Arbitrary Boundary Data Research Articles

Spin-foams as semiclassical vertices: Gluing constraints and a hybrid algorithm

Numerical methods in spin-foam models have significantly advanced in the last few years, yet challenges remain in efficiently extracting results for amplitudes with many quantum degrees of freedom. In this paper we sketch a proposal for a ``hybrid algorithm'' that would use both the full quantum amplitude and its asymptotic approximation in the relevant regimes. As a first step towards the algorithm, we derive a new representation of the partition function where each spin-foam vertex possesses its own coherent data, such that it can be individually asymptotically approximated. We do this through the implementation of gluing constraints between vertices, which we study numerically. We further derive an asymptotic expression for the constraints for arbitrary boundary data, including data for which there are no critical points. From this new representation we conjecture an intermediate quasi-geometric spin-foam regime describing a superposition of semi-classical vertices glued in a non-matching way via the gluing constraints.

On the Hilbert problem for semi-linear Beltrami equations

The presented paper is devoted to the study of the well-known Hilbert boundary-value problem for semi-linear Beltrami equations with arbitrary boundary data that are measurable with respect to logarithmic capacity. Namely, we prove here the corresponding results on the existence, regularity, and representation of its nonclassical solutions with a geometric interpretation of boundary values as the angular (along the nontangential paths) limits in comparison with the classical approach in PDE. For this purpose, we apply completely continuous operators by Ahlfors-Bers, first of all to obtain solutions of semi-linear Beltrami equations, generally speaking with no boundary conditions, and then to derive their representation through the solutions of the Vekua-type equations and the so-called generalized analytic functions with sources. Besides, we obtain similar results for nonclassical solutions of the Poincare boundary-value problem on directional derivatives and, in particular, of the Neumann problem with arbitrary measurable data to semi-linear equations of the Poisson type. The obtained results are applied to some problems of mathematical physics describing such phenomena as diffusion with physical and chemical absorption, plasma states, and stationary burning in anisotropic and inhomogeneous media.

Dirichlet problem with measurable data for quasilinear Poisson equations

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions in the unit disk $\mathbb D$ goes to the famous dissertation of Luzin, see e.g. its reprint \cite{L}. His result was formulated in terms of angular limits (along nontangent paths) that are a traditional tool for the research of the boundary behavior in the geometric function theory. Following this way, we proved in \cite{GNR1} Theorem 7 on the solvability of the Dirichlet problem for the Poisson equations $\triangle\, U=G$ with sources in classes $G\in L^p,$ $ p>1$, in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity. There we assumed that the domains satisfy the quasihyperbolic boundary condition by Gehring--Martio, generally speaking, without the known $(A)-$condition by Ladyzhenskaya--Ural'tseva and, in particular, without the outer cone condition that were standard for boundary-value problems in the PDE theory. Note that such Jordan domains cannot be even locally rectifiable. With a view to further development of the theory of boundary value problems for semi-linear equations, the present paper is devoted to the Dirichlet problem with arbitrary measurable (over logarithmic capacity) boundary data for quasilinear Poisson equations in such Jordan domains. For this purpose, it is first constructed completely continuous operators generating nonclassical solutions of the Dirichlet boundary-value problem with arbitrary measurable data for the Poisson equations $\triangle\, U=G$ with the sources $G\in L^p,$ $ p>1$. The latter makes it possible to apply the Leray-Schauder approach to the proof of theorems on the existence of regular nonclassical solutions of the measurable Dirichlet problem for quasilinear Poisson equations of the form $\triangle\, U(z)=H(z)\cdot Q(U(z))$ for multipliers $H\in L^p$ with $ p>1$ and continuous functions $Q: \mathbb R\to\mathbb R$ with $Q(t)/t\to 0$ as $t\to \infty$. As consequences, we give applications to some concrete quasilinear equations of mathematical physics, arising under modelling various phy\-si\-cal processes such as diffusion with absorption, plasma states, stationary burning etc. These results can be also applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

Logarithmic potential and generalized analytic functions

The study of the Dirichlet problem in the unit disk $\mathbb D$ with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin [31]. Later on, the known monograph of Vekua \cite{Ve} has been devoted to boundary-value problems (only with H\"older continuous data) for the generalized analytic functions, i.e., continuous complex valued functions $h(z)$ of the complex variable $z=x+iy$ with generalized first partial derivatives by Sobolev satisfying equations of the form $\partial_{\bar z}h\, +\, ah\, +\ b{\overline h}\, =\, c\, ,$ where it was assumed that the complex valued functions $a,b$ and $c$ belong to the class $L^{p}$ with some $p>2$ in smooth enough domains $D$ in $\mathbb C$. The present paper is a natural continuation of our previous articles on the Riemann, Hilbert, Dirichlet, Poincar\'{e} and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic, and the so-called $A-$harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here, we extend the corresponding results to the generalized analytic functions $h:D\to\mathbb C$ with the sources $g$ : $\partial_{\bar z}h\ =\ g\in L^p$, $p>2\,$, and to generalized harmonic functions $U$ with sources $G$ : $\triangle\, U=G\in L^p$, $p>2\,$. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary-value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary-values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincar\'{e} problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations $\triangle\, U=G$ with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

On boundary-value problems for generalized analytic and harmonic functions

The present paper is a natural continuation of our last articles on the Riemann, Hilbert, Dirichlet, Poincaré, and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic functions and the so-called A-harmonic functions with arbitrary boundary data that are measurable with respect to the logarithmic capacity. Here, we extend the corresponding results to generalized analytic functions h : D→C with sources g : ∂z-h = g ∈ Lp , p > 2, and to generalized harmonic functions U with sources G : ΔU =G ∈Lp , p > 2. Our approach is based on the geometric (functional-theoretic) interpretation of boundary values in comparison with the classical operator approach in PDE. Here, we will establish the corresponding existence theorems for the Poincaré problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations ΔU =G with arbitrary boundary data that are measurable with respect to the logarithmic capacity. A few mixed boundary-value problems are considered as well. These results can be also applied to semilinear equations of mathematical physics in anisotropic and inhomogeneous media.

Boundary value problems for the generalized analytic and harmonic functions

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin. Later on, the known monograph of Vekua has been devoted to boundary value problems (only with H\"older continuous data) for the generalized analytic functions, i.e., continuous complex valued functions $h(z)$ of the complex variable $z=x+iy$ with generalized first partial derivatives by Sobolev satisfying equations of the form $\partial_{\bar z}h\, +\, ah\, +\ bh\, =\, c\, ,$ where $\partial_{\bar z}\ :=\ \frac{1}{2}\left(\ \frac{\partial}{\partial x}\ +\ i\cdot\frac{\partial}{\partial y}\ \right),$ and it was assumed that the complex valued functions $a,b$ and $c$ belong to the class $L^{p}$ with some $p>2$ in the corresponding domains $D\subset \mathbb C$. The present paper is a natural continuation of our articles on the Riemann, Hilbert, Dirichlet, Poincare and, in particular, Neumann boundary value problems for quasiconformal, analytic, harmonic and the so-called $A-$harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here we extend the correspon\-ding results to the generalized analytic functions $h:D\to\mathbb C$ with the sources $g$ : $\partial_{\bar z}h\ =\ g\in L^p$, $p>2\,$, and to generalized harmonic functions $U$ with sources $G$ : $\triangle\, U=G\in L^p$, $p>2\,$. It was also given relevant definitions and necessary references to the mentioned articles and comments on previous results. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincare problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations $\triangle\, U=G$ with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

To the theory of semilinear equations in the plane

In two dimensions, we present a new approach to the study of the semilinear equations of the form div[A(z)∇u] = f(u), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions A(z), whereas its reaction term f(u) is a continuous non-linear function. Assuming that f(t)/t → 0 as t → ∞, we establish a theorem on existence of weak $$ C\left(\overline{D}\right)\cap {W}_{\mathrm{loc}}^{1,2}(D) $$ solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains D without degenerate boundary components. As consequences, we give applications to some concrete model semilinear equations of mathematical physics, arising from modeling processes in anisotropic and inhomogeneous media. With a view to the further development of the theory of boundary-value problems for the semilinear equations, we prove a theorem on the solvability of the Dirichlet problem for the Poisson equation in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity.

To the theory of semilinear equations in the plane

In two dimensions, we present a new approach to the study of the semilinear equations of the form \(\mathrm{div}[ A(z) \nabla u] = f(u)\), the diffusion term of which is the divergence uniform elliptic operator with measurable matrix functions \(A(z)\), whereas its reaction term \(f(u)\) is a continuous non-linear function. Assuming that \(f(t)/t\to 0\) as \(t\to\infty\), we establish a theorem on existence of weak \(C(\overline D)\cap W^{1,2}_{\rm loc}(D)\) solutions of the Dirichlet problem with arbitrary continuous boundary data in any bounded domains \(D\) without degenerate boundary components. As consequences, we give applications to some concrete model semilinear equations of mathematical physics, arising from modeling processes in anisotropic and inhomogeneous media. With a view to the further development of the theory of boundary-value problems for the semilinear equations, we prove a theorem on the solvability of the Dirichlet problem for the Poisson equation in Jordan domains with arbitrary boundary data that are measurable with respect to the logarithmic capacity.

Dirichlet problem for Poisson equations in Jordan domains

First, we study the Dirichlet problem for the Poisson equations \(\triangle u(z) = g(z)\) with \(g\in L^p\), \(p>1\), and continuous boundary data \(\varphi :\partial D\to\mathbb{R}\) in arbitrary Jordan domains \(D\) in \(\mathbb{C}\) and prove the existence of continuous solutions \(u\) of the problem in the class \(W^{2,p}_{\rm loc}\). Moreover, \(u\in W^{1,q}_{\rm loc}\) for some \(q>2\) and \(u\) is locally Hölder continuous. Furthermore, \(u\in C^{1,\alpha}_{\rm loc}\) with \(\alpha = (p-2)/p\) if \(p>2\). Then, on this basis and applying the Leray-Schauder approach, we obtain the similar results for the Dirichlet problem with continuous data in arbitrary Jordan domains to the quasilinear Poisson equations of the form \(\triangle u(z) = h(z)\cdot f(u(z))\) with the same assumptions on \(h\) as for \(g\) above and continuous functions \(f:\mathbb{R}\to\mathbb{R}\), either bounded or with nondecreasing \(|f\,|\) of \( |t\,|\) such that \(f(t)/t \to 0\) as \(t\to\infty\). We also give here applications to mathematical physics that are relevant to problems of diffusion with absorbtion, plasma and combustion. In addition, we consider the Dirichlet problem for the Poisson equations in the unit disk \(\mathbb{D}\subset\mathbb{C}\) with arbitrary boundary data \(\varphi :\partial\mathbb{D}\to\mathbb{R}\) that are measurable with respect to logarithmic capacity. Here we establish the existence of continuous nonclassical solutions \(u\) of the problem in terms of the angular limits in \(\mathbb D\) a.e. on \(\partial\mathbb D\) with respect to logarithmic capacity with the same local properties as above. Finally, we extend these results to almost smooth Jordan domains with qusihyperbolic boundary condition by Gehring-Martio.

Solvability of general boundary-value problems in a half-space for inhomogeneous differential equations with constant coefficients

We present sufficient conditions for the existence of solutions of general boundary-value problems in a half-space for inhomogeneous differential equations with constant coefficients and arbitrary boundary data in the space of tempered distributions.

Solvability of boundary value problems in a half-space for differential equations with constant coefficients in the class of tempered distributions

We present sufficient conditions for existence of solutions to general boundary value problems in a half-space for homogeneous differential equations with constant coefficients and arbitrary boundary data in the space of tempered distributions.

A Signalling Problem for Near-Equilibrium Flows in the Fickett-Majda Model of Combustion

The signalling problem for chemico-acoustic interactions in a simplified model of combustion and detonation is considered. The problem is examined in different limits, depending upon the sensitivity of the reaction rate with the degree of reaction coordinate and with the thermodynamic state. In one limit, the problem is singular, and singular perturbation techniques are used to obtain an approximate solution. In the general case, a combination Riemann function-Laplace transform method is used to obtain exact solutions with arbitrary boundary data. This combined method allows the Riemann method to be applied to certain wave problems with discontinuous data.

On steady plane flows in classical elasticity

A new class of boundary-value problems in mathematical elasticity is proposed, wherein the medium flows steadily relative to a non-embedded surface over which tractions or velocities are prescribed. Such flows are seen in metal forming operations where purely elastic streams enter and leave the working zone. The deformations are assumed here to be plane and isochoric. A general solution is formulated in terms of two complex potentials. Residual stress is accounted for in detail and a uniqueness theorem is proved. Some simple flows are examined, but it remains to develop a systematic procedure for matching the general solution to arbitrary boundary data.