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.

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

##### MSC:

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 |