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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, the authors consider the boundary value problem [ (\textup{E}) \qquad\qquad\quad x^{(2m)}(t)+(-1)^{m+1} f(x(t))=0, \quad 0

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]

In this paper, we deal with a class of semilinear elliptic equations in a bounded domain $\Omega\subset\mathbb{R}^N$, $N\geq 3$, with $C^{1,1}$ boundary. Using a new fixed point result of the Krasnoselskii type for the sum of two operators, an existence principle of strong solutions is proved. We give two examples where the nonlinearity can be critical. [ABSTRACT FROM AUTHOR]

We consider the nonlocal diffusion equation \[u_t-u_{xx}=u^p(t,x_0(t)),\] on the space interval $(0,1)$, with Dirichlet boundary conditions. It is known that if the curve $x_0(t)$ remains in a compact subset of $(0,1)$ for all times, then blow-up cannot occur in infinite time. The aim of this paper is to show that the assumption on $x_0$ is sharp: for a large class of functions $x_0(t)$ approaching the boundary as $t\to\infty$, blow-up in infinite time does occur for certain initial data. Moreover, the asymptotic behavior of the corresponding solution is precisely estimated and more general nonlinearities are also considered. [ABSTRACT FROM AUTHOR]

This paper deals with the blow-up properties of the solution to the degenerate nonlinear reaction diffusion equation with nonlocal source $x^{q}u_{t}-(x^{\gamma}u_{x})_{x}=\int_{0}^{a}u^{p}dx$ in $(0,a)\times (0,T)$ subject to the homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution exists globally or blows up in finite time are obtained. Furthermore, it is proved that under certain conditions the blow-up set of the solution is the whole domain. [ABSTRACT FROM AUTHOR]

In this paper we consider the Cauchy problem for the $n$-dimen\-sional Navier-Stokes equations and we prove a regularity criterion for weak solutions involving the summability of the pressure. Related results for the initial-boundary value problem are also presented. [ABSTRACT FROM AUTHOR]

In this paper we study numerical blow-up sets for semidicrete approximations of the heat equation with nonlinear boundary conditions. We prove that the blow-up set either concentrates near the boundary or is the whole domain. [ABSTRACT FROM AUTHOR]

A nonlinear convection-diffusion equation with boundary conditions that conserve the spatial integral of the solution is considered. Previous results on finite-time blowup of solutions and on decay of solutions to the corresponding Cauchy problem were based on the assumption that the nonlinearity obeyed a power law. In this paper, it is shown that assumptions on the growth rate of the nonlinearity, which take the form of weak superquadraticity and strong superlinearity criteria, are sufficient to imply that a large class of nonnegative solutions blow up in finite time. [ABSTRACT FROM AUTHOR]

Consider an independent random cascade acting on the positive Borel measures defined on the boundary of a Galton-Watson tree. Assuming an offspring distribution with finite moments of all orders, J. Peyrière computed the fine scale structure of an independent random cascade on Galton-Watson trees. In this paper we use developments in the cascade theory to relax and clarify the moment assumptions on the offspring distribution. Moreover a larger class of initial measures is covered and, as a result, it is shown that it is the Hölder exponent of the initial measure which is the critical parameter in the Peyrière theory. [ABSTRACT FROM AUTHOR]

In this paper we establish the existence of two nonnegative solutions to singular $(n,p)$ and singular $(p,n-p)$ focal boundary value problems. Our nonlinearity $f(t,y)$ may be singular at $y=0$, $t=0$ and/or $t=1$. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence and uniqueness of a solution for a class of fourth-order elliptic boundary value problems. The existence of a solution is proven by the method of upper and lower solutions without any monotone nondecreasing or nonincreasing property of the nonlinear function. Sufficient conditions for the uniqueness of a solution and some techniques for the construction of upper and lower solutions are given. All the existence and uniqueness results are directly applicable to fourth-order two-point boundary value problems. [ABSTRACT FROM AUTHOR]

In this paper we provide new necessary and sufficient conditions for a general class of eigenvalue completion problems. [ABSTRACT FROM AUTHOR]

In this paper, based on of the concept $q_0\in H_0(p,(0,1),\alpha,\beta)$, which is a generalized form of the first resonant point $\pi^2$ to the Picard problem $x''+\lambda x=0$, $x(0)=x(1)=0$, we study the solvability of second-order Sturm-Liouville boundary value problems at resonance $(p(t)x')'+q_0(t)x+g(t,x)=h(t)$, $x(0){\cos \alpha}-p(0)x'(0)\sin \alpha=0$, $x(1)\cos \beta-p(1)x'(1)\sin \beta=0$, and improve the previous results about problems $x''+\pi^2x+g(t,x)=h(t),x(0)=x(1)=0$ derived by Chaitan P. Gupta, R. Iannacci and M. N. Nkashama, and Ma Ruyun, respectively. [ABSTRACT FROM AUTHOR]

We prove estimates of Gaussian type for the kernel of the semigroup associated to the operator mΔ, where m is a positive function which may vanish at the boundary (and thus the operator may not be strongly elliptic). No regularity conditions either on the boundary of the domain or on the function m are posed. The optimality of the growth condition on m is discussed. [ABSTRACT FROM AUTHOR]

Using Szulkin's critical point theory, we prove that the relativistic forced pendulum with periodic boundary value conditions Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. has at least two solutions not differing by a multiple of 2π for any continuous Function h : [0, T] → &#x211D with Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. And any μ ≠ 0. The existence of at least one solution has been recently proved by Brezis and Mawhin. [ABSTRACT FROM AUTHOR]