Thesis--University of California, Berkeley, 1963. / "UC-32 Mathematics and Computers" -t.p. "TID-4500 (19th Ed.)" -t.p. Includes bibliographical references (p. 45).

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1993. / Includes bibliographical references (p. 49-50).

The classical Riemann-Roch theorem is extended to solutions of elliptic equations on manifolds with conical points.

Let Ω be a Lipschitz domain in Rᴺ,n ≥ 3, and L = divA∇ - B∇ be a second order elliptic operator in divergence form with real coefficients such that A is a bounded elliptic matrix and the vector field B ɛ L∞loc(Ω) is divergence free and satisfies the growth condition dist(X,∂Ω)|B(X)|≤ ɛ1 for ɛ1 small in a neighbourhood of ∂Ω. For these elliptic operators we will study on the basis of the theory for elliptic operators without drift terms the Dirichlet problem for boundary data in Lp(∂Ω), 1 < p < ∞, and the regularity problem for boundary data in W¹,ᵖ(∂Ω) and HS¹. The main result of this thesis is that the solvability of the regularity problem for boundary data in HS1 implies the solvability of the adjoint Dirichlet problem for boundary data in Lᵖ'(∂Ω) and the solvability of the regularity problem with boundary data in W¹,ᵖ(∂Ω for some 1 < p < ∞. In [KP93] C.E. Kenig and J. Pipher have proven for elliptic operators without drift terms that the solvability of the regularity problem with boundary data in W¹,ᵖ(∂Ω) implies the solvability with boundary data in HS1. Thus the result of C.E. Kenig and J. Pipher and our main result complement a result in [DKP10], where it was shown for elliptic operators without drift terms that the Dirichlet problem with boundary data in BMO is solvable if and only if it is solvable for boundary data in Lᵖ(∂Ω) for some 1 < p < ∞. In order to prove the main result we will prove for the elliptic operators L the existence of a Green's function, the doubling property of the elliptic measure and a comparison principle for weak solutions, which are well known results for elliptic operators without drift terms. Moreover, the solvability of the continuous Dirichlet problem will be established for elliptic operators L = div(A∇+B)+C∇+D with B,C,D ɛ L∞loc(Ω) such that in a small neighbourhood of ∂Ω we have that dist(X,∂Ω)(|B(X)| + |C(X)| + |D(X)|) ≤ ɛ1 for ɛ1 small and that the vector field B satisfies |∫B∇Ø| ≤ C∫|∇Ø| for all Ø ɛ Wₒ¹'¹ of that neighbourhood.

For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the 'eta' invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be non-zero.

Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We consider smooth symbols and ellipticity without additional boundary conditions which is the relevant case on a manifold with boundary. Starting from a class of symbols that has been investigated before for integer orders in boundary value problems with the transmission property we study operators of arbitrary real orders that play a similar role for operators without the transmission property. Moreover, we show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We finally investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary.

We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance. We provide a selfcontained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson's relative index theorem and a generalization of the cobordism theorem.

Let $M$ be a smooth compact manifold equipped with a co-oriented subbundle $E\subset TM$. We suppose that a compact Lie group $G$ acts on $M$ preserving $E$, such that the $G$-orbits are transverse to $E$. If the fibres of $E$ are equipped with a complex structure then it is possible to construct a $G$-invariant Dirac operator $\dirac$ in terms of the resulting almost CR structure. We show that there is a canonical equivariant differential form with generalized coefficients $\mathcal{J}(E,X)$ defined on $M$ that depends only on the $G$-action and the co-oriented subbundle $E$. Moreover, the group action is such that $\dirac$ is a $G$-transversally elliptic operator in the sense of Atiyah \cite{AT}. Its index is thus defined as a generalized function on $G$. Beginning with the equivariant index formula of Paradan and Vergne \cite{PV3}, we obtain an index formula for $\dirac$ computed as an integral over $M$ that is free of choices and growth conditions. This formula necessarily involves equivariant differential forms with generalized coefficients and we show that the only such form required is the canonical form $\mathcal{J}(E,X)$. In certain cases the index of $\dirac$ can be interpreted in terms of a CR analogue of the space of holomorphic sections, allowing us to view our index formula as a character formula for the $G$-equivariant quantization of the almost CR manifold $(M,E)$. In particular, we obtain the ``almost CR'' quantization of a contact manifold, in a manner directly analogous to the almost complex quantization of a symplectic manifold.

Let $M$ be a smooth compact manifold equipped with a co-oriented subbundle $E\subset TM$. We suppose that a compact Lie group $G$ acts on $M$ preserving $E$, such that the $G$-orbits are transverse to $E$. If the fibres of $E$ are equipped with a complex structure then it is possible to construct a $G$-invariant Dirac operator $\dirac$ in terms of the resulting almost CR structure. We show that there is a canonical equivariant differential form with generalized coefficients $\mathcal{J}(E,X)$ defined on $M$ that depends only on the $G$-action and the co-oriented subbundle $E$. Moreover, the group action is such that $\dirac$ is a $G$-transversally elliptic operator in the sense of Atiyah \cite{AT}. Its index is thus defined as a generalized function on $G$. Beginning with the equivariant index formula of Paradan and Vergne \cite{PV3}, we obtain an index formula for $\dirac$ computed as an integral over $M$ that is free of choices and growth conditions. This formula necessarily involves equivariant differential forms with generalized coefficients and we show that the only such form required is the canonical form $\mathcal{J}(E,X)$. In certain cases the index of $\dirac$ can be interpreted in terms of a CR analogue of the space of holomorphic sections, allowing us to view our index formula as a character formula for the $G$-equivariant quantization of the almost CR manifold $(M,E)$. In particular, we obtain the ``almost CR'' quantization of a contact manifold, in a manner directly analogous to the almost complex quantization of a symplectic manifold.

Thesis (Ph. D.)--University of Missouri-Columbia, 1999. / Typescript. Vita. Includes bibliographical references (leaves 162-169). Also available on the Internet.