In this dissertation, we consider various aspects of the boundary behavior of holomorphic functions of several complex variables. In dimension one, a characterization of the radial limit zero sets of nonconstant holomorphic functions on the disc has been given by Lusin, Privalov, McMillan, and Berman. In higher dimensions, no such characterization is known for holomorphic functions on the unit ball B. Rudin posed the question as to the existence of nonconstant holomorphic functions on the ball with radial limit zero almost everywhere. Hakim, Sibony, and Dupain showed that such functions exist. Because the characterization in dimension one involves both Lebesgue measure and Baire category, it is natural to also ask whether there exist nonconstant holomorphic functions on the ball having residual radial limit zero sets. We show here that such functions exist. We also prove a higher dimensional version of the Lusin-Privalov Radial Uniqueness Theorem, but we show that, in contrast to what is the case in dimension one, the converse does not hold. We show that any characterization of radial limit zero sets on the ball must take into account the "complex structure" on the ball by giving an example that shows that the family of these sets is not closed under orthogonal transformations of the underlying real coordinates. In dimension one, using the theorem of McMillan and Berman, it is easy to see that radial limit zero sets are not closed under unions (even finite unions). Since there is no analogous result in higher dimensions of the McMillan and Berman result, it is not obvious whether the radial limit zero sets in higher dimensions are closed under finite unions. However, we show that, as is the case in dimension one, these sets are not closed under finite unions. Finally, we show that there are smooth curves of finite length in S that are non-tangential limit uniqueness sets for holomorphic functions on B. This strengthens a result of M. Tsuji.

The Swedish defence agency (FOI) has developed a particle model (called Pello) that simulates the dispersion of aerosols and gases. At the boundaries, such as the ground, the particles can either reflect back into the domain (the atmosphere) or be absorbed. Which of the events that occurs is decided by a certain probability, which in the present model depends on mere physical properties. In this thesis we have investigated a newly proposed boundary behaviour which also depends on the time step used in the numerical simulations. We verified the accuracy of the new model by using a dispersion model with an explicit solution. To gain a better understanding of how important parameters at the boundary influence each other, we performed a sensitivity analysis. Simulations showed an overall improving concentration profile as the time step became smaller and the new model working well. The convergence order of the simulations was found to be close to 0.5. In this thesis we have shown that there exist an upper limit for the time step, which depends on the specific model. The present used time step at FOI does not have this versatile property. But having this upper limit for the time step close to the boundary, and a uniform time step can be time demanding. This lead us to the conclusion that an adaptive time step should be implemented.

Let D be a smoothly bounded domain in Cn , n> 1. For each point p _ D, we have the Green function G(z, p) associated to the standard sum-of-squares Laplacian Δ with pole at p and the Robin constant __ Λ(p) = lim G(z, p) −|z − p−2n+2 z→p | at p. The function p _→ Λ(p) is called the Robin function for D. Levenberg and Yamaguchi had proved that if D is a C∞-smoothly bounded pseudoconvex domain, then the function log(−Λ) is a real analytic, strictly plurisubharmonic exhaustion function for D and thus induces a metric ds2 = n∂2 log(−Λ)(z) dzα ⊗ dzβ z ∂zα∂zβ α,β=1 on D, called the Λ-metric. For an arbitrary C∞-smoothly bounded domain, they computed the boundary asymptotics of Λ and its derivatives up to order 3, in terms of a defining function for the domain. As a consequence it was shown that the Λ-metric is complete on a C∞-smoothly bounded strongly pseudoconvex domain or a C∞-smoothly bounded convex domain. In this thesis, we study the boundary behaviour of the function Λ and its derivatives of all orders near a C2-smooth boundary point of an arbitrary domain. We compute the boundary asymptotics of the Λ-metric on a C∞-smoothly bounded pseudoconvex domain and as a consequence obtain that on a C∞-smoothly bounded strongly pseudoconvex domain, the Λ-metric is comparable to the Kobayashi metric (and hence to the Carath´eodory and the Bergman metrics). Using the boundary asymptotics of Λ and its derivatives, we calculate the holomorphic sectional curvature of the Λ-metric on a C∞-smoothly bounded strongly pseudoconvex domain at points on the inner normals and along the normal directions. The unit ball in Cn is also characterised among all C∞-smoothly bounded strongly convex domains on which the Λ-metric has constant negative holomorphic sectional curvature. Finally we study the stability of the Λ-metric under a C2 perturbation of a C∞-smoothly bounded pseudoconvex domain. (For equation pl refer the abstract pdf file)

We investigate local properties of the Green function of the complement of a compact set E. First we consider the case E ⊂ [0, 1] in the extended complex plane. We extend a result of V. Andrievskii which claims that if the Green function satisfies the Hölder1/2 condition locally at the origin, then the density of E at 0, in terms of logarithmic capacity, is the same as that of the whole interval [0, 1]. We give an integral estimate on the density in terms of the Green function, which also provides a necessary condition for the optimal smoothness. Then we extend the results to the case E ⊂ [−1, 1]. In this case the maximal smoothness of the Green function is H¨older-1 and a similar integral estimate and necessary condition hold as well. In the second part of the paper we consider the case when E is a compact set in Rd , d > 2. We give a Wiener type characterization for the Hölder continuity of the Green function, thus extending a result of L. Carleson and V. Totik. The obtained density condition is necessary, and it is sufficient as well, provided E satisfies the cone condition. It is also shown that the Hölder condition for the Green function at a boundary point can be equivalently stated in terms of the equilibrium measure and the solution to the corresponding Dirichlet problem. The results solve a long standing open problem - raised by Maz’ja in the 1960’s - under the simple cone condition.

This thesis consists of a comprehensive summary and six scientific papers dealing with the boundary behaviour of functions related to parabolic partial differential equations and several complex variables. Paper I concerns solutions to non-linear parabolic equations of linear growth. The main results include a backward Harnack inequality, and the Hölder continuity up to the boundary of quotients of non-negative solutions vanishing on the lateral boundary of an NTA cylinder. It is also shown that the Riesz measure associated with such solutions has the doubling property. Paper II is concerned with solutions to linear degenerate parabolic equations, where the degeneracy is controlled by a weight in the Muckenhoupt class 1+2/n. Two main results are that non-negative solutions which vanish continuously on the lateral boundary of an NTA cylinder satisfy a backward Harnack inequality and that the quotient of two such functions is Hölder continuous up to the boundary. Another result is that the parabolic measure associated to such equations has the doubling property. In Paper III, it is shown that a bounded pseudoconvex domain whose boundary is α-Hölder for each 0<α<1, is hyperconvex. Global estimates of the exhaustion function are given. In Paper IV, it is shown that on the closure of a domain whose boundary locally is the graph of a continuous function, all plurisubharmonic functions with continuous boundary values can be uniformly approximated by smooth plurisubharmonic functions defined in neighbourhoods of the closure of the domain. Paper V studies  Poletsky’s notion of plurisubharmonicity on compact sets. It is shown that a function is plurisubharmonic on a given compact set if, and only if, it can be pointwise approximated by a decreasing sequence of smooth plurisubharmonic functions defined in neighbourhoods of the set. Paper VI introduces the notion of a P-hyperconvex domain. It is shown that in such a domain, both the Dirichlet problem with respect to functions plurisubharmonic on the closure of the domain, and the problem of approximation by smooth plurisubharmoinc functions in neighbourhoods of the closure of the domain have satisfactory answers in terms of plurisubharmonicity on the boundary.