Sharp Moser–Trudinger inequalities for the Laplacian without boundary conditions

We derive a sharp Moser–Trudinger inequality for the borderline Sobolev imbedding of $W^{2,n/2}(B_n)$ into the exponential class, where $B_n$ is the unit ball of ${\mathbb{R}}^n$. The corresponding sharp results for the spaces $W_0^{d,n/d}(\Omega )$ are well known, for general domains Ω, and are due to Moser and Adams. When the zero boundary condition is removed the only known results are for $d=1$ and are due to Chang–Yang, Cianchi and Leckband. The proof of our result is based on a new integral representation formula for the “canonical” solution of the Poisson equation on the ball, that is, the unique solution of the equation $\mathrm{\Delta }u=f$ which is orthogonal to the harmonic functions on the ball. The main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation, expressed in terms of its distribution function.     

Continuous dependence for the convective Brinkman–Forchheimer equations

In this article, we have considered the convective Brinkman–Forchheimer equations with Dirichlet's boundary conditions. The continuous dependence of solutions on the Forchheimer coefficient in H ¹ norm is proved.     

Non-uniform dependence on initial data for the μ − b equation

This paper is concerned with the non-uniform dependence on initial data for the μ?b equation on the circle. Using the approximate solution method, we construct two solution sequences to show that the data-to-solution map of the Cauchy problem of the μ?b equation is not uniformly continuous in ${H^s(\mathbb{S})}$ .     

Fast transforms for high order boundary conditions in deconvolution problems

We study strategies to increase the precision in deconvolution models, while maintaining the complexity of the related numerical linear algebra procedures (matrix-vector product, linear system solution, computation of eigenvalues, etc.) of the same order of the celebrated fast Fourier transform. The key idea is the choice of a suitable functional basis to represent signals and images. Starting from an analysis of the spectral decomposition of blurring matrices associated to the antireflective boundary conditions introduced in Serra Capizzano (SIAM J. Sci. Comput. 25(3):1307–1325, 2003), we extend the model for preserving polynomials of higher degree and fast computations also in the nonsymmetric case.     

The heat kernel of the Laplacian defined on a uniform grid

We adapt a method originally developed by E.B. Davies for second order elliptic operators to obtain an upper heat kernel bound for the Laplacian defined on a uniform grid on the plane.     

Eigenvalues of second-order linear equations with coupled boundary conditions on time scales

In this paper we consider coupled boundary value problems for second-order linear equations on time scales. By properties of eigenvalues of the Dirichlet boundary value problem and some oscillation results, existence of eigenvalues of this boundary value problem is proved, numbers of their eigenvalues are calculated, and their relationships are obtained. These results not only extend the existing ones of coupled boundary value problems for second-order difference equations but also contain more complicated time scales.     

Cosine families generated by second-order differential operators on W1,1(0, 1) with generalized Wentzell boundary conditions

Using the abstract framework [Bátkai, A. and Engel, K.-J., 2004, Abstract wave equations with generalized Wentzell boundary conditions. Journal of Differential Equations, 207, 1–20.] we show that certain second-order differential operators with generalized Wentzell boundary conditions generate cosine families and hence also analytic semigroups on W^1,1(0,1). This complements the main result [Favini, A., Ruiz Goldstein, G., Goldstein, J.A., Obrecht, E. and Romanelli, S., 2003, General Wentzell boundary conditions and analytic semigroups on W^{1, p} (0,1). Applicable Analysis, 82, 927–935.] on the generation of an analytic semigroup by the second derivative with generalized Wentzell boundary conditions on W^{1, p} (0,?1) for 1<p<∞.     

On the Wave Equation Associated to the Hermite and the Twisted Laplacian

The dispersive properties of the wave equation u _tt +Au=0 are considered, where A is either the Hermite operator −Δ+|x|² or the twisted Laplacian −(∇ _x −iy)²/2−(∇ _y +ix)²/2. In both cases we prove optimal L ¹−L ^∞ dispersive estimates. More generally, we give some partial results concerning the flows exp (itL ^ν ) associated to fractional powers of the twisted Laplacian for 0<ν<1.     

Hydrodynamic limit for the Ginzburg-Landau ∇φ interface model with boundary conditions

Hydrodynamic limit for the Ginzburg-Landau interface model was established in [6] under periodic boundary conditions. This paper studies the same problem on a bounded domain imposing Dirichlet boundary conditions. A nonlinear partial differential equation of second order with boundary conditions is derived as a macroscopic equation. The main tools are H ^–1-method used in [6] and the higher integrability of gradients in [2].The author was supported by DFG project De 663/2-2 and 2/3 from April, 2001 to March, 2002, and by JSPS reserch fellowship for young scientists from April, 2002. Mathematics Subject Classification (2000): 60K35, 82C24, 35K55     

ζ-determinant and the Quillen determinant on the Grassmannian of elliptic self-adjoint boundary conditions

We announce a proof of the equality of the ζ-determinant of a Dirac operator over an odd-dimensional manifold with boundary with the Quillen determinant section computed in a canonical trivialization over the Grassmannian of elliptic self-adjoint boundary conditions.     

Comparison theorems for nonlinear evolution equations and their application to the investigation of the dependence of the solutions of the equation ut=δϕ(u) ON ϕ and on the boundary conditions

One obtains estimates of the form, whereu. are generalized solutions of the equationsdu/dt=Au, du/dt=Bu whileA, B are non-linear,m-dissipative operators in a Banach space, and there exists an operatorP:D(A)D(B), such thatPw · W+BPw –Aw, uniformly on some setw. These results are applied to the investigation of the dependence of the solutions of the Cauchy, Dirichlet problems and of the problem with the boundary condition –du/dn=(u) for the equation u₁=(u) on the continuous nondecreasing functions and.Translated from Problemy Matematicheskogo Analiza, No. 9, pp. 183–198, 1984.The author is sincerely grateful to O. A. Ladyzhenskaya and N. N. Ural'tseva for their interest in this paper and for useful discussions.     

On the Navier–Stokes flows for heat-conducting fluids with mixed boundary conditions

We consider a coupled model for steady flows of viscous incompressible heat-conducting fluids with temperature dependent material coefficients in a fixed three-dimensional open cylindrical channel. We introduce the Banach spaces X and Y to be the space of possible solutions of this problem and the space of its data, respectively. We show that the corresponding operator of the problem acting between X and Y is Fréchet differentiable. Applying the local diffeomorphism theorem we get the local solvability results for a variational formulation.     

Critical Landscape Topology for Optimization on the Symplectic Group

Optimization problems over compact Lie groups have been studied extensively due to their broad applications in linear programming and optimal control. This paper analyzes an optimization problem over a noncompact symplectic Lie group Sp(2N,ℝ), i.e., minimizing the Frobenius distance from a target symplectic transformation, which can be used to assess the fidelity function over dynamical transformations in classical mechanics and quantum optics. The topology of the set of critical points is proven to have a unique local minimum and a number of saddlepoint submanifolds, exhibiting the absence of local suboptima that may hinder the search for ultimate optimal solutions. Compared with those of previously studied problems on compact Lie groups, such as the orthogonal and unitary groups, the topology is more complicated due to the significant nonlinearity brought by the incompatibility of the Frobenius norm with the pseudo-Riemannian structure on the symplectic group.     

Fourth order singular p-Laplacian differential equations with integral boundary conditions

In this paper, we investigate the existence of positive solutions for fourth order singular p-Laplacian differential equations with integral boundary conditions and non-monotonic function terms. Firstly, we establish a comparison theorem, then we define a partial ordering in E ₀ and construct lower and upper solutions to give a necessary and sufficient condition for the existence of C ²[0,1] as well as pseudo-C ³[0,1] positive solutions. Our nonlinearity f(t,x,y) may be singular at x=0, y=0, t=0 and t=1. Finally, we give some the dual results for the other cases of fourth order singular integral boundary value problems and an example to demonstrate the corresponding main results.     

On the solvability of the tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions

We study the existence of a regular (classical) solution of the Tricomi problem for the Lavrent’ev-Bitsadze equation with mixed boundary conditions. We find conditions under which the homogeneous problem has only the zero solution and give an example in which the homogeneous Tricomi problem has a nonzero solution. We also study the solvability of the inhomogeneous Tricomi problem.     

Selfadjoint Schrödinger operators on the half-space with compactly supported Robin boundary conditions

Schrödinger differential operators on a half-space with compactly supported Robin boundary conditions are studied. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)