The Boundary Value Problem of the Equations with Nonnegative Characteristic Form
© Limei Li and Tian Ma. 2010
Received: 22 May 2010
Accepted: 7 July 2010
Published: 25 July 2010
We study the generalized Keldys-Fichera boundary value problem for a class of higher order equations with nonnegative characteristic. By using the acute angle principle and the Hölder inequalities and Young inequalities we discuss the existence of the weak solution. Then by using the inverse Hölder inequalities, we obtain the regularity of the weak solution in the anisotropic Sobolev space.
Keldys  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  had discussed the Keldys-Fichera boundary value problem. In 1989, Ma and Yu  studied the existence of weak solution for the Keldys-Fichera boundary value of the nonlinear degenerate elliptic equations of second-order. Chen  and Chen and Xuan , Li , and Wang  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.
with and , where .
2. Formulation of the Boundary Value Problem
For second-order equations with nonnegative characteristic form, Keldys  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.
where is an open set, the coefficients of are bounded measurable functions, and
for all and , where .
has to be elliptic.
In order to illustrate the boundary value conditions (2.11)–(2.13), in the following we give an example.
In general the matrices and arranged are not unique, hence the boundary value conditions relating to the operator may not be unique.
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 ). In this case, for all . Besides, (2.13) run over all and , moreover is nondegenerate for any . Solving the system of equations, we get .
It shows that when , (2.12) and (2.13) are coincide with Keldys-Fichera boundary value condition.
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.
From the three equalities above we obtain (2.30).
Due to satisfying (2.36), hence satisfies (2.36). Thus (2.31) is verified.
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.
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.
Lemma 2.7 (see ).
then the equation has a solution in .
Theorem 2.8 (existence theorem).
where is the component of corresponding to , then the problem (2.10)–(2.13) has a weak solution in .
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.
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.
In next subsection, we can see that under certain assumptions, the weak solutions of degenerate elliptic equations are in .
3. Existence of Higher-Order Quasilinear Equations
and the eigenvalues are . We denote
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:
where or .
where is a critical embedding exponent from to . Let be defined by (2.27) and be the completion of under the norm
Under the conditions , if , then the problem (3.3) has a weak solution in .
Denote by the left part of (3.12). It is easy to verify that the inner product defines a bounded mapping by the condition .
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.
where is an unit ball in , see Figure 2
4. -Solutions of Degenerate Elliptic Equations
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 .
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.
The boundary value condition associated with (4.1) is given by (4.3)–(4.5). Suppose that is bounded, and the following assumptions hold.
is a measure zero set in , and there is a sequence of subdomains with cone property such that and
where is a constant, or for .
where is a constant, , is the critical embedding exponent from to .
then the weak solution
From (4.15) and (4.17), the estimates (4.12) follows. This completes the proof.
Suppose that the following holds.
where C is a constant, , are the critical embedding exponents from to .
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.
This project was supported by the National Natural Science Foundation of China (no. 10971148).
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