WELL-POSEDNESS AND REG . ULAITY RESULTS FOR A DYNAMIC VON KARMAN PLATE

We consider the problem of well-posedness and regularity of solutions for a dynamic von Khrmhn |)late which is clamped along one portion of the boundary and which experiences boundary damping through "free edge" condmons on the remainder of the boundary We prove the exmtence of unique strong solutions for this system

INTRODUCTION.In this paper, we consider the well-posedness of the yon Kirmn system given by (1.1) where we assume fl C R , with sufficiefftly smooth boundary F F0 U F1. Here, represents Poisson's ratio and the boundary operators B and B are given by (1.1) (5) Bw [(n n)w, + n,n(w,, w)] Also, F(w) satisfies the system of equations A2F --[W,W] ( F F 0 on E=F(O, cc) } where [' ] Oz --Oy --5 + Oy --Oz -OxOy OzOy" The well-posedness and regularity of such a system is both a delicate and interesting problem.
Such results are important in solving the problem of stabilization for system (1.1).Usual PDE techniques require the existence and uniqueness of "smooth" solutions to justify computations used in determining the stability and controllability of dynamical models.The stabilization of thin plates (and particularly the yon Krmn system)is of current interest in the literature (see (Ill, [2], [3], [4], [5])).The von Krmn nonlinearity poses many difficulties in obtaining the well-posedness and regularity results we seek.Difficulties also arise from the higher order boundary conditions on E.
To handle these difficulties we adapt abstract results proven in [6] to our more difficult boundary conditions.This paper will proceed follows.In Section 2 we state the main results of our paper.After this we state the appropriate abstract results from [6] which will be useful in the proofs of our results.
In Section 3 we prove the results stated in Section 2.
2. STATEMENT OF RESULTS.Before stating the results we i1tend 1o prove, we defillc meaning of "x('ak sollilions" through a variational equality.Let
The proofs of Theorems 2.1-2.3 will be based primarily on the work of Favini and Lasiecka [6].
That paper deals with abstract problems of the form (2.4) w(t 0) w0; wt(t O) (l,1 which will be described in detail shortly.Our intention in this paper is to recast system (1.1) in the abstract framework of (2.4).We will then show that the results of [6] may be applied directly to or may be adapted for our system.For the purpose of self-containment, we now state the necessary background and results fi'om [6] which will be useful in this present context.
Let .A be a closed, positive self-adjoint operator on a Hilbert space H with D(.A) C H. Let V be another (appropriately chosen) Hilbert space such that "D(.,41/2 We assume that kt V V' is both bounded and boundedly invertible so that the restriction 2 Mitt with domain D(kT)= {u V" Mu 11} gives that V D(lt/).
The operator G is defined on another Hilbert space, U.It is assumed that G U H is a bounded linear operator such that G'.A (D(.Aa/); H).
We note that for our purposes, f 0.
We now state the results from [6] which form the framework for Theorem 2.1-2.3.
Then the weak solution (w(t), wt(t)) is 91obal for any T > O.
Mw (I A)w + MG .
This will be useful to us later in the proof.PROOF OF THEOREM 2.1.To complete the proof, it suffices to show that (2.5) holds.
PROOF OF THEOREM 2.3.Here we would like to use the following strong regularity result from [6].holds for all > 0 on H.