Gauss-Green Theorem Research Articles

The Gauss-Green theorem for bounded vectorfields with divergence measure on sets of finite perimeter

The Gauss-Green theorem for bounded vectorfields with divergence measure on sets of finite perimeter

Capacity and the Corresponding Heat Semigroup Characterization from Dunkl-Bounded Variation

In this paper, we study some important basic properties of Dunkl-bounded variation functions. In particular, we derive a way of approximating Dunkl-bounded variation functions by smooth functions and establish a version of the Gauss–Green Theorem. We also establish the Dunkl BV capacity and investigate some measure theoretic properties, moreover, we show that the Dunkl BV capacity and the Hausdorff measure of codimension one have the same null sets. Finally, we develop the characterization of a heat semigroup of the Dunkl-bounded variation space, thereby giving its relation to the functions of Dunkl-bounded variation.

COMMENTS ON WASHEK PFEFFER’S CONTRIBUTIONS TO INTEGRATION THEORY

Hereunder, I describe some contributions of Washek F. Pfeffer and his collaborators, to integration theory, starting from his remarkable paper “The Gauss-Green Theorem,” published in 1991.

BV‐packing integral in Rn

AbstractWe introduce new integrals (called packing and integrals) which combine advantages of integrals developed by Pfeffer [17], Malý [12], Kuncová and Malý [10] and Malý and Pfeffer [13]. We prove Gauss–Green theorem in generality of the new integrals and provide comparison with the integrals mentioned above and some others (like by Ball and Preiss [2]).

The Gauss–Green theorem in stratified groups

We lay the foundations for a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups. Such vector fields form a new family of function spaces, which generalize in a sense the BV fields. They provide the most general setting to establish Gauss–Green formulas for vector fields of low regularity on sets of finite perimeter. We show several properties of divergence-measure fields in stratified groups, ultimately achieving the related Gauss–Green theorem.

Kinematic wave solutions for dam-break floods in non-uniform valleys

In non-uniform valleys, dam-break flood waves can be significantly affected by downstream variations in river width, slope, and roughness. To model these effects, we derive new analytical solutions to the kinematic wave equation, applicable to rating curves in the power law form and hydrographs of generic shape as long as they produce a single shock at the wave front. New results are first obtained for uniform channels, using the Gauss-Green theorem applied to characteristic-bounded regions of the (x,t) plane. The results are then extended to non-uniform valleys, using a change of variable that homogenizes river properties by rescaling the distance coordinate. The solutions are illustrated and validated for idealized cases, then applied to three historical dam failures: the 2008 breaching failure of the Tangjiashan landslide dam, the 1976 piping failure of Teton Dam, and the 1959 sudden failure of Malpasset Dam. In spite of the much reduced computational cost and data requirements, the results agree well with the field data and with more elaborate simulations. They also clarify how both river and hydrograph properties affect flood propagation and attenuation.

Anzellotti's pairing theory and the Gauss–Green theorem

In this paper we obtain a very general Gauss–Green formula for weakly differentiable functions and sets of finite perimeter. This result is obtained by revisiting Anzellotti's pairing theory and by characterizing the measure pairing (A,Du) when A is a bounded divergence measure vector field and u is a bounded function of bounded variation.

Summability on non-rectifiable Jordan curves

Abstract For a given parameterization of a Jordan curve, we define the notion of summability or classes of measurable functions on a contour where a new integral is introduced. It is shown that natural functional spaces defining summability for non-rectifiable Jordan curves are the Lebesgue spaces with the weighted norm. For non-rectifiable Jordan curves where an integral was previously defined for continuous (Hölder) functions [Y. Guseynov, Integrable boundaries and fractals for Hölder classes; the Gauss–Green theorem, Calc. Var. Partial Differential Equations 55 2016, 4, Article ID 103], a weight function is constructed which, in general, is not summable by parameter, and a weighted functional space (summability) is defined where the new integral exists.

The prescribed mean curvature equation in weakly regular domains

We show that the characterization of existence and uniqueness up to vertical translations of solutions to the prescribed mean curvature equation, originally proved by Giusti in the smooth case, holds true for domains satisfying very mild regularity assumptions. Our results apply in particular to the non-parametric solutions of the capillary problem for perfectly wetting fluids in zero gravity. Among the essential tools used in the proofs, we mention a \textit{generalized Gauss-Green theorem} based on the construction of the weak normal trace of a vector field with bounded divergence, in the spirit of classical results due to Anzellotti, and a \textit{weak Young's law} for $(\Lambda,r_{0})$-minimizers of the perimeter.

Integrable boundaries and fractals for Hölder classes; the Gauss–Green theorem

We define a surface integral over the boundary of open bounded sets in \( R^d,\ d\ge {}2\), and provide a criterion that completely describes all the boundaries where this integral exists for the \( (d-1\))-differential forms from Holder classes. The surface integral and the Gauss–Green formula are used in many areas of mathematics and physics, and we give the proof for the Gauss–Green formula under the same condition as the existence of the newly defined integral. This integration process substantially extends the class of boundaries where surface integrals were previously defined for Holder differential forms and includes highly irregular boundaries like non-rectifiable Jordan curves, fractals, sets of finite perimeter boundaries and flat chains. In all known generalizations, to prove the existence of the surface or contour integral for a differential form defined on the boundary, the differential of the form or the differential of its Whitney extension has to be summable in the region. As we show, this significant restriction is superfluous in the new integral definition.

The gauss–green theorem in clifford analysis and its applications

In this article, we establish the Gauss–Green type theorems for Clifford-valued functions in Clifford analysis. The boundary conditions in theorems obtained are very general by using the geometric measure theoretic method. The Cauchy–Pompeiu formula for Clifford-valued functions under the weak condition will be derived as their simple application. Furthermore, Cauchy formula for monogenic functions under the weak condition is derived directly from the Cauchy-Pompeiu formula.

Non-absolutely convergent integrals with respect to distributions

We define an integral of a function with respect to a distribution. In case that the underlying distribution is just the Lebesgue measure, the definition leads to a new non-absolutely convergent integral which is wider than the Denjoy–Perron integral. We present a version of the Gauss–Green theorem where the new integral is used for both interior and boundary terms. As a by-product, we characterize the predual Sobolev space \(W^{-1,1}\).

A conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed-sphere grid

A conservative multi-tracer transport algorithm on the cubed-sphere based on the semi-Lagrangian approach (CSLAM) has been developed. The scheme relies on backward trajectories and the resulting upstream cells (polygons) are approximated with great-circle arcs. Biquadratic polynomial functions are used for approximating the density distribution in the cubed-sphere grid cells. The upstream surface integrals associated with the conservative semi-Lagrangian scheme are computed as line-integrals by employing the Gauss–Green theorem. The line-integrals are evaluated using a combination of exact integrals and high-order Gaussian quadrature. The upstream cell (trajectories) information and computation of weights of integrals can be reused for each additional tracer. The CSLAM scheme is extensively tested with various standard benchmark test cases of solid-body rotation and deformational flow in both Cartesian and spherical geometry, and the results are compared with those of other published schemes. The CSLAM scheme is accurate, robust, and moreover, the edges and vertices of the cubed-sphere (discontinuities) do not affect the overall accuracy of the scheme. The CSLAM scheme exhibits excellent convergence properties and has an option for enforcing monotonicity. The advantages of introducing cross-terms in the fully two-dimensional biquadratic density distribution functions are also examined in the context of Cartesian as well as the cubed-sphere grid which has six local sub-domains with discontinuous edges and corners.

On Gauss–Green theorem and boundaries of a class of Hölder domains

The purpose of this paper is to show that, if α > 1 3 and ɛ > 0 , the boundary of an α-Hölder domain is a 1 α + ɛ geometric rough path; and as a direct application, we extend the classical Green–Gauss' formula to this class of fractal domains.

THE GAUSS–GREEN THEOREM IN THE CONTEXT OF LEBESGUE INTEGRATION

In the context of Lebesgue integration the Gauss–Green theorem is proved for bounded vector fields with substantial sets of singularities with respect to continuity and differentiability. The resulting integration by parts is applied to removable sets for the Cauchy–Riemann, Laplace, and minimal surface equations. A simple connection between the Gauss–Green theorem and distributional divergence is established.

A mesoscopic theory of damage and fracture in heterogeneous materials

Deformation patterns in solids are often characterized by self-similarity at the mesolevel. The framework for the mechanics of heterogeneous solids, deformable over fractal subsets, is briefly outlined. Mechanical quantities with noninteger physical dimensions are considered, i.e., the fractal stress [σ ∗] and the fractal strain [ε ∗] . By means of the local fractional calculus, the static and kinematic equations are obtained. The extension of the Gauss–Green Theorem to fractional operators permits to demonstrate the Principle of Virtual Work for fractal media. From the definition of the fractal elastic potential φ ∗ , the fractal linear elastic relation is derived. Beyond the elastic limit, peculiar mechanisms of energy dissipation come into play, providing the softening behaviour characterized by the fractal fracture energy G F ∗ . The entire process of deformation in heterogeneous bodies can thus be described by the fractal theory. In terms of the fractal quantities it is possible to define a scale-independent cohesive law which represents a true material property. It is also possible to calculate the size-dependence of the nominal quantities and, in particular, the scaling of the critical displacement w c, which explains the increasing tail of the cohesive law with specimen size, and that of the critical strain ε c, which explains the brittleness increase with specimen size.

The Gauss–Green theorem and removable sets for PDEs in divergence form

Applying a very general Gauss–Green theorem established for the generalized Riemann integral, we obtain simple proofs of new results about removable sets of singularities for the Laplace and minimal surface equations. We treat simultaneously singularities with respect to differentiability and continuity.

Extended Divergence-Measure Fields and the Euler Equations for Gas Dynamics

A class of extended vector fields, called extended divergence-measure fields, is analyzed. These fields include vector fields in L p and vector-valued Radon measures, whose divergences are Radon measures. Such extended vector fields naturally arise in the study of the behavior of entropy solutions of the Euler equations for gas dynamics and other nonlinear systems of conservation laws. A new notion of normal traces over Lipschitz deformable surfaces is developed under which a generalized Gauss-Green theorem is established even for these extended fields. An explicit formula is obtained to calculate the normal traces over any Lipschitz deformable surface, suitable for applications, by using the neighborhood information of the fields near the surface and the level set function of the Lipschitz deformation surfaces. As an application, we prove the uniqueness and stability of Riemann solutions that may contain vacuum in the class of entropy solutions of the Euler equations for gas dynamics.

Static–kinematic duality and the principle of virtual work in the mechanics of fractal media

The framework for the mechanics of solids, deformable over fractal subsets, is outlined. While displacements and total energy maintain their canonical physical dimensions, renormalization group theory permits to define anomalous mechanical quantities with fractal dimensions, i.e., the fractal stress [ σ *] and the fractal strain [ ε *]. A fundamental relation among the dimensions of these quantities and the Hausdorff dimension of the deformable subset is obtained. New mathematical operators are introduced to handle these quantities. In particular, classical fractional calculus fails to this purpose, whereas the recently proposed local fractional operators appear particularly suitable. The static and kinematic equations for fractal bodies are obtained, and the duality principle is shown to hold. Finally, an extension of the Gauss–Green theorem to fractional operators is proposed, which permits to demonstrate the Principle of Virtual Work for fractal media.

On the theory of divergence-measure fields and its applications

Divergence-measure fields are extended vector fields, including vector fields inL p and vector-valued Radon measures, whose divergences are Radon measures. Such fields arise naturally in the study of entropy solutions of nonlinear conservation laws and other areas. In this paper, a theory of divergence-measure fields is presented and analyzed, in which normal traces, a generalized Gauss-Green theorem, and product rules, among others, are established. Some applications of this theory to several nonlinear problems in conservation laws and related areas are discussed. In particular, with the aid of this theory, we prove the stability of Riemann solutions, which may contain rarefaction waves, contact discontinuities, and/or vacuum states, in the class of entropy solutions of the Euler equations for gas dynamics.