The Poincare Inequality on Smooth and Bounded Domains in R^d

In this thesis we study for which domain types the Poincare inequality holds for all functions having continuous first derivative. We first consider the classical Poincare inequality, which we prove holds for a very large class of open sets in R^d. We then constructively prove that bounded, open, and connected domains in R^d,...

On the trace embedding and its applications to evolution equations

In this paper, we consider traces at initial times for functions with mixed time-space smoothness. Such results are often needed in the theory of evolution equations. Our result extends and unifies many previous results. Our main improvement is that we can allow general interpolation couples. The abstract results are applied to regularity...

Multiple Markov properties for fractional parabolic SPDEs

Spatiotemporal stochastic processes have applications in various fields, but they can be difficult to numerically approximate in a reasonable time, in particular, in the context of statistical inference for large datasets. <br/>Recently, a new approach for efficient spatiotemporal statistical modeling has been proposed, where the space-time...

Stability properties of stochastic maximal Lp-regularity

In this paper we consider L^p-regularity estimates for solutions to stochastic evolution equations, which is called stochastic maximal L^p-regularity. Our aim is to find a theory which is analogously to Dore's theory for deterministic evolution equations. He has shown that maximal L^p-regularity is independent of...

Stochastic maximal regularity for rough time-dependent problems

We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only progressively measurable in the time variable. For 2m-th order systems with VMO regularity in space, we obtain...

Spectral element method for parabolic interface problems

In this paper, an h∕p spectral element method with least-square formulation for parabolic interface problem will be presented. The regularity result of the parabolic interface problem is proven for non-homogeneous interface data. The differentiability estimates and the main stability estimate theorem, using non-conforming spectral element...

Sobolev Algebras Through a 'Carré du Champ' Identity

We consider abstract Sobolev spaces of Bessel-type associated with an operator. In this work, we pursue the study of algebra properties of such functional spaces through the corresponding semigroup. As a follow-up to our previous work, we show that by making use of the property of a 'carré du champ' identity, this algebra property holds in a...

Complex interpolation with Dirichlet boundary conditions on the half line

We prove results on complex interpolation of vector-valued Sobolev spaces over the half-line with Dirichlet boundary condition. Motivated by applications in evolution equations, the results are presented for Banach space-valued Sobolev spaces with a power weight. The proof is based on recent results on pointwise multipliers in Bessel...

The Lp boundedness of the Riesz Transform

In this thesis we study theorems that will yield easier expressions for the Sobolev space W2,p.

The clamped elastic grid, a fourth order equation on a domain with corner