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We introduce new techniques to study the differential complexes associated to tube structures on M × T m of corank m, in which M is a compact manifold and T m is the m-torus. By systematically employing partial Fourier series, for complex tube structures, we completely characterize global solvability, in a given degree, in terms of a weak form of hypoellipticity, thus generalizing existing results and providing a broad answer to an open problem proposed by Hounie and Zugliani (Math Ann 369(3–4):1177–1209, 2017). We also obtain new results on the finiteness of the cohomology spaces in intermediate degrees. In the case of real tube structures, we extend an isomorphism for the cohomology spaces originally obtained by Dattori da Silva and Meziani (Math Nachr 289(17–18):2147–2158, 2016) in the case M = T n . Moreover, we establish necessary and sufficient conditions for the differential operator to have closed range in the first degree. [ABSTRACT FROM AUTHOR]

We present a method for the numerical approximation of distributed optimal control problems constrained by parabolic partial differential equations. We complement the first-order optimality condition by a recently developed space-time variational formulation of parabolic equations which is coercive in the energy norm, and a Lagrange multiplier. Our final formulation fulfills the Babuška–Brezzi conditions on the continuous as well as discrete level, without restrictions. Consequently, we can allow for final-time desired states, and obtain an a posteriori error estimator which is efficient and reliable up to an additional discretization error of the adjoint problem. Numerical experiments confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

We consider the first-order system space–time formulation of the heat equation introduced by Bochev and Gunzburger (in: Bochev and Gunzburger (eds) Applied mathematical sciences, vol 166, Springer, New York, 2009), and analyzed by Führer and Karkulik (Comput Math Appl 92:27–36, 2021) and Gantner and Stevenson (ESAIM Math Model Numer Anal 55(1):283–299 2021), with solution components (u 1 , u 2) = (u , - ∇ x u) . The corresponding operator is boundedly invertible between a Hilbert space U and a Cartesian product of L 2 -type spaces, which facilitates easy first-order system least-squares (FOSLS) discretizations. Besides L 2 -norms of ∇ x u 1 and u 2 , the (graph) norm of U contains the L 2 -norm of ∂ t u 1 + div x u 2 . When applying standard finite elements w.r.t. simplicial partitions of the space–time cylinder, estimates of the approximation error w.r.t. the latter norm require higher-order smoothness of u 2 . In experiments for both uniform and adaptively refined partitions, this manifested itself in disappointingly low convergence rates for non-smooth solutions u. In this paper, we construct finite element spaces w.r.t. prismatic partitions. They come with a quasi-interpolant that satisfies a near commuting diagram in the sense that, apart from some harmless term, the aforementioned error depends exclusively on the smoothness of ∂ t u 1 + div x u 2 , i.e., of the forcing term f = (∂ t - Δ x) u . Numerical results show significantly improved convergence rates. [ABSTRACT FROM AUTHOR]

We complete the work done by James A. Ward in the mid-twentieth century on a system of partial differential equations that defines an algebra 픸 for which this system is the generalized Cauchy–Riemann equations for the derivative introduced by Sheffers at the end of the nineteenth century with respect to 픸 , which is also known as the Lorch derivative with respect to 픸 , and recently simply called 픸 -differentiability. We get a characterization of finite-dimensional algebras, which are associative commutative with unity. [ABSTRACT FROM AUTHOR]

We provide a framework for the numerical approximation of distributed optimal control problems, based on least-squares finite element methods. Our proposed method simultaneously solves the state and adjoint equations and is inf – sup stable for any choice of conforming discretization spaces. A reliable and efficient a posteriori error estimator is derived for problems where box constraints are imposed on the control. It can be localized and therefore used to steer an adaptive algorithm. For unconstrained optimal control problems, i.e., the set of controls being a Hilbert space, we obtain a coercive least-squares method and, in particular, quasi-optimality for any choice of discrete approximation space. For constrained problems we derive and analyze a variational inequality where the PDE part is tackled by least-squares finite element methods. We show that the abstract framework can be applied to a wide range of problems, including scalar second-order PDEs, the Stokes problem, and parabolic problems on space-time domains. Numerical examples for some selected problems are presented. [ABSTRACT FROM AUTHOR]

Developing the comprehensive theory of the X-ray diffraction by distorted crystals remains to be topical of the mathematical physics. Up to now, the X-ray diffraction theory grounded on the Takagi–Taupin equations with the first-order partial derivatives over the two coordinates within the X-ray scattering plane. In the work, the theoretical approach based on the first-order fractional Takagi–Taupin equations with the 'quasi-time variable' of the order α ∈ (0 , 1 ] along the crystal depth has been suggested and the corresponding X-ray Cauchy issue is formulated. Accordingly, using the Green function method in the scope of the Cauchy issue, the fractional Takagi–Taupin equations in the integral form have been derived. In the case of the inhomogeneous incident X-ray beam, the solution of the Cauchy issue of the X-ray diffraction by perfect crystal has been obtained and compared with the corresponding one based on the solution of the conventional Takagi–Taupin equations, α = 1. In turn, notice that the value of order α may be adjusted from the experimental X-ray diffraction data. [ABSTRACT FROM AUTHOR]

We consider the de Rham complex over scales of weighted isotropic and anisotropic Hölder spaces with prescribed asymptotic behavior at infinity. Starting from theorems on the solvability of the system of operator equations generated by the de Rham differential d and the operator d ∗ formally adjoint to it, a description of the cohomology groups of the de Rham complex over these scales was obtained. It was also proved that in the isotropic case the cohomology space is finite-dimensional, and in the anisotropic case the general form of an element from the cohomology space is presented. [ABSTRACT FROM AUTHOR]

Motivated by strong versions of Green's theorem, we give an example of a differentiable vector field for which the curl is continuous but not all the partial derivatives are continuous. [ABSTRACT FROM AUTHOR]

Abstract: We are interested in the following question: when regularity properties of a linear differential operator imply solvability of its transpose in the sense of Gevrey ultradistributions? This question is studied for a class of abstract operators that contains the usual differential operators with real‐analytic coefficients. We obtain a new proof of a global result on compact manifolds (global Gevrey hypoellipticity implying global solvability of the transpose), as well as some results in the non‐compact case by means of the so‐called property of non‐confinement of singularities. We provide applications to Hörmander operators, to operators of constant strength and to locally integrable systems of vector fields. We also analyze a conjecture stated in a recent paper of Malaspina and Nicola, which asserts that, in differential complexes naturally arising from locally integrable structures, local solvability in the sense of ultradistributions implies local solvability in the sense of distributions. We establish the validity of the conjecture when the cotangent structure bundle is spanned by the differential of a single first integral. [ABSTRACT FROM AUTHOR]

We consider the inhomogeneous div-curl system (i.e. to find a vector field with prescribed div and curl) in a bounded star-shaped domain in $$\mathbb {R}^3$$ . An explicit general solution is given in terms of classical integral operators, completing previously known results obtained under restrictive conditions. This solution allows us to solve questions related to the quaternionic main Vekua equation $$DW=(Df/f)\overline{W}$$ in $$\mathbb {R}^3$$ , such as finding the vector part when the scalar part is known. In addition, using the general solution to the div-curl system and the known existence of the solution of the inhomogeneous conductivity equation, we prove the existence of solutions of the inhomogeneous double curl equation, and give an explicit solution for the case of static Maxwell's equations with only variable permeability. [ABSTRACT FROM AUTHOR]

In this article, we give a necessary and sufficient condition of Kalman type for the indirect controllability of systems of groups of linear operators, under some 'regularity and locality' conditions on the control operator that will be made precise later and fit very well the case of distributed controls. Moreover, in the case of first order in time systems, when the Kalman rank condition is not satisfied, we characterize exactly the initial conditions that can be controlled. Some applications to the control of systems of Schrödinger or wave equations are provided. The main tool used here is the fictitious control method coupled with the proof of an algebraic solvability property for some related underdetermined system and some regularity results. [ABSTRACT FROM AUTHOR]

To a system of closed one-forms on the torus, we associate a differential complex and compute the induced cohomology groups provided that a related matrix satisfies a Diophantine condition. [ABSTRACT FROM AUTHOR]

Inspired by the results of Ern et al. (Commun Partial Differ Equ 32:317-341, 2007) on the abstract theory for Friedrichs symmetric positive systems, we give the existence and uniqueness result for the initial- (boundary) value problem for the non-stationary abstract Friedrichs system. Despite the absence of the well-posedness result for such systems, there were already attempts for their numerical treatment by Burman et al. (SIAM J Numer Anal 48:2019-2042, 2010) and Bui-Thanh et al. (SIAM J Numer Anal 51:1933-1958, 2013). We use the semigroup theory approach and prove that the operator involved satisfies the conditions of the Hille-Yosida generation theorem. We also address the semilinear problem and apply the new results to a number of examples, such as the symmetric hyperbolic system, the unsteady div-grad problem, and the wave equation. Special attention was paid to the (generalised) unsteady Maxwell system. [ABSTRACT FROM AUTHOR]

In this work we show that if [ABSTRACT FROM AUTHOR]

The $$L^1$$ -Sobolev inequality states that for compactly supported functions u on the Euclidean n-space, the $$L^{n/(n-1)}$$ -norm of a compactly supported function is controlled by the $$L^1$$ -norm of its gradient. The generalization to differential forms (due to Lanzani and Stein and Bourgain and Brezis) is recent, and states that a the $$L^{n/(n-1)}$$ -norm of a compactly supported differential h-form is controlled by the $$L^1$$ -norm of its exterior differential du and its exterior codifferential $$\delta u$$ (in special cases the $$L^1$$ -norm must be replaced by the $$\mathcal H^1$$ -Hardy norm). We shall extend this result to Heisenberg groups in the framework of an appropriate complex of differential forms. [ABSTRACT FROM AUTHOR]

In this paper we consider functions $$f$$ defined on an open set $$U$$ of the Euclidean space $$\mathbb{R }^{n+1}$$ and with values in the Clifford Algebra $$\mathbb{R }_n$$ . Slice monogenic functions $$f: U \subseteq \mathbb{R }^{n+1} \rightarrow \mathbb{R }_n$$ belong to the kernel of the global differential operator with non constant coefficients given by $$ \mathcal{G }=|{\underline{x}}|^2\frac{\partial }{\partial x_0} \ + \ {\underline{x}} \ \sum _{j=1}^n x_j\frac{\partial }{\partial x_j}. $$ Since the operator $$\mathcal{G }$$ is not elliptic and there is a degeneracy in $$ {\underline{x}}=0$$ , its kernel contains also less smooth functions that have to be interpreted as distributions. We study the distributional solutions of the differential equation $$\mathcal{G }F(x_0,{\underline{x}})=G(x_0,{\underline{x}})$$ and some of its variations. In particular, we focus our attention on the solutions of the differential equation $$ ({\underline{x}}\frac{\partial }{\partial x_0} \ - E)F(x_0,{\underline{x}})=G(x_0,{\underline{x}}), $$ where $$E= \sum _{j=1}^n x_j\frac{\partial }{\partial x_j}$$ is the Euler operator, from which we deduce properties of the solutions of the equation $$ \mathcal{G }F(x_0,{\underline{x}})=G(x_0,{\underline{x}})$$ . [ABSTRACT FROM AUTHOR]

The Cauchy–Riemann type systems in ℝn+1are systems of first-order partial differential equations whose solutions satisfy the Laplace equation. The systems can be written in the form, whereE0 = Imis the identity matrix,Ei(i ≥ 1) arem × mconstant matrices satisfying the conditionEiEj + EjEi = − 2δijIm. We introduce a more general condition forEi, they are variable coefficients and satisfy the conditionEi(x) · Ej(x) + Ej(x) · Ei(x) = − 2aij(x)Im, where [aij] is a symmetric and positive matrix. Each 𝒞2-solution of the system now becomes a solution of a second-order elliptic system with the leading part. In this article, we introduce the Levi functions and construct fundamental solutions of the systems ‘in the large’. [ABSTRACT FROM AUTHOR]

Hypoellipticity for a class of germs of ℂ-valued, real-analytic 1-forms with an isolated singularity is studied. The main result states that in dimension ≥3 such forms are real-analytic hypoelliptic over L functions, but are not real-analytic hypoelliptic in dimension 2. The case of C-hypoellipticity for a class of rotation invariant forms in ℝ is also considered. [ABSTRACT FROM AUTHOR]