Parabolic Equations with Partially BMO Coefficients and Boundary Value Problems in Sobolev Spaces with Mixed Norms

Second order parabolic equations in Sobolev spaces with mixed norms are studied. The leading coefficients (except a ¹¹) are measurable in time and one spatial variable, and have small BMO (bounded mean oscillation) semi-norms as functions of the other spatial variables. The coefficient a ¹¹ is measurable in time and have a small BMO semi-norm in the spatial variables. The unique solvability of equations in the whole space is established. Then this result is applied to solving Dirichlet and Neumann boundary value problems for parabolic equations defined on a half-space or on a bounded domain.     

On the existence of solutions to the Navier–Stokes–Poisson equations of a two‐dimensional compressible flow

In this paper, we consider the Navier–Stokes–Poisson equations for compressible, barotropic flow in two space dimensions. We introduce useful tools from the theory of Orlicz spaces. Then we prove the existence of globally defined finite energy weak solutions for the pressure satisfying p(?)=a?log^d (?) for large ?. Here d>1 and a>0. Copyright © 2006 John Wiley & Sons, Ltd.     

On the spline collocation method for the single layer equation related to time-fractional diffusion

The boundary element spline collocation method is studied for the time-fractional diffusion equation in a bounded two-dimensional domain. We represent the solution as the single layer potential which leads to a Volterra integral equation of the first kind. We discretize the boundary integral equation with the spline collocation method on uniform meshes both in spatial and time variables. In the stability analysis we utilize the Fourier analysis technique developed for anisotropic pseudodifferential equations. We prove that the collocation solution is quasi-optimal under some stability condition for the mesh parameters. We have to assume that the mesh parameter in time satisfies (h_t=c h^\frac2a)(h_t=c h^{\frac{2}{\alpha}}), where (h) is the spatial mesh parameter.     

On reflecting diffusion processes and Skorokhod decompositions

Summary LetG be ad-dimensional bounded Euclidean domain, H¹ (G) the set off in L²(G) such that f (defined in the distribution sense) is in L²(G). Reflecting diffusion processes associated with the Dirichlet spaces (H¹(G), ) on L²(G, dx) are considered in this paper, where A=(a^ij is a symmetric, bounded, uniformly ellipticd×d matrix-valued function such thata ^ij H¹(G) for eachi,j, and H¹(G) is a positive bounded function onG which is bounded away from zero. A Skorokhod decomposition is derived for the continuous reflecting Markov processes associated with (H¹(G), ) having starting points inG under a mild condition which is satisfied when G has finite (d–1)-dimensional lower Minkowski content.     

Szeg? theorem, Carathéodory domains, and boundedness of calculating functionals

Suppose that G is a bounded simply connected domain on the plane with boundary í\subseteq òln( \tfracdm_a dw ) dw = - ￥\int {\ln \left( {\tfrac{{d\mu _a }}{{d\omega }}} \right)} d\omega = - \infty     

Persistence of bounded solutions of parabolic equations under domain perturbation

The aim of this paper is to study the behavior of bounded solutions of parabolic equations on the whole real line under perturbation of the underlying domain. We give the convergence of bounded solutions of linear parabolic equations in the L ² and the L ^p -settings. For the L ^p -theory, we also prove the H?lder regularity of bounded solutions with respect to time. In addition, we study the persistence of a class of bounded solutions which decay to zero at t → ±∞ of semilinear parabolic equations under domain perturbation.     

On the existence of solutions to the Navier–Stokes equations of a two‐dimensional compressible flow

We consider the Navier–Stokes equations for compressible, barotropic flow in two space dimensions. We introduce useful tools from the theory of Orlicz spaces. Then we prove the existence of globally defined finite energy weak solutions for the pressure satisfying p(?) = a?log^d(?) for large ?. Here d>1 and a > 0. Copyright © 2003 John Wiley & Sons, Ltd.     

On Bateman's method for second kind integral equations

Summary Under suitable conditions, we prove the convergence of the Bateman method for integral equations defined over bounded domains inR ^d ,d1. The proof makes use of Hilbert space methods, and requires the integral operator to be non-negative definite. For one-dimensional integral equations over finite intervals, estimated rates of convergence are obtained which depend on the smoothness of the kernel, but are independent of the inhomogeneous term. In particular, for aC kernel andn reasonably spaced Bateman points, the convergence is shown to be faster than any power of 1/n. Numerical calculations support this result.     

Regularity of weak solutions of Maxwell's equations with mixed boundary‐conditions

In this paper global H^s‐ and L^p‐regularity results for the stationary and transient Maxwell equations with mixed boundary conditions in a bounded spatial domain are proved. First it is shown that certain elements belonging to the fractional‐order domain of the Maxwell operator belong to H^s(Ω) for sufficiently small s > 0. It follows from this regularity result that H^s(Ω) is an invariant subspace of the unitary group corresponding to the homogeneous Maxwell equations with mixed boundary conditions. In the case that a possibly non‐linear conductivity is present a L^p‐regularity theorem for the transient equations is proved. Copyright © 1999 John Wiley & Sons, Ltd.     

Periodic solutions of the Navier�CStokes equations with the inhomogeneous time-dependent boundary data under the general flux condition

We consider the initial boundary value problem to the Navier–Stokes equations in a bounded domain with the inhomogeneous time-dependent data b(t) ? H^1/2(?W){\beta(t) \in H^{1/2}(\partial\Omega)} under the general flux condition. We establish a reproductive property for weak solutions of the Navier–Stokes equations. Here, the reproductive property is regarded as the generalization of the time periodicity. As an application, we can prove the existence of periodic weak solutions.     

Metastability of Nonreversible Random Walks in a Potential Field and the Eyring‐Kramers Transition Rate Formula

We consider nonreversible random walks in a bounded domain of ℝ^d evolving on a potential field. We describe the complete metastable behavior of the random walk among the landscape of valleys, and we derive the Eyring‐Kramers formula for the mean transition time from a metastable set to a stable set.© 2017 Wiley Periodicals, Inc.     

BLO spaces associated with the Ornstein–Uhlenbeck operator

Let (Rn,|⋅|,dγ) be the Gauss measure metric space, where Rn denotes the n-dimensional Euclidean space, |⋅| the Euclidean norm and for all x∈Rn the Gauss measure. In this paper, for any a∈(0,∞), the authors introduce some BLOa(γ) space, namely, the space of functions with bounded lower oscillation associated with a given class of admissible balls with parameter a. Then the authors prove that the noncentered local natural Hardy–Littlewood maximal operator is bounded from BMO(γ) of Mauceri and Meda to BLOa(γ). Moreover, a characterization of the space BLOa(γ), via the local natural maximal operator and BMO(γ), is given. The authors further prove that a class of maximal singular integrals, including the corresponding maximal operators of both imaginary powers of the Ornstein–Uhlenbeck operator and Riesz transforms of any order associated with the Ornstein–Uhlenbeck operator, are bounded from L^∞(γ) to BLOa(γ).     

Boundary Behavior of Harmonic Functions for Truncated Stable Processes

For any α∈(0,2), a truncated symmetric α-stable process in ℝ ^d is a symmetric Lévy process in ℝ ^d with no diffusion part and with a Lévy density given by c|x|^−d−α 1_{|x|<1} for some constant c. In (Kim and Song in Math. Z. 256(1): 139–173, [2007]) we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary Harnack principle is valid for the positive harmonic functions of this process in any bounded convex domain and showed that the Martin boundary of any bounded convex domain with respect to this process is the same as the Euclidean boundary. However, for truncated symmetric stable processes, the boundary Harnack principle is not valid in non-convex domains. In this paper, we show that, for a large class of not necessarily convex bounded open sets in ℝ ^d called bounded roughly connected κ-fat open sets (including bounded non-convex κ-fat domains), the Martin boundary with respect to any truncated symmetric stable process is still the same as the Euclidean boundary. We also show that, for truncated symmetric stable processes a relative Fatou type theorem is true in bounded roughly connected κ-fat open sets. The research of P. Kim is supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-331-C00037). The research of R. Song is supported in part by a joint US-Croatia grant INT 0302167.     

Solvability criterion and representation of solutions of <Emphasis Type="Italic">n</Emphasis>-normal and <Emphasis Type="Italic">d</Emphasis>-normal linear operator equations in a Banach space

On the basis of a generalization of the well-known Schmidt lemma to the case of n-normal and d-normal linear bounded operators in a Banach space, we propose constructions of generalized inverse operators. We obtain criteria for the solvability of linear equations with these operators and formulas for the representation of solutions of these equations.     

Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps

Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere $\mathbb{S}^{d}Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere \mathbbS^d\mathbb{S}^{d} of ℝ ^d+1, and let d(x,y) denote the geodesic distance arccos x⋅y between x,y ? \mathbbS^dx,y\in\mathbb{S}^{d} . Given a spherical cap

B(e,a)={x ? \mathbbSd:d(x,e) ￡ a}    (e ? \mathbbSd, a ? (0,p) is bounded awayfrom p),B(e,\alpha)=\big\{x\in\mathbb{S}^{d}:d(x,e)\leq\alpha\big\}\quad \bigl(e\in\mathbb{S}^{d},\ \alpha\in(0,\pi)\ \mbox{is bounded awayfrom}\ \pi\bigr),      16. Meromorphic representations of products of complex arithmetic progressions      Bruce W. Atkinson 《Aequationes Mathematicae》2012,83(3):295-311 Given a complex arithmetic sequence, a + nd, where a,d ? \mathbbC{a,d \in \mathbb{C}}, d ≠ 0, and n ? \mathbbZ+{n \in \mathbb{Z}^+}, define Pad(a+nd): = a(a+d)?(a+nd){\Pi^a_d(a+nd):= a(a+d)\cdots (a+nd)}. At first Pda{\Pi_d^a} is only defined on the terms of the arithmetic sequence. In this article, Pda{\Pi_d^a} is extended to a meromorphic function on \mathbbC \{-d,-2d,...}{\mathbb{C} \setminus\{-d,-2d,\dots\}} which satisfies the functional equation, Pda(z+d)=(z+d)Pda(z){\Pi_d^a(z+d)=(z+d)\Pi_d^a(z)}. This extension is represented in three ways: in terms of the classical P{\Pi} function; as a limit involving a Pochhammer-type symbol; as an infinite product involving a generalized Euler constant. The infinite product representation leads to a natural Multiplication Formula for the functions Pda{\Pi_d^a}, which, in turn, provides an easy way to prove Gauss’s Multiplication Formula for the Γ function.      17. Convex polytopes whose projection bodies and difference sets are polars      Horst Martini 《Discrete and Computational Geometry》1991,6(1):83-91 In a paper by the author and B. Weissbach it was proved that the projection body and the difference set of ad-simplex (d≥2) are polars. Obviously, ford=2 a convex domain has this property if and only if its difference set is bounded by a so-called Radon curve. A natural question emerges about further classes of convex bodies inR d (d≥3) inducing the mentioned polarity. The aim of this paper is to show that a convexd-polytope (d≥3) is a simplex if and only if its projection body and its difference set are polars.      18. Stationary solution of the Navier-Stokes equations in a 2d bounded domain for incompressible flow with discontinuous density      M. M. Santos 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2002,161(1):661-675 We consider the boundary value problem for the stationary Navier-Stokes equations describing an inhomogeneous incompressible fluid in a two dimensional bounded domain. We show the existence of a weak solution with boundary values for the density prescribed in L￥L^{\infty}.      19. G-convergence and the weak operator topology      Marcus Waurick 《PAMM》2016,16(1):883-884 We show that a bounded sequence (an)n of symmetric d × d-matrix valued functions is G-convergent if and only if ((ι∗︁anι)−1 )n converges in the weak operator topology. Here ι: R(grad0) ↪ L2(Ω)d denotes the (canonical) embedding from the range of the weak gradient grad0 defined on H10(Ω) into L2(Ω)d, where Ω ⊆ ℝd is open and bounded. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)      20. Spectral analysis of Pk Finite Element matrices in the case of Friedrichs–Keller triangulations via Generalized Locally Toeplitz technology      Ryma Imene Rahla  Stefano Serra‐Capizzano  Cristina Tablino‐Possio 《Numerical Linear Algebra with Applications》2020,27(4) In the present article, we consider a class of elliptic partial differential equations with Dirichlet boundary conditions and where the operator is div(?a( x )?·), with a continuous and positive over $\stackrel{￣}{\mathrm{\Omega }}$, Ω being an open and bounded subset of ${\mathbb{R}}^d$, d≥1. For the numerical approximation, we consider the classical ${\mathbb{P}}_k$ Finite Elements, in the case of Friedrichs–Keller triangulations, leading, as usual, to sequences of matrices of increasing size. The new results concern the spectral analysis of the resulting matrix‐sequences in the direction of the global distribution in the Weyl sense, with a concise overview on localization, clustering, extremal eigenvalues, and asymptotic conditioning. We study in detail the case of constant coefficients on Ω=(0,1)2 and we give a brief account in the more involved case of variable coefficients and more general domains. Tools are drawn from the Toeplitz technology and from the rather new theory of Generalized Locally Toeplitz sequences. Numerical results are shown for a practical evidence of the theoretical findings.

Meromorphic representations of products of complex arithmetic progressions

Given a complex arithmetic sequence, a + nd, where a,d ? \mathbbC{a,d \in \mathbb{C}}, d ≠ 0, and n ? \mathbbZ⁺{n \in \mathbb{Z}^+}, define P^a_d(a+nd): = a(a+d)?(a+nd){\Pi^a_d(a+nd):= a(a+d)\cdots (a+nd)}. At first P_d^a{\Pi_d^a} is only defined on the terms of the arithmetic sequence. In this article, P_d^a{\Pi_d^a} is extended to a meromorphic function on \mathbbC \{-d,-2d,...}{\mathbb{C} \setminus\{-d,-2d,\dots\}} which satisfies the functional equation, P_d^a(z+d)=(z+d)P_d^a(z){\Pi_d^a(z+d)=(z+d)\Pi_d^a(z)}. This extension is represented in three ways: in terms of the classical P{\Pi} function; as a limit involving a Pochhammer-type symbol; as an infinite product involving a generalized Euler constant. The infinite product representation leads to a natural Multiplication Formula for the functions P_d^a{\Pi_d^a}, which, in turn, provides an easy way to prove Gauss’s Multiplication Formula for the Γ function.     

Convex polytopes whose projection bodies and difference sets are polars

In a paper by the author and B. Weissbach it was proved that the projection body and the difference set of ad-simplex (d≥2) are polars. Obviously, ford=2 a convex domain has this property if and only if its difference set is bounded by a so-called Radon curve. A natural question emerges about further classes of convex bodies inR ^d (d≥3) inducing the mentioned polarity. The aim of this paper is to show that a convexd-polytope (d≥3) is a simplex if and only if its projection body and its difference set are polars.     

Stationary solution of the Navier-Stokes equations in a 2d bounded domain for incompressible flow with discontinuous density

We consider the boundary value problem for the stationary Navier-Stokes equations describing an inhomogeneous incompressible fluid in a two dimensional bounded domain. We show the existence of a weak solution with boundary values for the density prescribed in L^￥L^{\infty}.     

G-convergence and the weak operator topology

We show that a bounded sequence (a_n)_n of symmetric d × d-matrix valued functions is G-convergent if and only if ((ι^∗︁a_nι)⁻¹ )_n converges in the weak operator topology. Here ι: R(grad₀) ↪ L²(Ω)^d denotes the (canonical) embedding from the range of the weak gradient grad₀ defined on H¹₀(Ω) into L²(Ω)_d, where Ω ⊆ ℝ^d is open and bounded. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spectral analysis of Pk Finite Element matrices in the case of Friedrichs–Keller triangulations via Generalized Locally Toeplitz technology

In the present article, we consider a class of elliptic partial differential equations with Dirichlet boundary conditions and where the operator is div(?a( x )?·), with a continuous and positive over $\stackrel{￣}{\mathrm{\Omega }}$, Ω being an open and bounded subset of ${\mathbb{R}}^d$, d≥1. For the numerical approximation, we consider the classical ${\mathbb{P}}_k$ Finite Elements, in the case of Friedrichs–Keller triangulations, leading, as usual, to sequences of matrices of increasing size. The new results concern the spectral analysis of the resulting matrix‐sequences in the direction of the global distribution in the Weyl sense, with a concise overview on localization, clustering, extremal eigenvalues, and asymptotic conditioning. We study in detail the case of constant coefficients on Ω=(0,1)² and we give a brief account in the more involved case of variable coefficients and more general domains. Tools are drawn from the Toeplitz technology and from the rather new theory of Generalized Locally Toeplitz sequences. Numerical results are shown for a practical evidence of the theoretical findings.