Dirichlet Problem for Poisson and Bi-Poisson Equations in Clifford Analysis

Here, first we shall set together various basic statements on the periodicity of the solutions of first order linear difference equations, then we shall define various discrete functions which are in a sense connected with the concept of periodicity. Finally, we will formulate a decomposition theorem for the solutions of first order linear difference equations with periodic coefficients.

Numerous papers deal with asymptotic methods for boundary value problems in domains depending on a small parameter in a complicated way (perforated domains, partially perforated domains, framework structures, thin domains); e.g., see [1–9]. Boundary value problems in thick singularly degenerating junctions (the number of components of such junctions grows infinitely as the perturbation parameter tends to zero) have specific difficulties and require a separate consideration. It was shown in [10] that boundary value problems in thick junctions lose their coercivity under the passage to the limit, which substantially complicates asymptotic studies. The papers [11, 12] were the first in this direction. In [13–19], a classification of thick junctions was given and rigorous mathematical methods were developed for analyzing the main boundary value problems of mathematical physics in thick singularly degenerating junctions of various types. The study of boundary value problems in thick junctions is focused on the asymptotic behavior of their solutions as e → 0, i.e., as the number of thin attached domains grows infinitely and their thickness tends to zero. In the present paper, we consider a thick junction whose thin attached cylinders are divided into two levels depending on the boundary conditions posed on the lateral surface of these cylinders (inhomogeneous Neumann boundary conditions and homogeneous Dirichlet conditions). In addition, thin cylinders of each level e-periodically alternate along the junction zone. Such thick junctions will be called thick two-level junctions. A problem on a thick plane two-level junction was considered for the first time in [20], where the asymptotic behavior of eigenvalues and eigenfunctions of a spectral boundary value problem was analyzed. Other boundary value problems in plane two-level junctions were considered in [21–23]. Asymptotic analysis of boundary value problems with a periodic change of boundary conditions (Neumann and Dirichlet conditions) on the boundary of smooth unperturbed domains was carried out in [24–26], where it was shown that the first term of the asymptotics is mainly the solution of the corresponding boundary value problem with the Dirichlet conditions. The qualitatively new result of the present paper implies that the first term of the asymptotics is a vector function whose components are solutions of two independent boundary value problems (one in the junction body and another in a domain filled with thin cylinders in the limit) with homogeneous Dirichlet conditions in the junction zone. Note also that the second problem is a problem for a second-order ordinary differential equation with a new right-hand side that “remembers” the inhomogeneity in the Neumann boundary conditions of the original problem and the specific “packing density” of thin cylinders.

We introduce a generalized Cauchy integral formula for special classes of functions, holomorphic in several complex variables. These functions also satisfy a generalized Cachy-Riemann equation. The integral formula is then applied to demonstrate that these classes of holomorphic functions are Fréchet modules over complex Clifford algebras, and to provide maximal domains of convergence for Taylor expansions, with homogeneous terms grouped together, of real analytic solutions to Maxwell's equations in vacuo.

We treat the problem of determining the magnetic sources that have been shown to come into existence when an electric charge density abruptly starts an arbitrary motion and hence an infinite speed of light becomes necessary for the satisfaction of the Lorenz condition for the scalar and vector potentials. The change that the Poynting theorem equation undergoes when an infinite speed of light constraint is enforced is the basis for the development and also use is made of the fact that the required magnetic and electric charge density functions share the same support and the same velocity in their motion. Reference to Green's functions in the source region is made to obtain the solution for the magnetic sources. The magnetic sources are computed to be in the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m=Z\rho$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\vec{M}=Z\vec{J}$</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$Z$</tex> is the wave impedance of the lossless, simple medium which can be vacuum. Also, the solution of Maxwell's equations incorporating the magnetic charge and current densities is given in terms of potentials with an infinite speed of propagation for the waves emitted by the charge functions. The magnetic scalar potential is a summation in the form of an integral of monopole moments. For <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t &gt; 0$</tex> even though the resulting differential equations involve biharmonic operators, particular solutions of these equations are essentially solutions of Poisson's equations. When <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t &gt; 0$</tex> the same Lorenz condition for the case with no magnetic sources appears also in this case with magnetic sources. The solutions for the potentials at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t=0$</tex> where the Lorenz condition fails are obtained using the Helmholtz theorem.

Blow-up and regularization rates, loss and recovery of boundary conditions for the superquadratic viscous Hamilton-Jacobi equation

An irregular grid for the numerical solution of linear elliptic partial differential equations

Hydrostatic Stokes equations with non-smooth data for mixed boundary conditions

Let us consider the resonant boundary value problem $$ \begin{align} - &u''(x) - u(x) + g(u(x)) = h(x), \quad x \in [0,\pi], \\ &u(0) = u(\pi) = 0, \end{align} $$ where $ g: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous and $ T$-periodic function with zero mean value, not identically zero, and $ h \in C[0,\pi].$ If each $ h \in C[0,\pi] $ is written as $ h(x) = a\sin x + \tilde{h}(x), $ where $ a \in \mathbb{R} $ and $\int_{0}^{\pi} \tilde{h}(x) \sin x \ dx = 0,$ then, it is shown that for each $ \tilde{h},$ there are real numbers $ a_{1}(\tilde{h}) < 0 < a_{2}(\tilde{h})$ (which depend continuously on $ \tilde{h}$), such that there is solution if and only if $ a\in [a_{1}(\tilde{h}),a_{2}(\tilde{h})].$ In relation to the multiplicity, it is proved that the number of solutions increases to infinity as $ a $ goes to zero. The proof combines different tools such as Liapunov-Schmidt reduction and upper-lower solutions notions, together with a careful analysis of the connected subsets of the solution set of the auxiliary equation in the alternative method, as well as a detailed study of the oscillatory behavior of some integrals associated to the bifurcation equation of the previous problem.

We describe a systematic way of producing fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases which contain an emergent fermion. Our approach to fermion condensation can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions, and condensing pairs of physical and emergent fermions. There are two distinct types of objects in fermionic theories, which we call “m-type” and “q-type” particles. The endomorphism algebras of q-type particles are complex Clifford algebras, and they have no analogues in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations in fermionic topological phases. We then prove a series of results relating data in condensed theories to data in their parent theories; for example, if C is a modular tensor category containing a fermion, then the tube category of the condensed theory satisfies Tube(C/ψ) ≅ C × C/ψ. We also study how modular transformations, fusion rules, and coherence relations are modified in the fermionic setting, prove a fermionic version of the Verlinde dimension formula, construct a commuting projector lattice Hamiltonian for fermionic theories, and write down a fermionic version of the Turaev-Viro-Barrett-Westbury state sum.

In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras -- subspaces of fixed grades and subspaces determined by the reversion and the grade involution. We present groups of elements that define such inner automorphisms and study their properties. Some of these Lie groups can be interpreted as generalizations of Clifford, Lipschitz, and spin groups. We study the corresponding Lie algebras. Some of the results can be reformulated for the case of more general algebras -- graded central simple algebras or graded central simple algebras with involution.

We consider a class of elliptic-hyperbolic degenerate equations g(u) - Delta b(u) + div phi (u) = f with Dirichlet homogeneous boundary conditions and a class of elliptic-parabolic-hyperbolic degenerate equations g(u)(t) - Delta b(u) + div phi (u) = f with homogeneous Dirichlet conditions and initial conditions. Existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function phi satisfying rather general conditions. Comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for phi continuous.

Analytic solutions for unsteady flows over a superhydrophobic surface

The hydrodynamic impact problem is investigated within the framework of potential-flow theory. The vertical load acting on the rigid body is derived based on either momentum or energy conservation, and using the concept of added mass together with a homogeneous Dirichlet condition for the potential on the free surface as usually done to model an impact problem. It is demonstrated that the use of this simplified dynamic free-surface condition, instead of the fully nonlinear one, has a direct influence on the computation of the loads. In particular, the equivalence of momentum and energy analysis is in general not recovered. The situation is then highlighted by performing an asymptotic analysis of the two-dimensional blunt-body asymmetric impact problem. The asymptotic solution is given explicitly and validated through comparisons with experimental results. The energy distribution is then studied. It is shown that the contradiction between momentum and energy analysis can be removed, provided that the flux of energy through the jets is taken into account in the energy balance. If the simplified free-surface condition is indeed valid in the far-field, nonlinear terms must be retained near the body, in the spray-root domains. To leading order, the energy distribution during the gravity-free inertia stage does not depend on the blunt-body shape. The general analysis based on momentum or energy conservation suggests that this result also applies for arbitrary body shape as soon as a homogeneous Dirichlet condition can be applied as a dynamical free-surface boundary condition. In this case, and for a constant vertical impact velocity, half the work performed by the body would seem to be transferred to the fluid as kinetic energy within the spray.

Hermitean Clifford analysis focusses on h–monogenic functions taking values in a complex Clifford algebra or in a complex spinor space. Monogenicity is expressed here by means of two complex mutually adjoint Dirac operators, which are invariant under the action of a Clifford realisation of the unitary group. In this contribution we present a deeper insight in the transition from the orthogonal setting to the Hermitean one. Starting from the orthogonal Clifford setting, by simply introducing a so-called complex structure J ∈ SO(2n;\({\mathbb{R}}\)), the fundamental elements of the Hermitean setting arise in a quite natural way. Indeed, the corresponding projection operators 1/2 (1 ± iJ) project the initial basis (eα, α = 1, . . . , 2n) onto the Witt basis and moreover give rise to a direct sum decomposition of Open image in new window into two components, where the SO(2n;\({\mathbb{R}}\))-elements leaving those two subspaces invariant, commute with the complex structure J. They generate a subgroup which is doubly covered by a subgroup of Spin(2n;\({\mathbb{R}}\)), denoted Spin J (2n;\({\mathbb{R}}\)), being isomorphic with the unitary group U(n;\({\mathbb{C}}\)). Finally the two Hermitean Dirac operators are shown to originate as generalized gradients when projecting the gradient on the invariant subspaces mentioned, which actually implies their invariance under the action of Spin J (2n;\({\mathbb{R}}\)). The eventual goal is to extend the complex structure J to the whole Clifford algebra Open image in new window , in order to conceptually unravel the true meaning of Hermitean monogenicity and its connections to orthogonal monogenicity.

Abstract