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On the Dirichlet problem for higher-order equations. (English) Zbl 0532.35026
Harmonic analysis, Conf. in Honor A. Zygmund, Chicago 1981, Vol. 2, 620-633 (1983).
[For the entire collection see Zbl 0493.00009.]
This expository paper outlines the present state-of-art concerning Dirichlet problems for higher order elliptic equations related to non- smooth boundary regions \(G\subset R^ d\). In order to exhibit the rôle played by the domain G, the operator considered here is (for simplicity) assumed to be an iterate \(\Delta^ m\) of the Laplacian \(\Delta\). On the contrary the boundary \(\partial G\) of G is supposed to be very complicated. In fact the author assumes that it may be represented as follows: \(G=\cup^{n}_{k=1}M_ k\), where \(M_ k\) denotes a finite union of (d-k)-dimensional smooth manifolds, not forming cusps. Various formulations of the following Dirichlet problem \[ (1)\quad \Delta^ mu=0\quad in\quad G,\quad(2)\quad D^{\alpha}(u-g)|_ G=0,\quad 0\leq | \alpha | \leq m-1 \] are considered, g being an assigned function.
The first formulation is nothing but the (usual) variational (or weak) problem. It assures, as is well-known, the existence and the uniqueness of a solution \(u\in H^ m(G)\) to problem (1), (2) for any g prescribed in \(H^ m(G)\). Conditions (2) are meant as follows: \(u-g\in H^ m_ 0(G)\). The second formulation consists in assuming again \(g\in H^ m(G)\) and in interpreting conditions (2) in the sense of traces (following Sobolev): \(D^{\alpha}(u-g)|_{M_ k}=0\); \(1\leq k<2(m-| \alpha |)\), 0\(\leq | \alpha | \leq m-1\). The third formulation is based upon the concepts of (m,p)-capacity and of corrected functions (in the sense of (m,p)-capactity). Taking advantage of previous results by the author and of a new one by the author himself and T. H. Wolff [Ann. Inst. Fourier 33, No.4, 161-188 (1983; Zbl 0508.31008)] it is possible to show that the restrictions to G (in the sense of corrected functions) of solutions \(u\in H^ m(R^ d)\) to problem (1), (2) with \(g\in H^ m(R^ d)\) coincide, when boundary conditions (2) are also meant in the sense of corrected functions.
This result depends on the fact that every closed set \(F\subset R^ d\) admits (m,p)-spectral synthesis (i.e. \(f\in H^{m,p}(R^ d)\), \(D^{\alpha}f_ F=0\) (in the sense of corrected functions), 0\(\leq | \alpha | \leq m-1\), implies \(f\in H^ m_ 0(F^ c))\). Finally the superfluous hypothesis \(g\in H^ m(R^ d)\) is replaced by the more pertinent one \(g\in H^ m(G)\) by interpreting each condition in (2) in the sense of \((m-| \alpha |,2)\quad fine\) topology.
Reviewer: A.Lorenzi

35J40 Boundary value problems for higher-order elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35D05 Existence of generalized solutions of PDE (MSC2000)
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
35J35 Variational methods for higher-order elliptic equations