The Boundary Value Problem of the Equations with Nonnegative Characteristic Form

Keldys [1] studies the boundary problem for linear elliptic equations with degenerationg on the boundary. For the linear elliptic equations with nonnegative characteristic forms, Oleinik and Radkevich [2] had discussed the Keldys-Fichera boundary value problem. In 1989, Ma and Yu [3] studied the existence of weak solution for the Keldys-Fichera boundary value of the nonlinear degenerate elliptic equations of second-order. Chen [4] and Chen and Xuan [5], Li [6], and Wang [7] had investigated the existence and the regularity of degenerate elliptic equations by using different methods. In this paper, we study the generalized Keldys-Fichera boundary value problem which is a kind of new boundary conditions for a class of higher-order equations with nonnegative characteristic form. We discuss the existence and uniqueness of weak solution by using the acute angle principle, then study the regularity of solution by using inverse Hölder inequalities in the anisotropic Sobolev Space.

We firstly study the following linear partial differential operator

(1.1)

where is an open set, the coefficients of are bounded measurable, and the leading term coefficients satisfy

(1.2)

We investigate the generalized Keldys-Fichera boundary value conditions as follows:

(1.3)

(1.4)

(1.5)

with and , where .

The leading term coefficients are symmetric, that is, which can be made into a symmetric matrix . The odd order term coefficients can be made into a matrix is the outward normal at . and are the eigenvalues of matrices and , respectively. and are orthogonal matrix satisfying

(1.6)

The boundary sets are

(1.7)

At last, we study the existence and regularity of the following quasilinear differential operator with boundary conditions (1.3)–(1.5):

(1.8)

where and

This paper is a generalization of [3, 8–10].

For second-order equations with nonnegative characteristic form, Keldys [1] and Fichera presented a kind of boundary that is the Keldys-Fichera boundary value problem, with that the associated problem is of well-posedness. However, for higher-order ones, the discussion of well-posed boundary value problem has not been seen. Here we will give a kind of boundary value condition, which is consistent with Dirichlet problem if the equations are elliptic, and coincident with Keldys-Fichera boundary value problem when the equations are of second-order.

We consider the linear partial differential operator

(2.1)

where is an open set, the coefficients of are bounded measurable functions, and

Let be a series of functions with . If in certain order we put all multiple indexes with into a row , then can be made into a symmetric matrix . By this rule, we get a symmetric leading term matrix of (2.1), as follows:

(2.2)

Suppose that the matrix is semipositive, that is,

(2.3)

and the odd order part of (2.1) can be written as

(2.4)

where is the Kronecker symbol. Assume that for all , we have

(2.5)

We introduce another symmetric matrix

(2.6)

where is the outward normal at . Let the following matrices be orthogonal:

(2.7)

satisfying

(2.8)

where is the transposed matrix of are the eigenvalues of and are the eigenvalues of . Denote by

(2.9)

For multiple indices means that . Now let us consider the following boundary value problem,

(2.10)

(2.11)

(2.12)

(2.13)

for all and , where .

We can see that the item (2.13) of boundary value condition is determined by the leading term matrix (2.2), and (2.12) is defined by the odd term matrix (2.6). Moreover, if the operator is a not elliptic, then the operator

(2.14)

has to be elliptic.

In order to illustrate the boundary value conditions (2.11)–(2.13), in the following we give an example.

Example 2.1.

Given the differential equation

(2.15)

Here . Let and , then the leading and odd term matrices of (2.15) respectively are

(2.16)

and the orthogonal matrices are

(2.17)

We can see that , and as shown in Figure 1.The item (2.12) is

(2.18)

and the item (2.13) is

(2.19)

for all and . Since only , hence we have

(2.20)

however, and , therefore,

(2.21)

Thus the associated boundary value condition of (2.15) is as follows:

(2.22)

which implies that is free on

Remark 2.2.

In general the matrices and arranged are not unique, hence the boundary value conditions relating to the operator may not be unique.

Remark 2.3.

When all leading terms of are zero, (2.10) is an odd order one. In this case, only (2.11) and (2.12) remain.

Now we return to discuss the relations between the conditions (2.11)–(2.13) with Dirichlet and Keldys-Fichera boundary value conditions.

It is easy to verify that the problem (2.10)–(2.13) is the Dirichlet problem provided the operator being elliptic (see [11]). In this case, for all . Besides, (2.13) run over all and , moreover is nondegenerate for any . Solving the system of equations, we get .

When , namely, is of second-order, the condition (2.12) is the form

(2.23)

and (2.13) is

(2.24)

Noticing

(2.25)

thus the condition (2.13) is the form

(2.26)

It shows that when , (2.12) and (2.13) are coincide with Keldys-Fichera boundary value condition.

Next, we will give the definition of weak solutions of (2.10)–(2.13) (see [12]). Let

(2.27)

where is defined by

(2.28)

We denote by the completion of under the norm and by the completion of with the following norm

(2.29)

Definition 2.4.

is a weak solution of (2.10)–(2.13) if for any , the following equality holds:

(2.30)

We need to check the reasonableness of the boundary value problem (2.10)–(2.13) under the definition of weak solutions, that is, the solution in the classical sense are necessarily the solutions in weak sense, and conversely when a weak solution satisfies certain regularity conditions, it will surely satisfy the given boundary value conditions. Here, we assume that all coefficients of are sufficiently smooth.

Let be a classical solution of (2.10)–(2.13). Denote by

(2.31)

Thanks to integration by part, we have

(2.32)

Since , we have

(2.33)

Because satisfies (2.12),

(2.34)

From the three equalities above we obtain (2.30).

Let be a weak solution of (2.10)–(2.13). Then the boundary value conditions (2.11) and (2.13) can be reflected by the space . In fact, we can show that if , then satisfies

(2.35)

Evidently, when , we have

(2.36)

If we can verify that for any , (2.36) holds true, then we get

(2.37)

which means that (2.35) holds true. Since is dense in , for given, let and in . Then

(2.38)

Due to satisfying (2.36), hence satisfies (2.36). Thus (2.31) is verified.

Remark 2.5.

When (2.2) is a diagonal matrix, then (2.13) is the form

(2.39)

where In this case, the corresponding trace embedding theorem can be set, and the boundary value condition (2.13) is naturally satisfied. On the other hand, if the weak solution of (2.10)–(2.13) belongs to for some , then by the trace embedding theorems, the condition (2.13) also holds true.

It remains to verify the condition (2.12). Let satisfy (2.30). Since , hence we have

(2.40)

On the other hand, by (2.30), for any , we get

(2.41)

Because the coefficients of are sufficiently smooth, and is dense in , equality (2.41) also holds for any . Therefore, due to , we have

(2.42)

From (2.36), one drives

(2.43)

Furthermore,

(2.44)

From (2.30) and (2.42), one can see that

(2.45)

Noticing in , one deduces that satisfies (2.12) provided Finally, we discuss the well-posedness of the boundary value problem (2.10)–(2.13).

Let be a linear space, and be the completion of , respectively, with the norm . Suppose that is a reflexive Banach space and is a separable Banach space.

Definition 2.6.

A mapping is called to be weakly continuous, if for any in , one has

(2.46)

Lemma 2.7 (see [3]).

Suppose that is a weakly continuous, if there exists a bounded open set , such that

(2.47)

then the equation has a solution in .

Theorem 2.8 (existence theorem).

Let be an arbitrary open set, and . If there exist a constant and such that

(2.48)

where is the component of corresponding to , then the problem (2.10)–(2.13) has a weak solution in .

Proof.

Let be the inner product as in (2.31). It is easy to verify that defines a bounded linear operator . Hence is weakly continuous (see [3]). From (2.42), for we drive that

(2.49)

Hence we obtain

(2.50)

Thus by Hölder inequality (see [13]), we have

(2.51)

By Lemma 2.7, the theorem is proven.

Theorem 2.9 (uniqueness theorem).

Under the assumptions of Theorem 2.8 with in (2.48). If the problem (2.10)–(2.13) has a weak solution in , then such a solution is unique. Moreover, if in , , then the weak solution of (2.10)–(2.13) is unique.

Proof.

Let be a weak solution of (2.10)–(2.13). We can see that (2.30) holds for all . Hence is well defined. Let . Then from (2.49) it follows that , we obtain , which means that the solution of (2.10)–(2.13) in is unique. If all the odd terms of , then (2.30) holds for all , in the same fashion we known that the weak solution of (2.10)–(2.13) in is unique. The proof is complete.

Remark 2.10.

In next subsection, we can see that under certain assumptions, the weak solutions of degenerate elliptic equations are in .

Given the quasilinear differential operator

(3.1)

where and

Let , the odd order part of (3.1) be as that in (2.4), , and be the same as those in Section 2. The leading matrix is

(3.2)

and the eigenvalues are . We denote

We consider the following problem:

(3.3)

Denote the anisotropic Sobolev space by

(3.4)

whose norm is

(3.5)

when all for , then the space is denoted by . is termed the critical embedding exponent from to , if is the largest number of the exponent in where , and the embedding is continuous.

For example, when is bounded, the space with norm is an anisotropic Sobolev space, and the critical embedding exponents from to are .

Suppose that the following hold.

The coefficients of the leading term of satisfy one of the following two conditions:

()

()

There is a constant such that

(3.6)

There are functions with , such that

(3.7)

There is a constant such that

(3.8)

where or .

There is a constant such that

(3.9)

where is a critical embedding exponent from to . Let be defined by (2.27) and be the completion of under the norm

(3.10)

and be the completion of with the norm

(3.11)

where

is a weak solution of (3.3), if for any , we have

(3.12)

Theorem 3.1.

Under the conditions , if , then the problem (3.3) has a weak solution in .

Proof.

Denote by the left part of (3.12). It is easy to verify that the inner product defines a bounded mapping by the condition .

Let , by , one can deduce that

(3.13)

Noticing that , , , by Hölder and Young inequalities (see[13]), from (3.13) we can get

(3.14)

Ones can easily show that the mapping is weakly continuous. Here we omit the details of the proof. By Lemma 2.7, this theorem is proven.

In the following, we take an example to illustrate the application of Theorem 3.1.

Example 3.2.

We consider the boundary value problem of odd order equation as follows:

(3.15)

where is an unit ball in , see Figure 2

The odd term matrix is

(3.16)

It is easy to see that

(3.17)

The boundary value condition associated with (3.15) is

(3.18)

Applying Theorem 3.1, if , then the problem (3.15)–(3.18) has a weak solution .

We start with an abstract regularity result which is useful for the existence problem of -solutions of degenerate quasilinear elliptic equations of order . Let be the spaces defined in Definition 2.6, and be a reflective Banach space, at the same time .

Lemma 4.1.

Under the hypotheses of Lemma 2.7, there exists a sequence of in such that . Furthermore, if, we can derive that , is a constant, then the solution of belongs to .

In the following, we give some existence theorems of -solutions for the boundary value conditions (4.3)–(4.5) of higher-order degenerate elliptic equations.

First, we consider the quasilinear equations

(4.1)

where . Now, we consider the following problem

(4.2)

(4.3)

(4.4)

(4.5)

The boundary value condition associated with (4.1) is given by (4.3)–(4.5). Suppose that is bounded, and the following assumptions hold.

The condition (3.6) holds, and there is a continuous function on such that

(4.6)

where ,

is a measure zero set in , and there is a sequence of subdomains with cone property such that and

The positive definite condition is

(4.7)

where is a constant, or for .

The structure conditions are

(4.8)

where is a constant, , is the critical embedding exponent from to .

Let be defined by (2.27) and be the completion of with the norm

(4.9)

Definition 4.2.

is a weak solution of (4.2)–(4.5), if for any , the following equality holds:

(4.10)

Theorem 4.3.

Under the assumptions , if , then the problem and (4.2)–(4.5) has a weak solution . Moreover, if there is a real number , such that

(4.11)

then the weak solution

Proof.

According to Lemma 4.1, it suffices to prove that there is a constant such that for any ( is as that in Section 3) with , we have

(4.12)

From (4.10) we know

(4.13)

Due to and we have

(4.14)

Noticing that , and consequently we have

(4.15)

where the or , is the critical embedding exponent from to . By the reversed Hölder inequality (see [14])

(4.16)

Then we obtain

(4.17)

From (4.15) and (4.17), the estimates (4.12) follows. This completes the proof.

Next, we consider a quasilinear equation

(4.18)

where

Suppose that the following holds.

There is a real number such that

(4.19)

The structural conditions are

(4.20)

where C is a constant, , are the critical embedding exponents from to .

Theorem 4.4.

Let the conditions and be satisfied. If , then the problem (4.2)–(4.5) has a weak solution .

The proof of Theorem 4.4 is parallel to that of Theorem 4.3; here we omit the detail.