants, , , (depending on, ) withandsatisfying the slope conditions;
Thenis a Hölder function. (So is.)
The equation of the tangent plane of the graph of at reads
Then for and , can be rewritten as
From the slope conditions (14), (15) together with the observation that
On the other hand, we observe that every point on the surface is an elliptic point since the Gaussian curvature of a point on the surface is positive from the convexity hypotheses on and (we refer to p.162 in ). Hence the tangent planes touch the graph at and nowhere lie below the graph , that is, for any , ,
In fact, since the restriction of the tangent plane is a tangent line to the graph () in the - plane and is concave up on , we get . Furthermore, since is an elliptic point, a local neighborhood of in the surface belongs to the same side of (p.158 in ). So on a local neighborhood of , the graph lies below the tangent plane . This holds for all . Hence is concave up on , which, in turn, illustrates (20). Combining (19) and (20), we conclude that
where we set and . □
We now define the Lebesgue-Orlicz type function spaces: for a Hölder function ,
where we set
A Hölder type inequality and a Minkowski inequality on the new space are presented as follows:
Let be a Hölder function and be the corresponding Hölder conjugate function. Then for any and , we have
The name of Hölder functions originates from the Hölder inequality (24). So we briefly sketch the idea. Let (), (), and then there exist , such that and
Integration of both sides yields
As an important application of a Hölder inequality, we have the Minkowski inequality on . We omit the proof.
(Generalized Minkowski inequality)
Let be a -finite measure space. Suppose that is a Hölder function and is a nonnegative measurable function on satisfying for almost every . Then
In particular, for any , we have
We can also show that the metric space is complete with respect to the metric:
We now present some remarks on the dual space of . To each is associated a bounded linear functional on by
and the operator (inhomogeneous) norm of is at most :
For , if we put , we have and
This implies that the mapping is isomorphic from into the space of continuous linear functionals . Furthermore, it can be shown that the linear transformation is onto.
(Dual space of )
Let be the conjugate Hölder function of a Hölder function . Then the dual space is isomorphic to .
The proof is quite parallel to the classical Riesz representation theorem, so we omit the proof.
The two inequalities (31) and (32) explain the quasi-homogeneity of . That is, we have the following.
In particular, when, we have homogeneity:
The metric space and the classical Orlicz space differ by the choice of the conjugate function. In fact, for the Orlicz space , the complementary -function of is designed to satisfy the relation
which implies, in turn,
for some constants . Also, the Luxemburg norm
for the Orlicz space requires the convexity of the -function for the triangle inequality of the norm. On the other hand, the (inhomogeneous) norm for the space does not require the convexity of the Hölder function and it has indeed inherited the beautiful and convenient properties from the classical Lebesgue norm.
Trace operator on Sobolev type space
Let be an open subset of . The Sobolev type space is employed in
together with the norm
where . Then it can be shown that the function space is a separable complete metric space, and that is dense in .
The completion of the space with respect to the norm is denoted by , where is the space of smooth functions on with compact support.
We introduce the trace operator on , which is important by itself and also useful in Section 4. We want to point out that the trace operator we present here is an improvement and a completion of the one briefly introduced in .
We say that a pre-Hölder function satisfies a slope condition if there exists a positive constant for which
holds for almost every . The slope condition (40), in fact, corresponds to the -condition for Orlicz spaces.
We prove that the boundary trace on can be extended to the space as follows.
(Trace map on )
Letbe a Hölder pair obeying the slope condition (40) andbe a bounded open subset ofwith smooth boundary. Then the trace operatoris continuous and uniquely determined byfor.
We first prove the theorem for the special case of flat boundary, and by using it, we take care of the general cases.
A special case - . For the case and for a smooth function , we observe that
Owing to the identity (9), we have
On the other hand, using the identity (6), we can notice that the slope condition (40) is equivalent to
Reflecting (43) to the identity (42), we have
Therefore we have
Inserting this into the right side of (41), we obtain
for some positive constants (two ’s may be different). The comparable condition (13) has been used in the second inequality. Taking integrations on both sides over , we obtain
This inequality says that the trace on can be uniquely extended to the space .
The general case - being bounded open in. In this section we restrict our attention to the case of being a bounded open subset. However, can be more general, such as unbounded domains satisfying the uniform -regularity condition (p.84 in ).
Assume that is an dimensional -manifold. Letting
the last condition can be stated as follows. There is a finite collection of open bounded sets in ; with and corresponding which are bijections satisfying , , and mapping onto , , and , respectively, and each Jacobian is positive. Each pair is a coordinate patch.
Let . We can construct , , with , and
Thus, is a partition-of-unity subordinate to the open cover of , and is a partition-of-unity subordinate to the open cover of .
If is a function defined on , then we have
where and is the magnitude of the vector
at . Here represents the standard basis for . From the fact that
we notice that
Then by the smoothness property,
since . Finally, we construct the trace on as indicated. First, we represent
We take the trace operator on each component except for the first component , to say , :
We note that , . Finally, summing up all components, we obtain
The fact follows from the estimates
where is the maximum of all Jacobians, is the norm of the trace from half-space as in (47), and is the largest norm in under a change of variables . Clearly, if then . The proof is now completed. □
Ill-posedness of boundary value problems
In this section we investigate a nonlinear elliptic partial differential equation, namely the nonlinear boundary value problem:
Here we assume that is the derivative of a Hölder function satisfying a slope condition; . Hence there exists a positive constant for which
holds for almost every . Our goal is to demonstrate that the slope condition (60) with large constant implies that is the only strong solution of (59) under a certain geometric condition on . We are going to find such a constant explicitly. turns out to be exact for .
In the following discussion, is assumed to be a bounded open set with smooth boundary. We multiply the PDE (59) by and integrate over to find
The left side can be rewritten as
since on . In the above, represents the unit outward normal vector. Therefore we get
On the other hand, multiplying the PDE (59) by and integrating over , we get
We take a close look at the left side:
and we consider and separately which are defined as
We first take care of the second term . For it, we observe that
and the second term on the right side of (67) becomes
This says that can be rewritten as
The second term of (69) is, in turn,
Therefore we obtain
We now take care of . We rewrite it as
Since on , is parallel to the normal vector at each point . Thus we get . This identity makes it possible to rewrite as
where we count on the fact that . In all, the left side of (64) becomes
Now we consider the right side of (64):
The last equality follows from the fact that on . In view of (64) together with (74) and (75), we get
which can be written as
Hence if we suppose that is a connected convex domain containing the origin, for example, an open ball , then for all . From this, we see that (77) implies
which says, in turn, that the constant should be less than or equal to . We summarize.
Letbe a Hölder function with the slope condition (60) andand letbe a connected and bounded open convex subset of () containing the origin with smooth boundary. Then the nonlinear boundary value problem:
has only a trivial solution inif, whereis the constant appearing in (60).
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The first author was supported by the research fund of Dankook University in 2012.
The authors declare that they have no competing interests.
HCP organized and wrote this paper. YJP contributed to all the steps of the proofs in this research. All authors read and approved the final manuscript.
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Cite this article
Pak, H.C., Park, Y.J. Trace operator and a nonlinear boundary value problem in a new space. Bound Value Probl 2014, 153 (2014). https://doi.org/10.1186/s13661-014-0153-z
- trace operator
- boundary value problem
- function space