BIHARMONIC EIGEN-VALUE PROBLEMS AND Lp ESTIMATES

Biharmonic eigen-values arise in the study of static equilibrium of an elastic body which has been suitably secured at the boundary. This paper is concerned mainly with the existence of and Lp-estimates for the solutions of certain biharmonic boundary value problems which are related to the first eigen-values of the associated biharmonic operators. The methods used in this paper consist of the “a-priori” estimates due to Agmon-Douglas-Nirenberg and P. L. Lions along with the Fredholm theory for compact operators.


0
Here denotes the exterior normal derivative at a point on the boundary I" of f.The eigen- value problem (1.1) arises in the study of the static equilibrium of an elastic body f which is simply supported along the boundary F while problem (1.2) corresponds to the case when the boundary F is supported by sliding clamps.The eigen-value problem (1.3) arises when the boundary F of an elastic body is fixed or cantilevered.
The spectrum and the corresponding eigen-spaces for the problem (1.1) (respectively (1.2)) can be studied by considering the biharmonic operator associated to (1.1) (respectively (1.2)) as the squares of the second order Dirichlet (respectively Neumann) operator --A in L2(f).Indeed, >,2 is an eigen-value for (1.1) iff k is an eigen-value for the Dirichlet problem --Au ,u, on f u 0, on I', (1.1)* and q is an eigen-value for (1.2) if/ 7 is an eigen-value for the Neumann problem -Au "u, (1.2)* O, on I', (see [I]).
We note that the associated biharmonic operator in L(f) for the eigen-value problem (1.3)   cannot be obtained as the square of any operator in Lz(f) defined in terms of the Laplacian operator -A.Our approach then is to study the eigen-value problem (1.:3) through the notion of weak-solutions and the Fredhlom theory of compact operators.Accordingly, we define in section 2, a biharmonic operation A associated to (1.3) which has Wo2'(f 0 as its domain in L'(f).
The results concerning (1.3) in this paper (c.f.Theorem 2.2 (i), (ii) and (iii)) are about the general nature of the eigen-values and the-eigen-spaces and are for general f in RN.The other works mentioned give methods for the computation or estimation of the eigen-values and the eigen-functions for (1.3) (refer to section 2 for more details).In general, they just deal with either a square or a circular domain in R. Furthermore, our approach for problem (1.3) also works for (1.1) and (1.2) through straightforward adaptations.Besides, our approach yields as a by-product, some preliminary L 2- estimates ((2.4), (3.6) and (3.7)) which are needed in the proofs of the main results of this paper.
The main interest of this paper is to obtain LP-estimates on u, where u is a solution of one of the following linear problems A2u X12u f, or, f, (1.6) tlere k denotes the first positive eigen-value of the eigen-value problems (1.1), / the first eigen- value of the eigen-value problem (1.3), and f is in a certain subset of L'(ft), 2 _ p < oo (this will be specified later).
We call problems (1.1) and (1.4) the Biharmonic Dirichlet problems, (1.2) and (1.5) the Biharmonic Neumann problems, (1.3) and (1.6) the Mixed Biharmonic problems.The significance of these problems and their L p estimates lies on the consequence that they are fundamental to existence results for 4 tA order non-linear elliptic problems which are treated by the authors in a forthcoming paper [7].
The strategy of this paper is to first obtain LZ(fl) estimates for the weak solutions of (1.4), (1.5) and (1.6) through classical Fredholm theory for compact operators and then extrapolate to the L estimates for higher p by standard bootstrap techniques together with the use of estimates from Agmon-Douglis-Nirenberg [8] and P. L. Lions [9].Consequently, the weak solutions are infact proved to be strong solutions in W4'p(t2) together with the W4'-estimates on u.In the rest of this paper, we will simply focus our attention on (1.3) and (1.6).The proofs in the cases of the other two types of problems are parallel to that of (1.3) and (1.6), we will therefore, omit the details and only mention the necessary modifications for the other two cases whenever it is necessary.This will be done in section 3.
u W"(), p 2 tre.(Note that in this ce, even the wek-lution satfies the boundary conditions in the sense of tre since it belon W '2 ()).
We observe that for u, v (u,v), u v (2.3) defines an inner pruct on Ds{A2) and we denote by V the Hilbert space obtained by endowing Ds(A2) with the norm induced by the inner pruct {2.3): I1 I1 I I', v. (2.4) Futhermore, II I1 on (=) i uivMent the ,2() norm on Da(A2) in view of Theorem 1.1 We now get me preliminary L2 timat and spectrum rults for (1.3) and (1.6).
Hence the threm.// We now define a linear mapping Ls D(Ls) C V L2() by setting D(L3) {u V: f L2() such that u is the weak lution of (2.1) and for u D(L3), Hence, by Theorem 2.1, we have for u D(L), or II.ll,vo(a) S C(N,f2)IILsu llt,(a). (2.9) The following theorem investigages the spectrum of the linear mapping L defined by (2.8).THEOREM 2.2.The spectrum of L 3 is given by {0 where R(L) denotes the range of L in L2(G) and ( denotes the direct sum so that R(L (kr L) E in (2.11).
(v) Furthermore, for any f L2(f2)R(L), there exists an unique solution u ( D (L ]) CI R (L) such that u satisfies Lu Lsu Iu f and hence u is a weak solution of (1.6) in the sense that f An. b plfu ff , / E V. (2.12) Also we have the estimate PROOF.We first notice in view of Theorem 2.1 that, L -l L2(f)-.W02,2(f) exists as a bounded linear mapping.We next assert that Ls is a positive-definite compact Hermitian operator on L 2(f).
The uniqueness of the solution u of (2.10) in D(L3)CI R(L) is another simple consequence of the Fredholm theory.It now remains to prove estimate (2.13) (cf.Remark 2.1).
Au ttu f on f, u O, on r, ou O, on r, in the weak sense of (2.12).It follows that ffl IAu 12 P'f lu 12 ffu.
(2.19) Also, since u (/R(L)---(ker L) ---E, we have from the complete orthogonality of the sub-spaces{E,...,En, }and0 < /l < < /n < that Invoking (2.19), we have and consequently, fl.12_< 1---2 (_,) f I/12, thus estimate (2.13) follows and the theorem has been proved.//REMARK 2.1.We note that at this point, our solution u in Theorem 2.2 is a weak solution in W02'2(fl); but in Theorem 2.4 with f (/LP(f/), p _ 2, we will eventually show that u is a strong solution in W4'P(f).Let us remark that the spectrum of (1.6) is not related to the spectrums of (1.1)* and (1.2)* in any clear and precise manner as the spectrum of (1.1) is related to that of (1.1)* (respectively the spectrum of (1.2) to that of (1.2)*).Thus, it becomes an interesting topic to estimate the /l,-.,/,.and to investigate their relationship with the spectrum of the other problems.
One of the most well known approach is the method of the intermediate problem (c.f.[6]) which consists of starting from a base problem (usually one with simplified boundary conditions) and then approaching the problem of interest through a sequence of intermediate problems (of which the spectrums are better known).As a result, one can obtain lower bounds on/l pa, For instance, when f is a square, very good estimates have been obtained in [3] for the first four eigen-values: Another approach is due to Fichera through the construction of intermediate Green's operators (c.f.[5]).This approach again leads to lower bounds on the /1,..-,/n, for (1.3) when 12 is a square.

On
[2]u 2uzzv, t on 12, O, on 012 then both eigen-value problems for A [1] and A [21 are solved by separation of variables, say, when 12 is a rectangle.
The success of each of these approaches in getting explicit numerical estimates usually depends on the geometry of 12.A result which relates the spectrum of (1.3) to the spectrums of other problems in a most general setting is perhaps the following theorem.This result is obtained via the intermediate problem approach using (1.1)* as a base problem {for details refer to [3]).THEOREM (2.2)*.If /tl,..-,pn,-.-are the eigen-values of the vibrating clamped plate problem (1.3) and Xl,... ,Xn,.--are the eigen-values of problem (1.1)*, then the two spectrums are related to each other by the inequality, k, It is not the main concern for this paper to investigate such estimates.The above remarks are included for the sake of completeness.Now, we are ready to show that for p > 2, f ( L p (12) CI R (L), u in Theorem 2.2 in ft a strong lution nd derive the corrponding L estimate.But before we go further, we need prent a result from Agmon-DouglNirenberg [8].For rens of simplicity, we shall only give statement of th theorem needed for the special ce of this paper.).o, o. r, where B(x; ,..., u) nd Bx; u) e lynomils in N, for x F, of order m and m rpeetively.Furthermore, let denote ( ) and P(0 t + + k + 2 (2.21) be the rresnding chartertie polynomial of 2. We define the Agmon-Dougl- Nirenberg "mplementary Boundary Condition" follows: At any point x F, let n(x) denote the normal to F at x, and be any non-zero real veer parallel to the boundary at the point x.We require that the polynomials, in r, By(z; + rn(x)), j 1,2 be linearly independent modulo the polynomial (r--r/())(r r()), where r/() and r() are the roots of P( + rn(x)) with positive imaginary parts (here r is a scalar).Next, we need the following theorem from THEOREM 2.3.Let f ELP(12), p > 1, be given and let the complementary boundary condition be satisfied.Furthermore, let (2.20) has at most one solution.Then there exists a unique strong solution u for (2.20) satisfying the estimate I1., I1.,..,(.) < C(N,12,,p)ll/IIL,{.) (2.2o.)REMARK 2.3.Theorem 2.3 says that uniqueness of solution is sufficient for solvability.
Equipped with Theorem 2.3, we are ready to prove the following theorem for >2.
(2.23) PROOF.To show that u is a strong solution and to obtain higher L estimate (2.23), we must fit the problem into the setting of Theorem 2.3.Therefore, we note that the chrtertic polynomiu corrponding to Lu, Bu u, B2u are rpectively P(, )= : + + } + B(x; u) (the constant polynomial 1), x F (2.24) B; ) () + + "()N, e r where n(x)-----(n(x),..., n,(x)) is the unit normal at g (//F.These polynomials satisfy the complementary boundary condition of Theorem 2.3 when F is smooth.Consider now the elliptic problem where ] L'(12)n E, p k 2 and u e W0'2(12) are the same as that in Theorem 2.2.Obviously, ply + f G. L2(fl) and since /--0 is not an eigen-value of L:, we have the uniqueness of solutions for (2.25).On invoking Theorem 2.3 with --0, we get that there exists a unique strong solution v for (2.25) in L2(fl).But u being a weak solution of (1.4) must also satisfy (2.25) in the weak sense.We conclude by uniqueness of the weak solution in Theorem 2.1 that u ---v and hence u is infact a strong solution in W4'2(f/) of (1.4) in view of Theorem 2.3.Now we claim that u is also in L (12).Indeed using the Sobolev embeddings, (i) W'(12)C_.C/-/v/(12) for q > -, /vl N (ii) W"(12)CC__, L u--4(12) for q < --, N (iii) W'(12) CC__ L(12) <_ k < oo for q -, (2.26) and for (1.5), the characteristic polynomials corresponding to COu are respectively L2u A2u' Blu "n' B2u cOn 2 B(; I ,N) n](z)l + 4-nN(z)N, z ( r, (3.9) It is trivial to see that both (3.8) and (3.9) satisfy the complementary boundary condition of Theorem 2.3.Now we can proceed in exactly the same way as in the case the mixed Biharmonic problem to get the following theorems which are the analogues of theorem 2.4 for the other two types of problems.
THEOREM 3.3.Let f .L'(f)N E for p _> 2, and let u be the unique weak solution of (i.4) in D(A2)CIE (as in Theorem 3.1), then u is infact the strong unique solution of (1.4) in W 4,p (I) satisfying the estimate THEOREM 3.4.Let f E LP(f)CI E0 -t for p _> 2, and let u be the unique weak- solution of (I.5) in D(A=)C)E0 -t (as in Theorem 3.2), then = is in fact the strong unique solution of (1.5) in W 4' (f) satisfying the estimate REMARK 4.1.If u in Theorem 2.4 belonged to Ds(A2) instead of Ds(A2) C)E:, then estimate (2.23) may no longer hold.However, it is easy to see (by tracing back to the original proof of Theorem 2.4) that u is still a strong solution and it satisfies the following weaker estimate This is due to the fact that we no longer have estimate (2.13) and u is not unique in this case.Similar remarks apply to Theorem 3.3 and 3.4.
ACKNOWLEDGEMENT.The authors are grateful to the referee for the references [3], [4], [5], [6] and for his valuable comments leading to this revision.REFERENCF I.