Gregory C. Verchota

Maximum principles for the polyharmonic equation on Lipschitz domains

The Agmon-Miranda maximum principle for the polyharmonic equations of all orders is shown to hold in Lipschitz domains in ℝ3. In ℝn,n≥4, the Agmon-Miranda maximum principle andLp-Dirichlet estimates for certainp>2 are shown to fail in Lipschitz domains for these equations. In particular if 4≤n≤2m+1 theLp Dirichlet problem for Δm fails to be solvable forp>2(n−1)/(n−3).

Nonsymmetric systems on nonsmooth planar domains

We study boundary value problems, in the sense of Dahlberg, for second order constant coefficient strongly elliptic systems. In this class are systems without a variational formulation, viz. the nonsymmetric systems. Various similarities and differences between this subclass and the symmetrizable systems are examined in nonsmooth domains.

A multidirectional Dirichlet problem

B.E.J. Dahlberg’s theorems on the mutual absolute continuity of harmonic and surface measures, and on the unique solvability of the Dirichlet problem for Laplace’s equation with data taken in Lp spaces p > 2 − δ are extended to compact polyhedral domains of ℝn. Consequently, for q < 2 + δ, Dahlberg’s reverse Hölder inequality for the density of harmonic measure is established for compact polyhedra that additionally satisfy the Harnack chain condition. It is proved that a compact polyhedral domain satisfies the Harnack chain condition if its boundary is a topological manifold. The double suspension of the Mazur manifold is invoked to indicate that perhaps such a polyhedron need not itself be a manifold with boundary; see the footnote in Section 9. A theorem on approximating compact polyhedra by Lipschitz domains in a certain weak sense is proved, along with other geometric lemmas.

Boundary Coerciveness and the Neumann Problem for 4th Order Linear Partial Differential Operators

The relationship between the classical interior coercive estimate over W 2;2 (›) and a required boundary coercive estimate for solutions to the L 2 (@›) Neumann problem is discussed. A conditional lemma in which boundary coerciveness implies interior is proved. Hilberts theorem that el- liptic operators need not be sums of squares of difierential operators and therefore cannot, in general, have formally positive integro-difierential forms is discussed. An elliptic operator that is a sum of squares yet has no formally positive coercive form is displayed. The existence of coercive forms for elliptic operators far away from sums of squares is questioned.

Harmonic homeomorphisms of the closed disc to itself need be in w1,p, p < 2, but not w1,2

Harmonic maps u from the closed disc onto bounded convex sets of the plane obey u ∈ W 1,p,p < 2.

Agmon Coerciveness and the Analysis of Operators with Formally Positive Integro-Differential Forms

For Sobolev spaces in Lipschitz domains with no imposed boundary conditions, the Aronszajn–Smith theorem algebraically characterizes coercive formally positive integro-differential quadratic forms. Recently, linear elliptic differential operators with formally positive forms have been constructed with the property that no formally positive forms for these operators can be coercive in any bounded domain. In the present article 4th order operators of this kind are shown by perturbation to have coercive forms that are (necessarily) algebraically indefinite. The perturbation here from noncoercive formally positive forms to coercive algebraically indefinite forms requires Agmons characterization of coerciveness in smoother domains than Lipschitz.