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The Dirichlet problem for the biharmonic equation in \(C^ 1\) domains. (English) Zbl 0644.35039

The Dirichlet problem for the bilaplacian is solved in bounded \(C^ 1\) domains in \(R^ n\), \(n\geq 2\), with \(L^ p\) data, \(1<p<\infty\), in the sense of nontangential almost everywhere convergence. The following theorem is proved:
Let \(D\subset R^ n\), \(n\geq 2\), be a bounded \(C^ 1\) domain with connected boundary. Let \(1<p<\infty\). Then, given \(f\in L^ p_ 1(\partial D)\) and \(g\in L^ p(\partial D)\) there exists a unique function u in D such that
(i) \(\Delta \Delta u=0\forall X\in D,\)
(ii) \(\lim_{x\to Q,X\in \Gamma (Q)}u(X)=f(Q)\) almost everywhere (dQ),
(iii) \(\lim_{X\to Q,X\in \Gamma (Q)}<N_ Q,\nabla u(X)>=g(Q)\) almost everywhere (dQ),
(iv) \((\nabla u)^*\in L^ p(\partial D).\)
Here \(\Gamma\) (Q) form any fixed regular family of cones for D.
Reviewer: A.Bove

MSC:

35J40 Boundary value problems for higher-order elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
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