Recent zbMATH articles in MSC 35Nhttps://www.zbmath.org/atom/cc/35N2021-06-15T18:09:00+00:00WerkzeugHypothesis on unification of solution of the Cauchy problem for overdetermined systems of differential equations.https://www.zbmath.org/1460.352382021-06-15T18:09:00+00:00"Zaĭtsev, M. L."https://www.zbmath.org/authors/?q=ai:zaitsev.maxim-leonidovich"Akkerman, V. B."https://www.zbmath.org/authors/?q=ai:akkerman.v-bSummary: In this paper, we study the possibility of the existence of a universal solution of the Cauchy problem for the partial differential equation (PDE) systems in the case if this system is overdetermined so that the new overdetermined system of PDE contains all solutions of the initial PDE system and, in addition, reduces to the ordinary differential equation (ODE) systems, whose solution is then found. To do this, the article discusses the modification of the method of finding particular solutions for any overdetermined systems of differential equations by reduction to overdetermined systems of implicit equations. In the previous papers of the authors, a method was proposed for finding particular solutions for overdetermined PDE systems. In this method, in order to find solutions it is necessary to solve systems of ordinary implicit equations. In this case, it can be shown that the solutions that we need cannot depend on a continuous parameter, i.e. they are no more than countable. In advance, there is a need for such an overriding of the systems of differential equations, so that their general solutions are no more than countable. Such an initial overdetermination is rather difficult to achieve. However, the proposed method also allows to reduce the overdetermined systems of differential equations not only up to systems of implicit equations, but also up to the PDE systems of dimension less than that of the initial systems of PDE. In particular, under certain conditions, reduction to the ODE systems is possible. It is proposed to choose solutions for the overdetermined PDE systems using the parameterized Cauchy problem, which is posed for parameterized ODE systems under certain conditions. The solution of this Cauchy problem is some function of the initial data and their derivatives. In order to find the solution of any corresponding Cauchy problem for the initial system of PDE, it is sufficient to calculate the universal solver for the reduced ODE system once. In this case, the solution will not only exist and be unique, but will also depend continuously on the initial data, since this holds for ODE systems.
The purpose of this paper is to study the Cauchy problem with the possibility of its universalization and the parameterized Cauchy problem as a whole for arbitrary PDE systems.Optimal regularity for two-dimensional Pfaffian systems and the fundamental theorem of surface theory.https://www.zbmath.org/1460.580032021-06-15T18:09:00+00:00"Litzinger, Florian"https://www.zbmath.org/authors/?q=ai:litzinger.florianSummary: We prove that a Pfaffian system with coefficients in the critical space \(L^2_{\text{loc}}\) on a simply connected open subset of \(\mathbb{R}^2\) has a non-trivial solution in \(W^{1,2}_{\text{loc}}\) if the coefficients are antisymmetric and satisfy a compatibility condition. As an application of this result, we show that the fundamental theorem of surface theory holds for prescribed first and second fundamental forms of optimal regularity in the classes \(W^{1,2}_{\text{loc}}\) and \(L^2_{\text{loc}}\), respectively, that satisfy a compatibility condition equivalent to the Gauss-Codazzi-Mainardi equations. Finally, we give a weak compactness theorem for surface immersions in the class \(W^{2,2}_{\text{loc}}\).Tube structures of co-rank 1 with forms defined on compact surfaces.https://www.zbmath.org/1460.580142021-06-15T18:09:00+00:00"Hounie, J."https://www.zbmath.org/authors/?q=ai:hounie.j"Zugliani, G."https://www.zbmath.org/authors/?q=ai:zugliani.giuliano-angeloSummary: We study the global solvability of a locally integrable structure of tube type and co-rank 1 by considering a linear partial differential operator \(\mathbb{L}\) associated to a general complex smooth closed 1-form \(c\) defined on a smooth closed \(n\)-manifold. The main result characterizes the global solvability of \(\mathbb{L}\) when \(n=2\) in terms of geometric properties of a primitive of a convenient exact pullback of the form \(\mathfrak{Im}(c)\) as well as in terms of homological properties of \(\mathfrak{Re}(c)\) related to small divisors phenomena. Although the full characterization is restricted to orientable surfaces, some partial results hold true for compact manifolds of any dimension, in particular, the necessity of the conditions, and the equivalence when \(\mathfrak{Im}(c)\) is exact. We also obtain informations on the global hypoellipticity of \(\mathbb{L}\) and the global solvability of \(\mathbb{L}^{n-1}\) -- the last non-trivial operator of the complex when \(M\) is orientable.Dispersionless multi-dimensional integrable systems and related conformal structure generating equations of mathematical physics.https://www.zbmath.org/1460.170402021-06-15T18:09:00+00:00"Hentosh, Oksana Ye."https://www.zbmath.org/authors/?q=ai:hentosh.oksana-ye"Prykarpatsky, Yarema A."https://www.zbmath.org/authors/?q=ai:prykarpatsky.yarema-anatoliyovych"Blackmore, Denis"https://www.zbmath.org/authors/?q=ai:blackmore.denis-l"Prykarpatski, Anatolij K."https://www.zbmath.org/authors/?q=ai:prykarpatsky.anatoliy-karolevychSummary: Using diffeomorphism group vector fields on \(\mathbb{C}\)-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of mathematical physics. An interesting modification of the devised Lie-algebraic approach subject to spatial-dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied in proving complete integrability of some conformal structure generating equations. As examples, we analyze the Einstein-Weyl metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations and the inverse first Shabat reduction heavenly equation. We also analyze the modified Plebański heavenly equations, the Husain heavenly equation and the general Monge equation along with their multi-dimensional generalizations. In addition, we construct superconformal analogs of the Whitham heavenly equation.