Besov regularity of stochastic partial differential equations on bounded Lipschitz domains

This thesis is concerned with the regularity of (semi-)linear second order parabolic stochastic partial differential equations (SPDEs, for short) of Itô type on bounded Lipschitz domains. The so-called adaptivity scale of Besov spaces is used to measure the regularity of the solution with respect to the space variable. It determines the convergence rate of the so-called best m-term wavelet approximation, which is the benchmark for modern adaptive numerical methods based on wavelet bases or frames. The regularity with respect to the time variable is measured in the classical Hölder-norm. The analysis is put into the framework of the analytic approach for SPDEs initiated by Nicolai V. Krylov. Recent results by Kyeong-Hun Kim regarding the spatial weighted Sobolev regularity of the solutions to SPDEs on non-smooth domains (DOI:10.1007/s10959-012-0459-7) are the starting point of the investigations. General embeddings of weighted Sobolev spaces into the classical Sobolev spaces and into the Besov spaces from the adaptivity scale are proven. These embeddings together with a generalization of Kim's results to a class of semi-linear SPDEs yield the desired spatial Besov regularity results. In particular, it is shown that in specific situations the spatial Besov regularity of the solution in the adaptivity scale is generically higher than its classical Sobolev regularity. As it is well-known from approximation theory, this indicates that in many cases adaptive wavelet methods for solving SPDEs should be used instead of uniform alternatives. It is worth noting that the aforementioned embeddings are proven independently of the SPDE context and are relevant also in other mathematical fields. In order to prove space time regularity of the solution, techniques from the analytic approach are combined with results obtained from the semigroup approach of Da Prato/Zabczyk (ISBN:9780521059800). This procedure yields an Lq(Lp)-theory for the heat equation with additive noise on general bounded Lipschitz domains. The integrability parameter q with respect to the time variable can be chosen to be strictly greater than the spatial integrability parameter p. As a consequence, Hölder-Besov regularity of the solution can be established

Spatial Besov Regularity for Stochastic Partial Differential Equations on Lipschitz Domains

We use the scale of Besov spaces B^\alpha_{\tau,\tau}(O), \alpha>0, 1/\tau=\alpha/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains O\subset R^d. The Besov smoothness determines the order of convergence that can be achieved by nonlinear approximation schemes. The proofs are based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions.Comment: 32 pages, 3 figure

Lax-Phillips scattering theory for PT-symmetric \rho-perturbed operators

The S-matrices corresponding to PT-symmetric \rho-perturbed operators are defined and calculated by means of an approach based on an operator-theoretical interpretation of the Lax-Phillips scattering theory

Linear-nonlinear stiffness responses of carbon fiber-reinforced polymer composite materials and structures: a numerical study

The stiffness response or load-deformation/displacement behavior is the most important mechanical behavior that frequently being utilized for validation of the mathematical-physical models representing the mechanical behavior of solid objects in numerical method, compared to actual experimental data. This numerical study aims to investigate the linear-nonlinear stiffness behavior of carbon fiber-reinforced polymer (CFRP) composites at material and structural levels, and its dependency to the sets of individual/group elastic and damage model parameters. In this regard, a validated constitutive damage model, elastic-damage properties as reference data, and simulation process, that account for elastic, yielding, and damage evolution, are considered in the finite element model development process. The linear-nonlinear stiffness responses of four cases are examined, including a unidirectional CFRP composite laminate (material level) under tensile load, and also three multidirectional composite structures under flexural loads. The result indicated a direct dependency of the stiffness response at the material level to the elastic properties. However, the stiffness behavior of the composite structures depends both on the structural configuration, geometry, lay-ups as well as the mechanical properties of the CFRP composite. The value of maximum reaction force and displacement of the composite structures, as well as the nonlinear response of the structures are highly dependent not only to the mechanical properties, but also to the geometry and the configuration of the structures