<p>In this thesis we study approximation of negative plurisubharmonic functions by functions defined on strictly larger domains. We show that, under certain conditions, every function <i>u</i> that is defined on a bounded hyperconvex domain Ω in C<i>n</i><i> </i>and has essentially boundary values zero and bounded Monge-Ampère mass, can be approximated by an increasing sequence of functions {<i>u</i><i>j</i>} that are defined on strictly larger domains, has boundary values zero and bounded Monge-Ampère mass. We also generalize this and show that, under the same conditions, the approximation property is true if the function u has essentially boundary values G, where G is a plurisubharmonic functions with certain properties. To show these approximation theorems we use subextension. We show that if Ω_1 and Ω_2 are hyperconvex domains in C<i>n</i> and if u is a plurisubharmonic function on Ω_1 with given boundary values and with bounded Monge-Ampère mass, then we can find a plurisubharmonic function û defined on Ω_2, with given boundary values, such that û <= u on Ω and with control over the Monge-Ampère mass of û.</p>

In this thesis we study approximation of negative plurisubharmonic functions by functions defined on strictly larger domains. We show that, under certain conditions, every function u that is defined on a bounded hyperconvex domain Ω in Cn and has essentially boundary values zero and bounded Monge-Ampère mass, can be approximated by an increasing sequence of functions {uj} that are defined on strictly larger domains, has boundary values zero and bounded Monge-Ampère mass. We also generalize this and show that, under the same conditions, the approximation property is true if the function u has essentially boundary values G, where G is a plurisubharmonic functions with certain properties. To show these approximation theorems we use subextension. We show that if Ω_1 and Ω_2 are hyperconvex domains in Cn and if u is a plurisubharmonic function on Ω_1 with given boundary values and with bounded Monge-Ampère mass, then we can find a plurisubharmonic function û defined on Ω_2, with given boundary values, such that û &lt;= u on Ω and with control over the Monge-Ampère mass of û.

In this thesis, we study two different kinds of approximation of plurisubharmonic functions. The first one is a Mergelyan type approximation for plurisubharmonic functions. That is, we study which domains in C^n have the property that every continuous plurisubharmonic function can be uniformly approximated with continuous and plurisubharmonic functions defined on neighborhoods of the domain. We will improve a result by Fornaess and Wiegerinck and show that domains with C^0-boundary have this property. We will also use the notion of plurisubharmonic functions on compact sets when trying to characterize those continuous and plurisubharmonic functions that can be approximated from outside. Here a new kind of convexity of a domain comes in handy, namely those domains in C^n that have a negative exhaustion function that is plurisubharmonic on the closure. For these domains, we prove that it is enough to look at the boundary values of a plurisubharmonic function to know whether it can be approximated from outside. The second type of approximation is the following: we want to approximate functions u that are defined on bounded hyperconvex domains Omega in C^n and have essentially boundary values zero and bounded Monge-Ampère mass, with increasing sequences of certain functions u_j that are defined on strictly larger domains. We show that for certain conditions on Omega, this is always possible. We also generalize this to functions with given boundary values. The main tool in the proofs concerning this second approximation is subextension of plurisubharmonic functions.

In this thesis we focus on Dirichlet's problem for the complex Monge-Ampère equation. That is, for a given non-negative Radon measure µ we are interested in the conditions under which there exists a plurisubharmonic function u such that (ddcu)n=µ, where (ddc)n is the complex Monge-Ampère operator. If this function u exists, then can it be chosen with given boundary values? Is this solution uniquely determined within a given class of functions?

This thesis consists of three papers concerning problems related to plurisubharmonic functions on bounded hyperconvex domains, in particular boundary values of such functions. The papers summarized in this thesis are:* Paper I Urban Cegrell and Berit Kemppe, Monge-Ampère boundary measures, Ann. Polon. Math. 96 (2009), 175-196.* Paper II Berit Kemppe, An ordering of measures induced by plurisubharmonic functions, manuscript (2009).* Paper III Berit Kemppe, On boundary values of plurisubharmonic functions, manuscript (2009).In the first paper we study a procedure for sweeping out Monge-Ampère measures to the boundary of the domain. The boundary measures thus obtained generalize measures studied by Demailly. A number of properties of the boundary measures are proved, and we describe how boundary values of bounded plurisubharmonic functions can be associated to the boundary measures.In the second paper, we study an ordering of measures induced by plurisubharmonic functions. This ordering arises naturally in connection with problems related to negative plurisubharmonic functions. We study maximality with respect to the ordering and a related notion of minimality for certain plurisubharmonic functions. The ordering is then applied to problems of weak*-convergence of measures, in particular Monge-Ampère measures.In the third paper we continue the work on boundary values in a more general setting than in Paper I. We approximate measures living on the boundary with measures on the interior of the domain, and present conditions on the approximation which makes the procedure suitable for defining boundary values of certain plurisubharmonic functions.

<p>In the theory of holomorphic functions of one complex variable it is often useful to study subharmonic functions. The subharmonic can be described using the Laplace operator. When one studies holomorphic functions of several complex variables one should study the plurisubharmonic functions instead. Here the complex Monge--Ampère operator has a role similar to that of the Laplace operator in the theory of subharmonic functions. The complex Monge--Ampère operator is nonlinear and therefore it is not as well understood as the Laplace operator. We consider two types of boundary value problems for the complex Monge--Ampere equation in certain pseudoconvex domains. In this thesis the right-hand side in the Monge--Ampère equation will always be smooth, strictly positive and meet a monotonicity condition. The first type of boundary value problem we consider is a Dirichlet problem where we look for plurisubharmonic solutions which are zero on the boundary of the domain. We show that this problem has a unique smooth solution if the domain has a smooth bounded plurisubharmonic exhaustion function which is globally Lipschitz and has Monge--Ampère mass larger than one everywhere. We obtain some results on which domains have such a bounded exhaustion function. The second type of boundary value problem we consider is a boundary blow-up problem where we look for plurisubharmonic solutions which tend to infinity at the boundary of the domain. Here we also assume that the right-hand side in the Monge--Ampère equation satisfies a growth condition. We study this problem in strongly pseudoconvex domains with smooth boundary and show that it has solutions which are Hölder continuous with arbitrary Hölder exponent α, 0 ≤ α < 1. We also show a uniqueness result. A result on the growth of the solutions is also proved. This result is used to describe the boundary behavior of the Bergman kernel.</p>

In the theory of holomorphic functions of one complex variable it is often useful to study subharmonic functions. The subharmonic can be described using the Laplace operator. When one studies holomorphic functions of several complex variables one should study the plurisubharmonic functions instead. Here the complex Monge--Ampère operator has a role similar to that of the Laplace operator in the theory of subharmonic functions. The complex Monge--Ampère operator is nonlinear and therefore it is not as well understood as the Laplace operator. We consider two types of boundary value problems for the complex Monge--Ampere equation in certain pseudoconvex domains. In this thesis the right-hand side in the Monge--Ampère equation will always be smooth, strictly positive and meet a monotonicity condition. The first type of boundary value problem we consider is a Dirichlet problem where we look for plurisubharmonic solutions which are zero on the boundary of the domain. We show that this problem has a unique smooth solution if the domain has a smooth bounded plurisubharmonic exhaustion function which is globally Lipschitz and has Monge--Ampère mass larger than one everywhere. We obtain some results on which domains have such a bounded exhaustion function. The second type of boundary value problem we consider is a boundary blow-up problem where we look for plurisubharmonic solutions which tend to infinity at the boundary of the domain. Here we also assume that the right-hand side in the Monge--Ampère equation satisfies a growth condition. We study this problem in strongly pseudoconvex domains with smooth boundary and show that it has solutions which are Hölder continuous with arbitrary Hölder exponent α, 0 ≤ α &lt; 1. We also show a uniqueness result. A result on the growth of the solutions is also proved. This result is used to describe the boundary behavior of the Bergman kernel.

Cette thèse est consacrée à l'étude de la régularité des solutions des équations de Monge-Ampère complexes ainsi que des équations hessiennes complexes dans un domaine borné de Cn. Dans le premier chapitre, on donne des rappels sur la théorie du pluripotentiel. Dans le deuxième chapitre, on étudie le module de continuité des solutions du problème de Dirichlet pour les équations de Monge-Ampère lorsque le second membre est une mesure à densité continue par rapport à la mesure de Lebesgue dans un domaine strictement hyperconvexe lipschitzien. Dans le troisième chapitre, on prouve la continuité hölderienne des solutions de ce problème pour certaines mesures générales. Dans le quatrième chapitre, on considère le problème de Dirichlet pour les équations hessiennes complexes plus générales où le second membre dépend de la fonction inconnue. On donne une estimation précise du module de continuité de la solution lorsque la densité est continue. De plus, si la densité est dans Lp , on démontre que la solution est Hölder-continue jusqu'au bord. / In this thesis we study the regularity of solutions to the Dirichlet problem for complex Monge-Ampère equations and also for complex Hessian equations in a bounded domain of Cn. In the first chapter, we give basic facts in pluripotential theory. In the second chapter, we study the modulus of continuity of solutions to the Dirichlet problem for complex Monge-Ampère equations when the right hand side is a measure with continuous density with respect to the Lebesgue measure in a bounded strongly hyperconvex Lipschitz domain. In the third chapter, we prove the Hölder continuity of solutions to this problem for some general measures. In the fourth chapter, we consider the Dirichlet problem for complex Hessian equations when the right hand side depends on the unknown function. We give a sharp estimate of the modulus of continuity of the solution as the density is continuous. Moreover, for the case of Lp-density we demonstrate that the solution is Hölder continuous up to the boundary.