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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 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]

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 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]

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]

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 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]

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]

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 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]

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]

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]

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]

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]

Let $M$ be an invariant subspace of the Bergman space $L_a^2(mathbb {D})$ and $S_M$ be the compression of the coordinate multiplication operator $M_z$ to the co-invariant subspace $L_a^2(mathbb {D})ominus M$. The present paper determines when the identity representation of $C^*(S_M)$ is a boundary representation for the Banach subalgebra $mathcal {B}(S_M)$. The paper also considers boundary representations on the co-invariant subspaces of $L_a^2(mathbb {B}_n)$. [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]

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]

In this paper we establish a gap phenomenon for immersed surfaces with arbitrary codimension, topology and boundaries that satisfy one of a family of systems of fourth-order anisotropic geometric partial differential equations. Examples include Willmore surfaces, stationary solitons for the surface diffusion flow, and biharmonic immersed surfaces in the sense of Chen. On the boundary we enforce either umbilic or flat boundary conditions: that the tracefree second fundamental form and its derivative or the full second fundamental form and its derivative vanish. For the umbilic boundary condition we prove that any surface with small L²-norm of the tracefree second fundamental form or full second fundamental form must be totally umbilic, that is, a piece of a round sphere or flat plane. We prove that the stricter smallness condition allows consideration for a broader range of differential operators. For the flat boundary condition we prove the same result with weaker hypotheses, allowing more general operators, and a stronger conclusion: only a piece of a flat plane is allowed. The method used relies only on the smallness assumption and thus holds without requiring the imposition of additional symmetries. The result holds in the class of surfaces with any genus and irrespective of the number or shape of the boundaries. [ABSTRACT FROM AUTHOR]

In this paper, we establish boundary Holder gradient estimates for solutions to the linearized Monge-Ampere equations with Lp (n < p ≤ ∞) right-hand side and C¹, γ boundary values under natural assumptions on the domain, boundary data and the Monge-Ampere measure. These estimates extend our previous boundary regularity results for solutions to the linearized Monge-Ampere equations with bounded right-hand side and C1,1 boundary data. [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]

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 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]

This paper gives an asymptotic expansion for the solution of the boundary problem with respect to the Laplace-Beltrami operator. We also consider some examples in the domain whose boundary is real ellipsoid, where the boundary problem does not have a $ C^n$ [ABSTRACT FROM AUTHOR]

In this paper, a solution of the Neumann problem on a half-space for a slowly growing continuous boundary function is constructed by the generalized Neumann integral with this boundary function. The relation between this particular solution and certain general solutions is discussed. A solution of the Neumann problem for any continuous boundary function is also given explicitly by the Neumann integral with the generalized Neumann kernel depending on this boundary function. [ABSTRACT FROM AUTHOR]

In this paper we investigate the existence of positive solutions of the $q$-difference equation $-D_q^2u(t)=a(t) f(u(t))$ with some boundary conditions by applying a fixed point theorem in cones. [ABSTRACT FROM AUTHOR]

For some $ggeq 3$, let $Gamma $ be a finite index subgroup of the mapping class group of a genus $g$ surface (possibly with boundary components and punctures). An old conjecture of Ivanov says that the abelianization of $Gamma $ should be finite. In the paper we prove two theorems supporting this conjecture. For the first, let $T_x$ denote the Dehn twist about a simple closed curve $x$. For some $ngeq 1$, we have $T_x^nin Gamma $. We prove that $T_x^n$ is torsion in the abelianization of $Gamma $. Our second result shows that the abelianization of $Gamma $ is finite if $Gamma $ contains a ``large chunk'' (in a certain technical sense) of the Johnson kernel, that is, the subgroup of the mapping class group generated by twists about separating curves. [ABSTRACT FROM AUTHOR]

In this paper, we extend the complex-valued Ray-Singer torsion, introduced by Burghelea-Haller, to compact connected Riemannian manifolds with boundary. We also compare it with the refined analytic torsion. [ABSTRACT FROM AUTHOR]

In this paper we study the incompressible Navier-Stokes equations in ${mathbb {T}}^3=[0,1]^3$ with periodic boundary conditions. We show that a weak solution of the Navier-Stokes equations that is small in $L^{infty }(0,T;Phi (2))$ is also smooth. We also give elementary proofs of some classical regularity results for the Navier-Stokes equations involving the Sobolev space $H^{frac {1}{2}}({mathbb {T}}^3)$. [ABSTRACT FROM AUTHOR]

In this paper, we study stable constant mean curvature $H$ surfaces in $mathbb {R}^3$. We prove that, in such a surface, the distance from a point to the boundary is less than or equal to $pi /(2H)$. This upper bound is optimal and is extended to stable constant mean curvature surfaces in space forms. [ABSTRACT FROM AUTHOR]

This paper concerns minimization and maximization of the first eigenvalue in problems involving the $p$-Laplacian, under homogeneous Dirichlet boundary conditions. Physically, in the case of $N=2$ and $p$ close to $2$, our equation models the vibration of a nonhomogeneous membrane $Omega $ which is fixed along the boundary. Given several materials (with different densities) of total extension $|Omega |$, we investigate the location of these material inside $Omega $ so as to minimize or maximize the first mode in the vibration of the membrane. [ABSTRACT FROM AUTHOR]