Abstract
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.
Highlights
Let [l be a bounded domain in Rv with smooth boundary r and k E IL Let A denote the Laplace operator on R2v
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 L P-estimates for the solutions of certain biharmonic boundary value problems which are related to the first eigen-values of the associated biharmonic operators
Summary
Let [l be a bounded domain in Rv with smooth boundary r and k E IL Let A denote the Laplace operator on R2v. Consider the following eigen-value problems for the biharmonic operator. 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). Our approach is to study the eigen-value problem (1.:3) through the notion of weak-solutions and the Fredhlom theory of compact operators. 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) They just deal with either a square or a circular domain in R. 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. 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:
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