A Remark on the Sobolev Regularity of Classical Solutions to Uniformly Parabolic Equations

We prove that C^2+α,1+α/2 (Q?) solutions of problem (1.6) below are in a subspace H_c^m+2(Q) of H^{m+2,(m+2)/2(Q)} for all m ∈ ?, if f and the coefficients are in H_c^m(Q)∪C^α,α/2 (Q?). We apply this result to obtain global existence of Sobolev solutions to the quasilinear problem (1.26) below.     

Heat Flow and Perimeter in \boldsymbol{{\mathbb{R}}^m}

Let Ω be an open set in Euclidean space ? ^m with finite perimeter ${\mathcal{P}}(\Omega),$ and with m-dimensional Lebesgue measure |Ω|. It was shown by M. Preunkert that if T(t) is the heat semigroup on L ²(? ^m ) then $H_{\Omega}(t):=\int_{\Omega}T(t)\textbf{1}_{\Omega}(x)dx=|\Omega|-\pi^{-1/2}{\mathcal{P}}(\Omega)t^{1/2}+o(t^{1/2}), \ t\downarrow 0$ . H _Ω(t) represents the amount of heat in Ω if Ω is at initial temperature 1 and if ? ^m ???Ω is at initial temperature 0. In this paper we will compare the quantitative behaviour of H _Ω(t) with the usual heat content Q _Ω(t) associated to the Dirichlet heat semigroup on Ω. We analyse the heat content for horn-shaped open sets of the form Ω(α, Σ)?=?{(x, x′)?∈?? ^m : x′?∈?(1?+?x)^???α Σ, x?>?0}, where α?>?0, and where Σ is an open set in ? ^m???1 with finite perimeter in ? ^m???1, which is star-shaped with respect to 0. For m?≥?3 we find that there are four regimes with very different behaviour depending on α, and a further two limiting cases where logarithmic corrections appear.     

Schauder Estimates for the Single Layer Potential in Hydrodynamics

Let Ω be an open set in ?^N(N ? 3), with compact boundary ?Ω of type C^1,α(?(0,1)). We show that the single layer potential Ef, related to the stationary Stokes system on Ω, belongs to C^1,α(?Ω)^N, provided the source density f belongs to C^α(?Ω)^N. In addition, we prove a related estimate of the function E(f)_?Ω and its tangential derivatives.     

Finite volume method for the variational inequalities of first and second kinds

We propose and analyze the finite volume method for solving the variational inequalities of first and second kinds. The stability and convergence analysis are given for this method. For the elliptic obstacle problem, we derive the optimal error estimate in the H¹‐norm. For the simplified friction problem, we establish an abstract H¹‐error estimate, which implies the convergence if the exact solution u∈H¹(Ω) and the optimal error estimate if u∈H^{1 + α}(Ω),0 < α≤2. Copyright © 2015 John Wiley & Sons, Ltd.     

Approximation by herglotz wave functions

By a general argument, it is shown that Herglotz wave functions are dense (with respect to the C^∞(Ω)‐topology) in the space of all solutions to the reduced wave equation in Ω. This is used to provide corresponding approximation results in global spaces (eg. in L2‐Sobolev‐spaces H^m(Ω)) and for boundary data. Copyright © 2004 John Wiley & Sons, Ltd.     

Cohomology of locally q-complete sets in Stein manifolds

We prove that if X is a Stein complex manifold of dimension n and Ω???X a locally q-complete open set in X with q?≤?n?2, then the cohomology groups H ^p (Ω , OΩ) vanish if p?≥?q and OΩ is the sheaf of germs of holomorphic functions on Ω.     

Approximation by Maxwell–Herglotz‐fields

By a general argument, it is shown that Maxwell–Herglotz‐fields are dense (with respect to the C^∞(Ω)‐topology) in the space of all solutions to Maxwell's equations in Ω. This is used to provide corresponding approximation results in global spaces (e.g. in L²‐Sobolev‐spaces H^m(Ω)) and for boundary data. Proofs are given within the framework of generalized Maxwell's equations using differential forms. Copyright © 2004 John Wiley & Sons, Ltd.     

The Dirichlet problem for the minimal surface system in arbitrary dimensions and codimensions

Let Ω be a bounded C² domain in ?ⁿ and ? ?Ω → ?^m be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map f : Ω → ?^m with f|_?Ω = ? and with the graph of f a minimal submanifold in ?^n+m. For m = 1, the Dirichlet problem was solved more than 30 years ago by Jenkins and Serrin [12] for any mean convex domains and the solutions are all smooth. This paper considers the Dirichlet problem for convex domains in arbitrary codimension m. We prove that if ψ : ¯Ω → ?^m satisfies 8nδ sup_Ω |D²ψ| + √2 sup_?Ω |Dψ| < 1, then the Dirichlet problem for ψ|_?Ω is solvable in smooth maps. Here δ is the diameter of Ω. Such a condition is necessary in view of an example of Lawson and Osserman [13]. In order to prove this result, we study the associated parabolic system and solve the Cauchy‐Dirichlet problem with ψ as initial data. © 2003 Wiley Periodicals, Inc.     

Schauder estimates for oblique derivative problems

We prove that the solution of the oblique derivative parabolic problem in a noncylindrical domain Ω^T belongs to the anisotropic Holder space C^{2+α, 1+α/2}(gwT) 0 < α < 1, even if the nonsmooth “lateral boundary” of Ω^T is only of class C^{1+α, (1+α)/2}). As a corollary, we also obtain an a priori estimate in the Hölder space C^2+α(Ω₀) for a solution of the oblique derivative elliptic problem in a domain Ω₀ whose boundary belongs only to the classe C^1+α.     

La monotonia di rappresentazioni del gruppo libero

Let Γ be a free nonabelian group on finitely many generators. Let Ω be the boundary of Γ, letC(Ω) be theC ^*-algebra of continuous functions on Ω, and let λ be the natural action of Γ onC(Ω). Aboundary representation is a representation of the crossed productC ^*-algebra Γ×_λ C(Ω). Given a unitary representation π of Γ onH, aboundary realization of π is an isometric Γ-inclusion ofH into the space of a boundary representation whose image is cyclic for that boundary representation. If the Γ-inclusion is bijective, we call, the realizationperfect. We prove below that if π admits an imperfect boundary realization, then there exists a nonzero vectorv ₀∈H satisfying $$\sum\limits_{|x| = n} {|\left\langle {v,\pi (x)v_0 } \right\rangle |^2 \leqslant |v|^2 } for each v \in {\mathcal{H}} (GVB)$$ If π is irreducible and weakly contained in the regular representation, and if no suchv ₀ exists, it follows that π satisfiesmonotony: up to equivalence, there exists exactly one realization of π, and that realization is perfect.     

Kernel estimates for perturbations of positive semigroups

Given a positive C⁰-semigroup T⁰ on L₂(Ω, m) with a kernel k⁰, where (Ω, m) is a σ-finite measure space, we study a suitably perturbed semigroup T and prove existence of a kernel k for T and an estimate of the k in terms of k⁰. In this way we extend a heat kernel estimate proven by Barlow, Grigor’yan and Kumagai [4] for Dirichlet forms perturbed by jump processes. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Unitary Equivalence, Similarity and Calculation of K0-Group

In this paper, we are concerned with the classification of operators on complex separable Hilbert spaces, in the unitary equivalence sense and the similarity sense, respectively. We show that two strongly irreducible operators A and B are unitary equivalent if and only if W＊（A＋B）′≈M2（C）, and two operators A and B in B1（Ω） are similar if and only if A′（AGB）/J≈M2（C）. Moreover, we obtain V（H^∞（Ω,μ）≈N and Ko（H^∞（Ω,μ）≈Z by the technique of complex geometry, where Ω is a bounded connected open set in C, and μ is a completely non-reducing measure on Ω.     

Approximation by harmonic functions in theC m -Norm and harmonicC m -capacity of compact sets in ℝ n

We study the function Λ ^m (X), 0<m<1, of compact setsX in ℝ ⁿ , n≥2, defined as the distance in the spaceC ^m (X)≡lip ^m(X) from the function |x|² to the subspaceH ^m (X) which is the closure inC ^m (X) of the class of functions harmonic in the neighborhood ofX (each function in its own neighborhood). We prove the equivalence of the conditions Λ ^m (X)=0 andC ^m (X)=H ^m (X). We derive an estimate from above that depends only on the geometrical properties of the setX (on its volume). Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 372–382, September, 1997. Translated by I. P. Zvyagin     

A MANN ITERATIVE REGULARIZATION METHOD FOR ELLIPTIC CAUCHY PROBLEMS

We investigate the Cauchy problem for linear elliptic operators with C ^∞–coefficients at a regular set Ω ? R ², which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold Γ ? ?Ω and our goal is to reconstruct the trace of the H ¹(Ω) solution of an elliptic equation at ?Ω/Γ. The method proposed here composes the segmenting Mann iteration with a fixed point equation associated with the elliptic Cauchy problem. Our algorithm generalizes the iterative method developed by Maz'ya et al., who proposed a method based on solving successive well-posed mixed boundary value problems. We analyze the regularizing and convergence properties both theoretically and numerically.

     

Lower-semicontinuity result for the two-scale convergence

Let (m, n) ∈ ℕ², Ω an open bounded domain in ℝ^m , Y = [0, 1]^m ; u^ε in (L²(Ω))ⁿ which is two-scale converges to some u in (L²(Ω × Y))ⁿ . Let φ: Ω × ℝ^m × ℝⁿ → ℝ such that: φ(x, ·, ·) is continuous a.e. x ∈ Ω φ(·, y, z) is measurable for all (y, z) in ℝ^m × ℝⁿ , φ(x, ·, z) is 1-periodic in y, φ(x, y, ·) is convex in z. Assume that there exist a constant C₁ > 0 and a function C₂ ∈ L²(Ω) such that

     

Extinction and non-extinction for a polytropic filtration equation with a nonlocal source

In this article, the authors establish the conditions for the extinction of solutions, in finite time, of the fast diffusive polytropic filtration equation u _t ?=?div(|?u ^m | ^p?2?u ^m )?+?a∫_Ω u ^q (y,?t)dy with a, q, m?>?0, p?>?1, m(p???1)?R ^N (N?>?2). More precisely speaking, it is shown that if q?>?m(p???1), any non-negative solution with small initial data vanishes in finite time, and if 0?q?m(p???1), there exists a solution which is positive in Ω for all t?>?0. For the critical case q?=?m(p???1), whether the solutions vanish in finite time or not depends on the comparison between a and μ, where μ?=?∫?_Ωφ ^p?1(x)dx and φ is the unique positive solution of the elliptic problem ?div(|?φ| ^p?2?φ)?=?1, x?∈?Ω; φ(x)?=?0, x?∈??Ω.     

Global attractor and decay estimates of solutions to a class of nonlinear evolution equations

In this work, we prove the existence of global attractor for the nonlinear evolution equation u_tt?Δu?Δu_t?Δu_tt + g(x, u)=f(x) in X=(H²(Ω)∩H(Ω)) × (H²(Ω)∩H(Ω)). This improves a previous result of Xie and Zhong in (J. Math. Anal. Appl. 2007; 336 :54–69.) concerning the existence of global attractor in H(Ω) × H(Ω) for a similar equation. Further, the asymptotic behavior and the decay property of global solution are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

Existence and comparison results for nonlinear volterra integral equations in a banach space

Let L ⁰(ΩA,P) be the space of equivalent classes of random variables defined on a probability space (Ω,A,P). Let H be the closed subspace of L ⁰(Ω,A,P) spanned by a sequence of i.i.d. (independent and identically distributed) random variables having the symmetric nondegenerate law F. It is proved that H is linearly homeomorphic to l _p for 0<p≤2 if F belongs to the domain of normal attraction of symmetric stable law withexponent p.     

Perturbation theorems for linear hyperbolic mixed problems and applications to the compressible euler equations

The main result of this paper (which is completely new, apart from our previous and less general result proved in reference [9]) states that the nonlinear system of equations (1.11) (or, equivalently, (1.10)) that describes the motion of an inviscid, compressible (barotropic) fluid in a bounded domain Ω, gives rise to a strongly well-posed problem (in the Hadamard classical sense) in spaces H^k(Ω), k ≧ 3; see Theorem 1.4 below. Roughly speaking, if (a_n, ?_n) → (a, ?) in H^k × H^k and if f_n → f in ??²(0, T;H^k), then (v_n, g_n) → (v, g) in ?? (0, T; H^k × H^k). The method followed here (see also [8]) also applies to the non-barotropic case p = p(ρ, s) (see [10]) and to other nonlinear problems. These results are based upon an improvement of the structural-stability theorem for linear hyperbolic equations. See Theorem 1.2 below. Added in proof: The reader is referred to [29], Part I, for a concise explanation of some fundamental points in the method followed here. © 1993 John Wiley & Sons, Inc.