Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I

We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with $L_2$ boundary data. The coefficients $A$ may depend on all variables, but are assumed to be close to coefficients $A_0$ that are independent of the coordinate transversal to the boundary, in the Carleson sense $\|A-A_0\|_C$ defined by Dahlberg. We obtain a number of {\em a priori} estimates and boundary behaviour results under finiteness of $\|A-A_0\|_C$. Our methods yield full characterization of weak solutions, whose gradients have $L_2$ estimates of a non-tangential maximal function or of the square function, via an integral representation acting on the conormal gradient, with a singular operator-valued kernel. Also, the non-tangential maximal function of a weak solution is controlled in $L_2$ by the square function of its gradient. This estimate is new for systems in such generality, and even for real non-symmetric equations in dimension 3 or higher. The existence of a proof {\em a priori} to well-posedness, is also a new fact. As corollaries, we obtain well-posedness of the Dirichlet, Neumann and Dirichlet regularity problems under smallness of $\|A-A_0\|_C$ and well-posedness for $A_0$, improving earlier results for real symmetric equations. Our methods build on an algebraic reduction to a first order system first made for coefficients $A_0$ by the two authors and A. McIntosh in order to use functional calculus related to the Kato conjecture solution, and the main analytic tool for coefficients $A$ is an operational calculus to prove weighted maximal regularity estimates.Comment: This is an extended version of the paper, containing some new material and a road map to proofs on suggestion from the referee

Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems

We prove that the Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half space are well posed in $L_2$ for small complex $L_\infty$ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be independent of the transversal coordinate. We solve the Neumann, Dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order Dirac type operator. We establish quadratic estimates for this Dirac operator, which implies that the associated Hardy projection operators are bounded and depend continuously on the coefficient matrix. We also prove that certain transmission problems for $k$-forms are well posed for small perturbations of block matrices.Comment: Some changes made in the introduction of the pape

On a quadratic estimate related to the Kato conjecture and boundary value problems

We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with $L^2$ boundary data. We develop the application to the Kato conjecture and to a Neumann problem. This quadratic estimate enjoys some equivalent forms in various settings. This gives new results in the functional calculus of Dirac type operators on forms.Comment: Text of the lectures given at the El Escorial 2008 conference. Revised after the suggestions of the referee. Some historical material added. A short proof of the main result added under a further assumption. To appear in the Proceeding

Quadratic estimates and functional calculi of perturbed Dirac operators

We prove quadratic estimates for complex perturbations of Dirac-type operators, and thereby show that such operators have a bounded functional calculus. As an application we show that spectral projections of the Hodge--Dirac operator on compact manifolds depend analytically on $L_\infty$ changes in the metric. We also recover a unified proof of many results in the Calder\'on program, including the Kato square root problem and the boundedness of the Cauchy operator on Lipschitz curves and surfaces.Comment: To appear in Inventiones Mathematicae. Minor final changes added 4/7 200

Hilbert transforms and the Cauchy integral in euclidean space

We generalize the notion of harmonic conjugate functions and Hilbert transforms to higher dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These conjugate functions are in general far from being unique, but under suitable boundary conditions we prove existence and uniqueness of conjugates. The proof also yields invertibility results for a new class of generalized double layer potential operators on Lipschitz surfaces and boundedness of related Hilbert transforms.Comment: Some minor corrections mad

The Impact of Sea State Condition on Airborne Lidar Bathymetry Measurements

Due to a large number of available Airborne Lidar Bathymetry (ALB) survey datasets and scheduled future surveys, there is a growing need from coastal mapping communities to estimate the accuracy of ALB as a function of the survey system and environmental conditions. Knowledge of ALB accuracy can also be used to evaluate the quality of products derived from ALB surveying. This paper presents theoretical and experimental results focused on the relationship between sea surface conditions and the accuracy of ALB measurements. The simulated environmental conditions were defined according to the typical conditions under which successful ALB surveys can be conducted. The theoretical part of the research included simulations, where the ray-path geometry of the laser beam was monitored below the water surface. Wave-tank experiments were conducted to support the simulations. A cross section of the laser beam was monitored underwater using a green laser with and without wind-driven waves. The results of the study show that capillary waves and small gravity waves distort the laser footprint. Because sea-state condition is related to wind at a first-order approximation, it is possible to suggest wind speed thresholds for different ALB survey projects that vary in accuracy requirements. If wind or wave information were collected during an ALB survey, then it is possible to evaluate the change in accuracy of ALB survey due to different sea surface conditions

Market survey for Södra skogsägarna within Vimmerby management district

This thesis is based on a market survey sent to the passive members of Södra skogsägarna within the scope of practice of Vimmerby. A member of Södra skogsägarna is considered passive if the member not has delivered timber to the association during the past five years. The purpose of this thesis was to obtain as much information as possible about the forest owners, to be able to interpret why the member did not choose to deliver their timber to the association. The survey was worked out in close collaboration with Södra skogsägarna to answer this question. The responses indicate that economic returns are important to the individual landowner. The price of timber is most important in a timber business and the majority of forest owners want to have a personal meeting if you can choose how you wish to be contacted by the inspector. A couple of the questions dealt with why they did not choose to sell their timber to the association. Some of the reasons for that were preserving the local industry, better prices at competitors, good contact with the buyer at another corporation, small acreage or that something has gone wrong during harvesting

Continuous gastric saline perfusion elicits cardiovascular responses in freshwater rainbow trout (Oncorhynchus mykiss)

When in seawater, rainbow trout (Oncorhynchus mykiss) drink to avoid dehydration and display stroke volume (SV) mediated elevations in cardiac output (CO) and an increased proportion of CO is diverted to the gastrointestinal tract as compared to when in freshwater. These cardiovascular alterations are associated with distinct reductions in systemic and gastrointestinal vascular resistance (R-Sys and R-GI, respectively). Although increased gastrointestinal blood flow (GBF) is likely essential for osmoregulation in seawater, the sensory functions and mechanisms driving the vascular resistance changes and other associated cardiovascular changes in euryhaline fishes remain poorly understood. Here, we examined whether internal gastrointestinal mechanisms responsive to osmotic changes mediate the cardiovascular changes typically observed in seawater, by comparing the cardiovascular responses of freshwater-acclimated rainbow trout receiving continuous (for 4 days) gastric perfusion with half-strength seawater (1/2 SW, similar to 17 ppt) to control fish (i.e., no perfusion). We show that perfusion with 1/2 SW causes significantly larger increases in CO, SV and GBF, as well as reductions in R-Sys and R-GI, compared with the control, whilst there were no significant differences in blood composition between treatments. Taken together, our data suggest that increased gastrointestinal luminal osmolality is sensed directly in the gut, and at least partly, mediates cardiovascular responses previously observed in SW acclimated rainbow trout. Even though a potential role of mechano-receptor stimulation from gastrointestinal volume loading in eliciting these cardiovascular responses cannot be excluded, our study indicates the presence of internal gastrointestinal milieu-sensing mechanisms that affect cardiovascular responses when environmental salinity changes