A family of interpolation inequalities is derived, which differ from estimates of classical Gagliardo–Nirenberg type through the appearance of certain logarithmic deviations from standard Lebesgue norms in zero-order expressions. Optimality of the obtained inequalities is shown. A subsequent application reveals that when posed under homogeneous Neumann boundary conditions in smoothly bounded planar domains and with suitably regular initial data, for any choice of \alpha>0 the Keller–Segel-type migration–consumption system u_{t} = \Delta (uv^{-\alpha}) , v_{t} = \Delta v-uv , admits a global classical solution.

Abstract Using the logarithmic capacity, we give quantitative estimates of the Green function, as well as precise lower bounds of the Bergman kernel for bounded pseudoconvex domains in and the Bergman distance for bounded planar domains. In particular, it is shown that the Bergman kernel satisfies the sharp estimate if is a bounded pseudoconvex domain with ‐boundary.

We consider the first and second initial–boundary value problems for inhomogeneous second-order parabolic systems with Dini continuous coefficients under nonzero initial conditions in bounded domains on the plane with nonsmooth lateral boundaries that, in particular, admit cusps. Theorems are proved on the unique classical solvability of these problems in the space of functions that are continuous together with their first spatial derivatives in the closure of these domains.

We consider the first and second initial–boundary value problems for inhomogeneous second-order parabolic systems with Dini continuous coefficients under nonzero initial conditions in bounded domains on the plane with nonsmooth lateral boundaries that, in particular, admit cusps. Theorems are proved on the unique classical solvability of these problems in the space of functions that are continuous together with their first spatial derivatives in the closure of these domains.

The purpose of this article is to develop a Hadamard–Riesz-type parametrix for the wave propagator in bounded planar domains with smooth, strictly convex boundary. This parametrix then allows us to rederive an oscillatory integral representation for the wave trace appearing in the work by Marvizi and Melrose (1982) and compute its principal symbol explicitly in terms of geometric data associated to the billiard map. This results in new formulas for the wave invariants. The order of the principal symbol, which appears to be inconsistent in the works by Marvizi and Melrose (1982) and Popov (1994), is also corrected. In those papers, the principal symbol was never actually computed and to our knowledge, this paper contains the first explicit formulas for the principal symbol of the wave trace. The wave trace formulas we provide are localized near both simple lengths corresponding to nondegenerate periodic orbits and degenerate lengths associated to one parameter families of periodic orbits tangent to a single rational caustic. Existence of a Hadamard–Riesz-type parametrix with explicit symbol and phase calculations in the interior appears to be new in the literature, with the exception of the author’s previous work [J. Geom. Anal. 31 (2021), 2238–2295] in the special case of elliptical domains. This allows us to circumvent the symbol calculus used by Duistermaat and Guillemin (1975) and Hezari and Zelditch (2012) when computing trace formulas, which are instead derived from integrating our explicit parametrix over the diagonal.

We investigate regularization methods for solving the problem of crack detection in bounded planar domains from electrical measurements on the boundary. Based on the multiple level-set approach introduced in Álvarez et al (2009 J. Comput. Phys. 228 5710–21) and on the regularization strategy devised in De Cezaro et al (2009 Inverse Problems 25 035004), we propose a Tikhonov type method for stabilizing the inverse problem. Convergence and stability results for this Tikhonov method are proven. An iterative method of (multiple) level-set type is derived from the optimality conditions for the Tikhonov functional, and a relation between this method and the iterated Tikhonov method is established. The proposed level-set method is tested on the same benchmark problem considered in Álvarez et al (2009 J. Comput. Phys. 228 5710–21). The numerical experiments demonstrate its ability to identify cracks in different scenarios with high accuracy even in the presence of noise.

In this paper, we prove that in bounded planar domains with $C^{2,\alpha}$ boundary, for almost every initial condition in the sense of the Lebesgue measure, the point-vortex system has a global solution, meaning that there is no collision between two point-vortices or with the boundary. This extends the work previously done in [C. Marchioro and M. Pulvirenti, Vortex Methods in Two-Dimensional Fluid Dynamics, Springer-Verlag, 1984] for the disk. The proof requires the construction of a regularized dynamics that approximates the real dynamics and some strong inequalities for the Green's function of the domain. The establishment of some useful estimates is discussed and the details of the proof are given in the original article [M. Donati, SIAM J. Math. Anal., 54 (2022), pp. 79--113].

The SIGEST article in this issue, “Improbability of Collisions of Point-Vortices in Bounded Planar Domains,” by Martin Donati, is based on the 2022 SIAM Journal on Mathematics Analysis article “Two-Dimensional Point Vortex Dynamics in Bounded Domains: Global Existence for Almost Every Initial Data.” This work concerns point-vortex dynamics in a very general bounded domain in the plane. The main result is that the set of initial configurations which lead to finite-time collision, although nonempty, has Lebesgue measure zero. This is an elegant and highly nontrivial extension to general bounded domains with $C^{2, \alpha}$ boundary of a result previously known only for three domains: the full plane, a disk, and the complement of a disk. It is notable that the result also extends to point-vortex dynamics in multiply connected domains. In preparing this SIGEST version, the author has added a new introductory section that explains the context of the new results. Moreover, some of the technical details from the original article have been replaced by more accessible high-level explanations. Recent references on the topics of vortex collisions and desingularization problems are also included. This SIGEST article will be of particular interest to researchers in fluid mechanics, partial differential equations, and applied analysis.

We develop the theory of torsional rigidity—a quantity routinely considered for Dirichlet Laplacians on bounded planar domains—for Laplacians on metric graphs with at least one Dirichlet vertex. Using a variational characterization that goes back to Pólya, we develop surgical principles that, in turn, allow us to prove isoperimetric-type inequalities: we can hence compare the torsional rigidity of general metric graphs with that of intervals of the same total length. In the spirit of the Kohler-Jobin inequality, we also derive sharp bounds on the ground-state energy of a quantum graph in terms of its torsional rigidity: this is particularly attractive since computing the torsional rigidity reduces to inverting a matrix whose size is the number of the graph’s vertices and is, thus, much easier than computing eigenvalues.

Consider Yudovich solutions to the incompressible Euler equations with bounded initial vorticity in bounded planar domains or in $\mathbb{R}^2$. We present a purely Lagrangian proof that the solution map is strongly continuous in $L^p$ for all $p\in [1, \infty)$ and is weakly-$*$ continuous in $L^\infty$.

In this paper two inequalities are presented for the torsional rigidity of homogeneous monoclinic piezoelectric beams. All results of the paper are based on the Saint-Venant theory of uniform torsion. The cross section of the considered elastic and piezoelectric beams may be simply connected or multiply connected two-dimensional bounded plane domain. Examples illustrate the proven inequality relations.

We initiate the study of the higher-order Escobar constants I_k(M), kge 3, on bounded planar domains M. The Escobar constants I_k of the unit disk and a family of polygons are provided.

The wave equation is considered, for all times , in a bounded plane domain with an internal crack. The distance from one of the crack tips to the external boundary of is proportional to a small parameter . Dirichlet or Neumann condition is given on the whole boundary of . Near the crack tip, the first derivatives of solutions have square-root () singularities. The asymptotics of the ‘stress intensity factors’ of such singularities are deduced as .

We consider the first initial boundary value problem for a Petrovsky second order parabolic system with variable coefficients in a bounded plane domain with nonsmooth lateral boundary. We prove the uniqueness of a solution in a Holder class.

Many earlier works were devoted to the study of the breakdown of superconductivity in type-II superconducting bounded planar domains, submitted to smooth magnetic fields. In the present contribution, we consider a new situation where the applied magnetic field is piecewise-constant, and the discontinuity jump occurs along a smooth curve meeting the boundary transversely. To handle this situation, we perform a detailed spectral analysis of a new effective model. Consequently, we establish the monotonicity of the transition from a superconducting to a normal state. Moreover, we determine the location of superconductivity in the sample just before it disappears completely. Interestingly, the study shows similarities with the case of corner domains subjected to constant fields.

In a bounded planar domain varOmega with smooth boundary, the initial-boundary value problem of homogeneous Neumann type for the Keller-Segel-fluid system \t\t\t{nt+∇⋅(nu)=Δn−∇⋅(n∇c),x∈Ω,t>0,0=Δc−c+n,x∈Ω,t>0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\left \\{ \\textstyle\\begin{array}{l@{\\quad }l} n_{t} + \\nabla \\cdot (nu) = \\Delta n - \\nabla \\cdot (n\\nabla c), & x\\in \\varOmega , \\ t>0, \\\\ 0 = \\Delta c -c+n, & x\\in \\varOmega , \\ t>0, \\end{array}\\displaystyle \\right . \\end{aligned}$$ \\end{document} is considered, where u is a given sufficiently smooth velocity field on overline {varOmega }times [0,infty ) that is tangential on partial varOmega but not necessarily solenoidal.It is firstly shown that for any choice of n_{0}in C^{0}(overline {varOmega }) with int _{varOmega}n_{0}<4pi , this problem admits a global classical solution with n(cdot ,0)=n_{0}, and that this solution is even bounded whenever u is bounded and int _{varOmega}n_{0}<2pi . Secondly, it is seen that for each m>4pi one can find a classical solution with int _{varOmega}n(cdot ,0)=m which blows up in finite time, provided that varOmega satisfies a technical assumption requiring partial varOmega to contain a line segment.In particular, this indicates that the value 4pi of the critical mass for the corresponding fluid-free Keller-Segel system is left unchanged by any fluid interaction of the considered type, thus marking a considerable contrast to a recent result revealing some fluid-induced increase of critical blow-up masses in a related Cauchy problem in the entire plane.

The fully parabolic Keller--Segel system is coupled to the incompressible Navier--Stokes equations through transport and buoyancy. It is shown that when posed with no-flux/no-flux/Dirichlet boundary conditions in smoothly bounded planar domains and along with appropriate assumptions on regularity of the initial data, under a smallness condition exclusively involving the total initial population mass $m$ an associated initial-boundary value problem admits a globally defined generalized solution; in particular, this hypothesis is fully explicit and independent of the initial size of further solution components. Moreover, the obtained solution is seen to enjoy a certain temporally averaged boundedness property which, inter alia, rules out any finite-time collapse into persistent Dirac-type measures, as well as convergence to such singular profiles in the large time limit.

Problems of multiple covering (k-covering) of a bounded set G with equal circles of a given radius are well known. They are thoroughly studied under the assumption that G is a finite set. There are several papers concerned with studying this problem in the case where G is a connected set. In this paper, we study the problem of minimizing the number of circles that form a k-covering, k 1, provided that G is a bounded convex plane domain.

The repulsive Keller–Segel–Navier–Stokes system $$\begin{aligned} \left\{ \begin{array}{llll} n_t + u\cdot \nabla n &{}=&{} \Delta n + \nabla \cdot (n\nabla c), &{}\quad x\in \Omega , \; t>0, \\ c_t + u\cdot \nabla c &{}=&{} \Delta c -c+n, &{}\quad x\in \Omega , \; t>0, \\ u_t + (u\cdot \nabla ) u &{}=&{} \Delta u + \nabla P + n\nabla \Phi , \quad \nabla \cdot u=0, &{}\quad x\in \Omega , \; t>0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$is considered in smoothly bounded planar domains, where $$\Phi \in W^{2,\infty }(\Omega )$$ is given. It is well-known that the corresponding fluid-free analogue, when posed under homogeneous no-flux boundary conditions, admits global classical solutions for arbitrarily large initial data, thus substantially differing from the classical two-dimensional Keller–Segel system featuring chemoattraction-driven finite-time blow-up for some initial data. The literature on such chemorepulsion systems, however, strongly relies on the presence of an associated energy structure which is apparently destroyed by the fluid interaction mechanism in ($$\star $$). By making use of appropriate functional inequalities involving certain logarithmic expressions arising due to the planarity of the considered setting, it is shown that nevertheless an initial-boundary value problem for ($$\star $$) admits globally defined classical solutions for all reasonably regular initial data.

We propose a method for determining the number of sensors, their arrangement, and approximate lower bounds for the number of sensors for the multiple covering of an arbitrary closed bounded convex area in a plane. The problem of multiple covering is considered with restrictions on the minimal possible distances between the sensors and without such restrictions. To solve these problems, some 0–1 linear programming (LP) problems are constructed.We use a heuristic solution algorithm for 0–1 LP problems of higher dimensions. The results of numerical implementation are given and for some particular cases it is obtained that the number of sensors found can not be decreased.