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Subject to given boundary data, nonexistence of solution to the one-dimensional Kirchhoff-like equation \begin{equation*} -M\Big (\big (a*|u|^q\big)(1)\Big)u''(t)=\lambda f\big (t,u(t)\big),\ 0\lambda _0, where \lambda _0 is defined in terms of initial data, the boundary value problem has no nontrivial positive solution. [ABSTRACT FROM AUTHOR]

In this work, we consider Dirac-type operators with a constant delay less than two-fifths of the interval and not less than one-third of the interval. For our considered Dirac-type operators, an incomplete inverse spectral problem is studied. Specifically, when two complex potentials are known a priori on a certain subinterval, reconstruction of the two potentials on the entire interval is studied from complete spectra of two boundary value problems with one common boundary condition. The uniqueness of the solution of the inverse problem is proved. A constructive method is developed for the solution of the inverse problem. [ABSTRACT FROM AUTHOR]

In this paper, we prove that there exists a unique weak solution to the Neumann boundary value problem for second order elliptic operators whose coefficients are signed measures. We will give a probabilistic representation of the solution. The heat kernel estimates and reflected diffusion processes play important roles. [ABSTRACT FROM AUTHOR]

The Restricted Additive Schwarz method with impedance transmission conditions, also known as the Optimised Restricted Additive Schwarz (ORAS) method, is a simple overlapping one-level parallel domain decomposition method, which has been successfully used as an iterative solver and as a preconditioner for discretised Helmholtz boundary-value problems. In this paper, we give, for the first time, a convergence analysis for ORAS as an iterative solver—and also as a preconditioner—for nodal finite element Helmholtz systems of any polynomial order. The analysis starts by showing (for general domain decompositions) that ORAS is an unconventional finite element approximation of a classical parallel iterative Schwarz method, formulated at the PDE (non-discrete) level. This non-discrete Schwarz method was recently analysed in [Gong, Gander, Graham, Lafontaine, and Spence, Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation ], and the present paper gives a corresponding discrete version of this analysis. In particular, for domain decompositions in strips in 2-d, we show that, when the mesh size is small enough, ORAS inherits the convergence properties of the Schwarz method, independent of polynomial order. The proof relies on characterising the ORAS iteration in terms of discrete 'impedance-to-impedance maps', which we prove (via a novel weighted finite-element error analysis) converge as h\rightarrow 0 in the operator norm to their non-discrete counterparts. [ABSTRACT FROM AUTHOR]

This paper serves as a corrigenda for the article P. W. Eloe and J. Henderson, "Two-point boundary value problems for ordinary differential equations, uniqueness implies existence", Proc. Amer. Math. Soc. 148 (2020), 4377–4387. In particular, the proof the authors give in that paper of Theorem 3.3 is incorrect, and so, that alleged theorem remains a conjecture. In this corrigenda, the authors state and prove a correct theorem. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial and boundary value problem of Ericksen-Leslie system modeling nematic liquid crystal flows in dimension three. Two examples of singularity at finite time are constructed. The first example is constructed in a special axisymmetric class with suitable axisymmetric initial and boundary data, while the second example is constructed for initial data with small energy but nontrivial topology. A counter example of maximum principle to the system is constructed by utilizing the Poiseuille flow in dimension one. [ABSTRACT FROM AUTHOR]

In this paper, we prove nonlinear orbital stability for steady vortex patches that maximize the kinetic energy among isovortical rearrangements in a planar bounded domain. As a result, nonlinear stability for an isolated vortex patch is proved. The proof is based on conservation of energy and vorticity, which is an analogue of the classical Liapunov function method. [ABSTRACT FROM AUTHOR]

In this paper, we provide a sufficient condition for a curve on a surface in R3 to be given by an orthogonal intersection with a sphere. This result makes it possible to express the boundary condition entirely in terms of the Weierstrass data without integration when dealing with free boundary minimal surfaces in a ball B3. Moreover, we show that the Gauss map of an embedded free boundary minimal annulus is one to one. By using this, the Fraser-Li conjecture can be translated into the problem of determining the Gauss map. On the other hand, we show that the Liouville type boundary value problem in an annulus gives some new insight into the structure of immersed minimal annuli orthogonal to spheres. It also suggests a new PDE theoretic approach to the Fraser-Li conjecture. [ABSTRACT FROM AUTHOR]

The shifted boundary method (SBM) is an approximate domain method for boundary value problems, in the broader class of unfitted/embedded/immersed methods. It has proven to be quite efficient in handling problems with complex geometries, ranging from Poisson to Darcy, from Navier-Stokes to elasticity and beyond. The key feature of the SBM is a shift in the location where Dirichlet boundary conditions are applied—from the true to a surrogate boundary—and an appropriate modification (again, a shift) of the value of the boundary conditions, in order to reduce the consistency error. In this paper we provide a sound analysis of the method in smooth domains and in domains with corners, highlighting the influence of geometry and distance between exact and surrogate boundaries upon the convergence rate. We consider the Poisson problem with Dirichlet boundary conditions as a model and we first detail a procedure to obtain the crucial shifting between the surrogate and the true boundaries. Next, we give a sufficient condition for the well-posedness and stability of the discrete problem. The behavior of the consistency error arising from shifting the boundary conditions is thoroughly analyzed, for smooth boundaries and for boundaries with corners and edges. The convergence rate is proven to be optimal in the energy norm, and is further enhanced in the L2-norm. [ABSTRACT FROM AUTHOR]

In this paper, we consider the semi-linear oblique boundary value problem for the prescribed mean curvature equation. We find a suitable auxiliary function and use the maximum principle to get the gradient estimate. As a consequence, we obtain the corresponding existence theorem for a class of mean curvature equations with semi-linear oblique derivative problems [ABSTRACT FROM AUTHOR]

This paper is concerned with homogenization of systems of linear elasticity with rapidly oscillating periodic coefficients. We establish sharp convergence rates in L² for the mixed boundary value problems with bounded measurable coefficients. [ABSTRACT FROM AUTHOR]

This paper develops and analyzes two local discontinuous Galerkin (LDG) methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the high frequency regime. It is shown that the proposed LDG methods are stable for all positive wave number k and all positive mesh size h. Energy norm and L2-norm error estimates are derived for both LDG methods in all mesh parameter regimes including pre-asymptotic regime (i.e., k2h ≳ 1). To analyze the proposed LDG methods, they are recast and treated as (nonconforming) mixed finite element methods. The crux of the analysis is to show that the sesquilinear form associated with each LDG method satisfies a coercivity property in all mesh parameter regimes. These coercivity properties then easily infer the desired discrete stability estimates for the solutions of the proposed LDG methods. In return, the discrete stabilities not only guarantee the well-posedness of the LDG methods but also play a crucial role in the error analysis. Numerical experiments are also presented in the paper to validate the theoretical results and to compare the performance of the proposed two LDG methods. [ABSTRACT FROM AUTHOR]

This paper is devoted to a canonical construction of a family of piecewise polynomials with the minimal degree capable of providing a consistent approximation of Sobolev spaces Hm in Rn (with n ≥ m ≥ 1) and also a convergent (nonconforming) finite element space for 2m-th-order elliptic boundary value problems in Rn. For this class of finite element spaces, the geometric shape is n-simplex, the shape function space consists of all polynomials with a degree not greater than m, and the degrees of freedom are given in terms of the integral averages of the normal derivatives of order m - k on all subsimplexes with the dimension n - k for 1 ≤ k ≤ m. This sequence of spaces has some natural inclusion properties as in the corresponding Sobolev spaces in the continuous cases. The finite element spaces constructed in this paper constitute the only class of finite element spaces, whether conforming or nonconforming, that are known and proven to be convergent for the approximation of any 2m-th-order elliptic problems in any Rn, such that n ≥ m ≥ 1. Finite element spaces in this class recover the nonconforming linear elements for Poisson equations (m = 1) and the well-known Morley element for biharmonic equations (m = 2). [ABSTRACT FROM AUTHOR]

In this paper, we consider the following discrete nonlinear problem which is subject to a periodic nonlinear forcing term: A u = λ u +p(u) + h, where A is an n × n matrix with real components, p: Rn → Rn is a periodic forcing term, and ⟨h,φ⟩ = 0, where φ is an eigenvector of AT, the transpose of A, corresponding to a simple real eigenvalue λ. Conditions on these terms will be provided such that this problem will have infinitely many distinct solutions. The results here are motivated by some recent results for discrete systems and by results obtained for analogous boundary value problems for semilinear elliptic problems at resonance. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ RN be an open bounded connected set. We consider the eigenvalue problem -Δpu = λρ∣u∣p-2u in Ω with homogeneous Dirichlet boundary condition, where Δp is the p-Laplacian operator and ρ is an arbitrary function that takes only two given values 0 < α < β and that is subject to the constraint ∫Ω ρdx = αγ +β(∣Ω∣-γ) for a fixed 0 < γ < ∣Ω∣. The optimization of the map ρ ↔ λ1(ρ), where λ1 is the first eigenvalue, has been studied by Cuccu, Emamizadeh and Porru. In this paper we consider a Steiner symmetric domain Ω and we show that the minimizers inherit the same symmetry. [ABSTRACT FROM AUTHOR]

In this paper we develop new geometric techniques to deal with the Dirichlet problem for a p-harmonic map from a compact manifold with boundary to a Cartan-Hadamard target manifold which is either 2-dimensional or rotationally symmetric. [ABSTRACT FROM AUTHOR]

In this paper we characterize extremal polynomials and the best constants for the Szeg˝o-Markov-Bernstein-type inequalities, associated with iterated weight functions ρk (x)=A(x + h) ρk−1 (x + h) of any classical weight ρ0 (x) = ρ (x) of discrete variable x = a + ih, which is defined to be the solution of a difference boundary value problem of the Pearson type. It yields the effective way to compute numerical values of the best constants for all six basic discrete classical weights of the Charlier, Meixner, Kravchuk, Hahn I, Hahn II, and Chebyshev kind. In addition, it enables us to establish the generic identities between the Lagrange barycentric coefficients and Christoffel numbers of Gauss quadratures for these classical discrete weight functions, which extends to the discrete case the recent results due to Wang et al. and the authors, published in [Math. Comp. 81 (2012) and 83 (2014), pp. 861-877 and 2893-2914, respectively] and [Math. Comp. 86 (2017), pp. 2409-2427]. [ABSTRACT FROM AUTHOR]

In this paper we analyze the capacitary potential due to a charged body in order to deduce sharp analytic and geometric inequalities, whose equality cases are saturated by domains with spherical symmetry. In particular, for a regular bounded domain Ω ⊂ Rn, n ≥3, we prove that if the mean curvature H of the boundary obeys the condition -[1/Cap(Ω)]1/n-2 ≤ H/n-1 ≤ [1/Cap(Ω)]1/n-2 then Ω is a round ball. [ABSTRACT FROM AUTHOR]

For a gradient flow of the Landau–de Gennes energy, the unique global weak solution of initial and boundary value problem in dimension two has been constructed by Iyer–Xu–Zarnescu [Math. Models Methods Appl. Sci. 25 (2015), no. 8, 1477–1517] with small initial data. We investigate the regularity of such a solution, and prove that the weak small solution constructed in that paper is actually regular. [ABSTRACT FROM AUTHOR]

In this paper we compute asymptotics of solutions of the kinetic Fokker- Planck equation with inelastic boundary conditions which indicate that the solutions are nonunique if r < rc. The nonuniqueness is due to the fact that different solutions can interact in a different manner with a Dirac mass which appears at the singular point (x,v) = (0,0). In particular, this nonuniqueness explains the different behaviours found in the physics literature for numerical simulations of the stochastic differential equation associated to the kinetic Fokker-Planck equation. The asymptotics obtained in this paper will be used in a companion paper (Nonuniqueness for the kinetic-Fokker- Planck equation with inelastic boundary conditions) to prove rigorously nonuniqueness of solutions for the kinetic Fokker-Planck equation with inelastic boundary conditions. [ABSTRACT FROM AUTHOR]

We develop further our strategy from our 2014 paper showing that flat or Lipschitz-free boundaries of two-phase problems with forcing terms are locally C2,γ. [ABSTRACT FROM AUTHOR]

We are interested in the asymptotic behavior of the solution to the nonlinear damped Schrodinger equation (NLDS). In the present paper, we discuss when the global solutions to NLDS exponentially scatter to linear damped solutions. Moreover, we also show the additional time decay order. [ABSTRACT FROM AUTHOR]

In this paper we prove the rectifiability of and measure bounds on the singular set of the free-boundary for minimizers of a functional first considered by Alt-Caffarelli [J. Reine Angew. Math. 325 (1981), pp. 105-144]. Our main tools are the Quantitative Stratification and Rectifiable-Reifenberg framework of Naber-Valtorta [Ann. of Math. (2) 185 (2017), pp. 131-227], which allow us to do a type of "effective dimension-reduction". The arguments are sufficiently robust that they apply to a broad class of related free-boundary problems as well. [ABSTRACT FROM AUTHOR]

This paper deals with the compressible subsonic flows in an infinitely long asymmetric branching channel with two exhaust ducts. The flow satisfies the slip boundary conditions on the nozzle walls, and the total mass flux is prescribed in the inlet of the nozzle. We first established the existence of smooth subsonic irrotational flows through the branching channel for given sufficiently small total mass flux in the inlet. Several results on uniqueness are also obtained. In particular, imposing the location of the branching point on the nose of the channel, the uniqueness and the asymptotic behavior of the subsonic flow in upstream and downstream are shown, provided that the total mass flux is less than some critical value. Due to the asymmetric geometrics, the location of the branching point of the fluids has to be considered here. Of particular interest, it is observed that the location of the branching point on the nozzle wall is monotonic and continuously dependent on the ratio of the mass fluxes in the two exhaust ducts, and the branching point is the unique stagnation point in the fluid field and its closure. Finally, as a direct application, some results on subsonic-sonic divided flows are established. [ABSTRACT FROM AUTHOR]

This paper extends two recent improvements in the Hilbert space setting of the well-known Katznelson-Tzafriri theorem by establishing both a version of the result valid for bounded representations of a large class of abelian semigroups and a quantified version for contractive representations. The paper concludes with an outline of an improved version of the Katznelson-Tzafriri theorem for individual orbits, whose validity extends even to certain unbounded representations. [ABSTRACT FROM AUTHOR]

This paper studies global existence of cross diffusion systems on 2-dimensional domains. We assume a quadratic growth on the reaction part and show that a solution exists globally if and only if its total reaction energy does not blow up in finite time. Applications to cross diffusion systems with Lotka-Volterra reaction are presented. [ABSTRACT FROM AUTHOR]

In this paper we prove a characterization theorem on the existence of one non-zero strong solution for elliptic equations defined on the Sierpiński gasket. More generally, the validity of our result can be checked studying elliptic equations defined on self-similar fractal domains whose spectral dimension ν ∈ (0, 2). Our theorem can be viewed as an elliptic version on fractal domains of a recent contribution obtained in a recent work of Ricceri for a two-point boundary value problem. [ABSTRACT FROM AUTHOR]

This paper studies randomized quasi-Monte Carlo (QMC) sampling for discontinuous integrands having singularities along the boundary of the unit cube [0, 1]d. Both discontinuities and singularities are extremely common in the pricing and hedging of financial derivatives and have a tremendous impact on the accuracy of QMC. It was previously known that the root mean square error of randomized QMC is only o(n1/2) for discontinuous functions with singularities. We find that under some mild conditions, randomized QMC yields an expected error of O(n-1/2-1/(4d-2)+ϵ) for arbitrarily small ϵ > 0. Moreover, one can get a better rate if the boundary of discontinuities is parallel to some coordinate axes. As a by-product, we find that the expected error rate attains O(n-1+ϵ) if the discontinuities are QMC-friendly, in the sense that all the discontinuity boundaries are parallel to coordinate axes. The results can be used to assess the QMC accuracy for some typical problems from financial engineering. [ABSTRACT FROM AUTHOR]

When using a finite difference method to solve an initial-boundaryvalue problem, the truncation error is often of lower order at a few grid points near boundaries than in the interior. Normal mode analysis is a powerful tool to analyze the effect of the large truncation error near boundaries on the overall convergence rate, and has been used in many research works for different equations. However, existing work only concerns problems in one space dimension. In this paper, we extend the analysis to problems in two space dimensions. The two dimensional analysis is based on a diagonalization procedure that decomposes a two dimensional problem to many one dimensional problems of the same type. We present a general framework of analyzing convergence for such one dimensional problems, and explain how to obtain the result for the corresponding two dimensional problem. In particular, we consider two kinds of truncation errors in two space dimensions: the truncation error along an entire boundary, and the truncation error localized at a few grid points close to a corner of the computational domain. The accuracy analysis is in a general framework, here applied to the second order wave equation. Numerical experiments corroborate our accuracy analysis [A basic convergence result for conforming adaptive finite elements, Math. Models Methods Appl. Sci., 2008, 18(5), 707-737]. [ABSTRACT FROM AUTHOR]

In this paper we develop an hp-adaptive procedure for the numerical solution of general second-order semilinear elliptic boundary value problems, with possible singular perturbation. Our approach combines both adaptive Newton schemes and an hp-version adaptive discontinuous Galerkin finite element discretisation, which, in turn, is based on a robust hp-version a posteriori residual analysis. Numerical experiments underline the robustness and reliability of the proposed approach for various examples. [ABSTRACT FROM AUTHOR]

We study the convexity of the area functional for the graphs of maps with respect to the singular values of their differentials. Suppose that f is a solution to the Dirichlet problem for the minimal surface system and the area functional is convex at f. Then the graph of f is stable. New criteria for the stability of minimal graphs in any co-dimension are derived in the paper by this method. Our results in particular generalize the co-dimension one case, and improve the condition in the 2003 paper of the first author and M.-T. Wang from ∣Λ² df∣ ≤ 1/p-1 to ∣ Λ² df∣ ≤ 1/√ p-1 , where p is an upper bound of the rank of df , and the condition in the 2008 paper of the first author and M.-T. Wang from √det(I + (df )T df ) ≤ 43/40 to √det(I + (df )T df ) ≤ 2. [ABSTRACT FROM AUTHOR]

In this paper, we establish existence and uniqueness of solutions of a class of mixed boundary value problems for elliptic operators with singular coefficients. Our approach is probabilistic. The theory of Dirichlet forms plays an important role. [ABSTRACT FROM AUTHOR]

This paper is concerned with an inverse random source problem for the one-dimensional stochastic Helmholtz equation, which is to reconstruct the statistical properties of the random source function from boundary measurements of the radiating random electric field. Although the emphasis of the paper is on the inverse problem, we adapt a computationally more efficient approach to study the solution of the direct problem in the context of the scattering model. Specifically, the direct model problem is equivalently formulated into a two-point spatially stochastic boundary value problem, for which the existence and uniqueness of the pathwise solution is proved. In particular, an explicit formula is deduced for the solution from an integral representation by solving the two-point boundary value problem. Based on this formula, a novel and efficient strategy, which is entirely done by using the fast Fourier transform, is proposed to reconstruct the mean and the variance of the random source function from measurements at one boundary point, where the measurements are assumed to be available for many realizations of the source term. Numerical examples are presented to demonstrate the validity and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence and multiplicity of positive periodic solutions for first-order vector differential equations. By using the Leray-Schauder alternative theorem and the Kransnosel'skii fixed point theorem, we show that the differential equations under the periodic boundary value conditions have at least two positive periodic solutions. [ABSTRACT FROM AUTHOR]

This paper deals with the fully parabolic 1d chemotaxis system with diffusion 1/(1 + u). We prove that the above mentioned nonlinearity, despite being a natural candidate, is not critical. It means that for such a diffusion any initial condition, independently on the magnitude of mass, generates the global-in-time solution. In view of our theorem one sees that the one-dimensional Keller-Segel system is essentially different from its higher-dimensional versions. In order to prove our theorem we establish a new Lyapunov-like functional associated to the system. The information we gain from our new functional (together with some estimates based on the well-known classical Lyapunov functional) turns out to be rich enough to establish global existence for the initial-boundary value problem. [ABSTRACT FROM AUTHOR]

In this paper, we prove uniqueness of solutions of mean field equations with general boundary conditions for the critical and subcritical total mass regime, extending the earlier results for null Dirichlet boundary condition. The proof is based on new Bol's inequalities for weak radial solutions obtained from rearrangement of the solutions. [ABSTRACT FROM AUTHOR]

This paper concerns the asymptotic behavior of solutions to a semilinear parabolic equation with boundary degeneracy. It is proved that for the problem in a bounded domain with a homogeneous boundary condition, there exist both nontrivial global and blowing-up solutions if the degeneracy is not strong, while the nontrivial solution must blow up in a finite time if the degeneracy is strong enough. For the problem in an unbounded domain, blowing-up theorems of Fujita type are established and the critical Fujita exponent is finite in the not strong degeneracy case, while infinite in the other case. Furthermore, the behavior of solutions at the degenerate point is studied, and it is shown that for the nontrivial initial datum vanishing at the degenerate point, the solution always vanishes at the degenerate point if the degeneracy is strong enough, while never if it is not. [ABSTRACT FROM AUTHOR]

For the contractive iterated function system Skz = e2πik/m + ρ(z - e2πik/m) with 0 < ρ < 1, k = 0, · · ·, m - 1, we let K ⊂ C be the attractor, and let μ be a self-similar measure defined by μ = 1/m Σk=0m-1 μoSk-1. We consider the Cauchy transform F of μ. It is known that the image of F at a small neighborhood of the boundary of K has very rich fractal structure, which is coined the Cantor boundary behavior. In this paper, we investigate the behavior of F away from K; it has nice geometry and analytic properties, such as univalence, starlikeness and convexity. We give a detailed investigation for those properties in the general situation as well as certain classical cases of self-similar measures. [ABSTRACT FROM AUTHOR]

In this paper we present a unified approach to establish the existence of renormalized solutions and a comparison result for a class of nonuniformly parabolic initial-boundary value problems. As a consequence, the uniqueness of renormalized solutions and the equivalence between entropy and renormalized solutions for such equations are obtained. The results extend the well-posedness results for the classical p-Laplacian type equations to a larger class of non-linear elliptic and parabolic PDEs including the nearly linear growth operators. [ABSTRACT FROM AUTHOR]

Slice regular functions have been extensively studied over the past decade, but much less is known about their boundary behavior. In this paper, we initiate the study of Julia theory for slice regular functions. More specifically, we establish the quaternionic versions of the Julia lemma, the Julia-Carathéodory theorem, the boundary Schwarz lemma, and the Burns-Krantz rigidity theorem for slice regular self-mappings of the open unit ball B and of the right half-space H+. Our quaternionic boundary Schwarz lemma involves a Lie bracket reflecting the non-commutativity of quaternions. Together with some explicit examples, it shows that the slice derivative of a slice regular self-mapping of B at a boundary fixed point is not necessarily a positive real number, in contrast to that in the complex case, meaning that its commonly believed version turns out to be totally wrong. [ABSTRACT FROM AUTHOR]

In this paper we study the convergence of the perfectly matched layer (PML) method for solving the time harmonic elastic wave scattering problems. We introduce a simple condition on the PML complex coordinate stretching function to guarantee the ellipticity of the PML operator. We also introduce a new boundary condition at the outer boundary of the PML layer which allows us to extend the reflection argument of Bramble and Pasciak to prove the stability of the PML problem in the truncated domain. The exponential convergence of the PML method in terms of the thickness of the PML layer and the strength of PML medium property is proved. Numerical results are included. [ABSTRACT FROM AUTHOR]

The paper is a comprehensive study of Lp and Schauder estimates for higher-order divergence type parabolic systems with discontinuous coefficients on a half space and cylindrical domains with the conormal derivative boundary conditions. For the Lp estimates, we assume that the leading coefficients are only bounded and measurable in the t variable and have vanishing mean oscillations (VMOx) with respect to x. We also prove the Schauder estimates in two situations: the coefficients are Holder continuous only in the x variable; the coefficients are Hölder continuous in the t variable as well on the lateral boundary. [ABSTRACT FROM AUTHOR]

The main result of this paper is a characterization of the minimal surface hull of a compact set K in ℝ³ by sequences of conformal minimal discs whose boundaries converge to K in the measure theoretic sense, and also by 2-dimensional minimal currents which are limits of Green currents supported by conformal minimal discs. Analogous results are obtained for the null hull of a compact subset of C³. We also prove a null hull analogue of the Alexander-Stolzenberg-Wermer theorem on polynomial hulls of compact sets of finite linear measure, and a polynomial hull version of classical Bochner's tube theorem. [ABSTRACT FROM AUTHOR]

In this paper, making use of the method developed by Catlin, we study the L²-estimate for the ∂-equation on a lunar manifold with mixed boundary conditions. [ABSTRACT FROM AUTHOR]

We propose a new numerical 2-point flux for a quasilinear convection–diffusion equation. This numerical flux is shown to be an approximation of the numerical flux derived from the solution of a two-point Dirichlet boundary value problem for the projection of the continuous flux onto the line connecting neighboring collocation points. The later approach generalizes an idea first proposed by Scharfetter and Gummel [IEEE Trans. Electron Devices 16 (1969), pp. 64–77] for linear drift-diffusion equations. We establish first that the new flux satisfies sufficient properties ensuring the convergence of the associate finite volume scheme, while respecting the maximum principle. Then, we pay attention to the long time behavior of the scheme: we show relative entropy decay properties satisfied by the new numerical flux as well as by the generalized Scharfetter-Gummel flux. The proof of these properties uses a generalization of some discrete (and continuous) log-Sobolev inequalities. The corresponding decay of the relative entropy of the continuous solution is proved in the appendix. Some 1D numerical experiments confirm the theoretical results. [ABSTRACT FROM AUTHOR]

We present an error analysis for an unconditionally energy stable, fully discrete finite difference scheme for the Cahn-Hilliard-Hele-Shaw equation, a modified Cahn-Hilliard equation coupled with the Darcy flow law. The scheme, proposed by S. M. Wise, is based on the idea of convex splitting. In this paper, we rigorously prove first order convergence in time and second order convergence in space. Instead of the (discrete) Ls∞ (0, T;Lh²)∩Ls²(0, T;Hh²) error estimate, which would represent the typical approach, we provide a discrete Ls∞ (0, T;Hh¹)∩Ls² (0, T;Hh³) error estimate for the phase variable, which allows us to treat the nonlinear convection term in a straightforward way. Our convergence is unconditional in the sense that the time step s is in no way constrained by the mesh spacing h. This is accomplished with the help of an Ls²(0, T;Hh³) bound of the numerical approximation of the phase variable. To facilitate both the stability and convergence analyses, we establish a finite difference analog of a Gagliardo-Nirenberg type inequality. [ABSTRACT FROM AUTHOR]

In this paper we show that every sufficiently large family of convex bodies in the plane has a large subfamily in convex position provided that the number of common tangents of each pair of bodies is bounded and every subfamily of size five is in convex position. (If each pair of bodies has at most two common tangents it is enough to assume that every triple is in convex position, and likewise, if each pair of bodies has at most four common tangents it is enough to assume that every quadruple is in convex position.) This confirms a conjecture of Pach and Tóth and generalizes a theorem of Bisztriczky and Fejes Tóth. Our results on families of convex bodies are consequences of more general Ramsey-type results about the crossing patterns of systems of graphs of continuous functions ƒ : [0, 1] → R. On our way towards proving the Pach-Tóth conjecture we obtain a combinatorial characterization of such systems of graphs in which all subsystems of equal size induce equivalent crossing patterns. These highly organized structures are what we call regular systems of paths, and they are natural generalizations of the notions of cups and caps from the famous theorem of Erdős and Szekeres. The characterization of regular systems is combinatorial and introduces some auxiliary structures which may be of independent interest. [ABSTRACT FROM AUTHOR]

This paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one, reflected at its infimum. First we derive the forward equation using the theory of sun-dual semigroups. The resulting forward equation is a boundary value problem on the positive half-line that involves a negative Riemann-Liouville fractional derivative in space, and a fractional reflecting boundary condition at the origin. Then we apply numerical methods to explicitly compute the transition density of this space-inhomogeneous Markov process, for any starting point, to any desired degree of accuracy. Finally, we discuss an application to fractional Cauchy problems, which involve a positive Caputo fractional derivative in time. [ABSTRACT FROM AUTHOR]

In this paper we consider a minimization problem for the functional.... [ABSTRACT FROM AUTHOR]

In this paper, we consider the equations of linear elasticity in an exterior domain. We exhibit artificial boundary conditions on a circle, which lead to a non-coercive second order boundary value problem. In the particular case of an axisymmetric geometry, explicit computations can be performed in Fourier series proving the well-posedness except for a countable set of parameters. A perturbation argument allows us to consider near-circular domains. We complete the analysis by some numerical simulations. [ABSTRACT FROM AUTHOR]