Identifiability at the boundary for first-order terms

Let Ω be a domain in R ⁿ whose boundary is C ¹ if n≥3 or C ^1,β if n=2. We consider a magnetic Schrödinger operator L _{W , q} in Ω and show how to recover the boundary values of the tangential component of the vector potential W from the Dirichlet to Neumann map for L _{W , q} . We also consider a steady state heat equation with convection term Δ+2W·? and recover the boundary values of the convection term W from the Dirichlet to Neumann map. Our method is constructive and gives a stability result at the boundary.     

Sum of weighted Lebesgue spaces and nonlinear elliptic equations

We study the sum of weighted Lebesgue spaces, by considering an abstract measure space (W,A,m){(\Omega ,\mathcal{A},\mu)} and investigating the main properties of both the Banach space

On coercivity and irregularity for some nonlinear degenerate elliptic systems

We study the ‘universal’ strong coercivity problem for variational integrals of degenerate p-Laplacian type by mixing finitely many homogenous systems. We establish the equivalence between universal p-coercivity and a generalized notion of p-quasiconvex extreme points. We then give sufficient conditions and counterexamples for universal coercivity. In the case of noncoercive systems we give examples showing that the corresponding variational integral may have infinitely many non-trivial minimizers in W ₀^1,p which are nowhere C ¹ on their supports. We also give examples of universally p-coercive variational integrals in W ₀^1,p for p ⩾ with L ^∞ coefficients for which uniqueminimizers under affine boundary conditions are nowhere C ¹.      

Diffusion polynomial frames on metric measure spaces

We construct a multiscale tight frame based on an arbitrary orthonormal basis for the L² space of an arbitrary sigma finite measure space. The approximation properties of the resulting multiscale are studied in the context of Besov approximation spaces, which are characterized both in terms of suitable K-functionals and the frame transforms. The only major condition required is the uniform boundedness of a summability operator. We give sufficient conditions for this to hold in the context of a very general class of metric measure spaces. The theory is illustrated using the approximation of characteristic functions of caps on a dumbell manifold, and applied to the problem of recognition of hand-written digits. Our methods outperforms comparable methods for semi-supervised learning.     

A basic norm equivalence for the theory of multilevel methods

Summary Subspace decompositions of finite element spaces based onL ₂-like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods. Recently several authors have proved the equivalence of the associated discrete norms with theH ¹-norm. The present paper gives an elementary, self-contained derivation of this result which is based on the use ofK-functionals known from the theory of interpolation spaces.     

On lower bounds for the widths of classes of functions defined by integral moduli of continuity

We establish lower bounds for the Kolmogorov widths d _2n-1(W ^r H ₁^ω.L _p ) and Gel’fand widths d ^2n-1(W ^r H ₁^ω.L _p ) of the classes of functions W ^r H ₁^ω with a convex integral modulus of continuity ω(t).     

ENSS' Theory in Long Range Scattering: Second Order Hyperbolic and Parabolic Operators

We prove asymptotic completeness using ENSS' method for h₀(P) + W_S(Q) + W_L(Q) where h₀: Rⁿ → R is a polynomial of degree 2 with lim lim_|ξ|→∞h₀(ξ)| + |Δh₀(ξ)| = ∞, W_S a short range potential and W_L a smooth long range potential.     

Abelian real W*-algebras

In this paper, we point out that most results on abelian (complex)W ^*-algebras hold in the real case. Of course, there are differences in homeomorphisms of period 2. Moreover, an abelian real Von Neumann algebra not containing any minimal projection on a separable real Hilbert space is * isomorphic toL _τ ^∞ ([0, 1]) (all real functions inL ^∞([0, 1])), orL ^∞([0, 1]) (as a realW ^*-algebra), orL _τ ^∞ ([0, 1]) ⋇L ^∞ ([0, 1]) (as a realW ^*-algebra), and it is different from the complex case. Partially supported by the NNSF     

Model problem for the motion of a compressible, viscous flow with the no-slip boundary condition

We consider the Navier–Stokes equations for the motion of a compressible, viscous, pressureless fluid in the domain W = \mathbbR³₊{\Omega = \mathbb{R}^3_+} with the no-slip boundary conditions. We construct a global in time, regular weak solution, provided that initial density ρ ₀ is bounded and the magnitude of the initial velocity u ₀ is suitably restricted in the norm ||?{r₀(·)}u₀(·)||_L²(W) + ||?u₀(·)||_L²(W){\|\sqrt{\rho_0(\cdot)}{\bf u}_0(\cdot)\|_{L^2(\Omega)} + \|\nabla{\bf u}_0(\cdot)\|_{L^2(\Omega)}}.     

Resolvent estimates for the linearized operator in L p associated with motion of compressible viscous fluids

We consider the resolvent problem for the linearized system of equations that describe motion of compressible viscous barotropic fluids in a bounded domain with the Navier boundary condition. This problem has uniquely a solution in [(W)\dot]¹_p ×(W²_p)ⁿ{\dot{W}^{1}_{p} \times (W^{2}_{p})^{n}} satisfying L _p estimates for any 1 < p < ∞. Moreover, resolvent estimates for the linearized operator of the above system in [(W)\dot]¹_p ×(L_p)ⁿ{\dot{W}^{1}_{p} \times (L_{p})^{n}} are established. Our main results yield clearly that the linearized operator is the infinitesimal generator of a uniformly bounded analytic semigroup on [(W)\dot]¹_p ×(L_p)ⁿ{\dot{W}^{1}_{p} \times (L_{p})^{n}}.     

Stability Constants for Courant Approximations

We study stability constants for Courant approximations in the metric of the spaces L₂()and W ₂ ¹ () in classes of triangular grids with uniformly bounded angles and with angles approaching zero. In the first case, we give sufficiently precise numerical estimates for the constants, and in the second case, we give asymptotics of the behavior of the boundaries for the corresponding quadratic forms.     

Constant Scalar Curvature Equation and Regularity of Its Weak Solution

In this paper we study the constant scalar curvature equation (CSCK), a nonlinear fourth-order elliptic equation, and its weak solutions on Kähler manifolds. We first define the notion of a weak solution of CSCK for an L^∞ Kähler metric. The main result is to show that such a weak solution (with uniform L^∞ bound) is smooth. As an application, this answers in part a conjecture of Chen regarding the regularity of K-energy minimizers. The new technical ingredient is a W^{2, 2} regularity result for the Laplacian equation Δ_gu=f on Kähler manifolds, where the metric has only L^∞ coefficients. It is well-known that such a W^{2, 2} regularity (W^{2, p} regularity for any p > 1) fails in general (except for dimension 2) for uniform elliptic equations of the form for a^ij ∊ L^∞ without certain smallness assumptions on the local oscillation of a^ij. We observe that the Kähler condition plays an essential role in obtaining a W^{2, 2} regularity for elliptic equations with only L^∞ elliptic coefficients on compact manifolds. © 2017 Wiley Periodicals, Inc.     

Superconvergence of tetrahedral quadratic finite elements for a variable coefficient elliptic equation

For a variable coefficient elliptic boundary value problem in three dimensions, using the properties of the bubble function and the element cancelation technique, we derive the weak estimate of the first type for tetrahedral quadratic elements. In addition, the estimate for the W^1,1‐seminorm of the discrete derivative Green's function is also given. Finally, we show that the derivatives of the finite element solution u_h and the corresponding interpolant Π₂u are superclose in the pointwise sense of the L^∞‐norm. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Nonlinear Filtering for Diffusions in Random Environments

Suppose that the signal X to be estimated is a diffusion process in a random medium W and the signal is correlated with the observation noise. We study the historical filtering problem concerned with estimating the signal path up until the current time based upon the back observations. Using Dirichlet form theory, we introduce a filtering model for general rough signal X ^W and establish a multiple Wiener integrals representation for the unnormalized pathspace filtering process. Then, we construct a precise nonlinear filtering model for the process X itself and give the corresponding Wiener chaos decomposition.     

Geometrization of 3-dimensional Coxeter orbifolds and Singer’s conjecture

Associated to any Coxeter system (W, S), there is a labeled simplicial complex L and a contractible CW-complex Σ _L (the Davis complex) on which W acts properly and cocompactly. Σ _L admits a cellulation under which the nerve of each vertex is L. It follows that if L is a triangulation of , then Σ _L is a contractible n-manifold. In this case, the orbit space, K _L := Σ _L /W, is a Coxeter orbifold. We prove a result analogous to the JSJ-decomposition for 3-dimensional manifolds: Every 3-dimensional Coxeter orbifold splits along Euclidean suborbifolds into the characteristic suborbifold and simple (hyperbolic) pieces. It follows that every 3-dimensional Coxeter orbifold has a decomposition into pieces which have hyperbolic, Euclidean, or the geometry of . (We leave out the case of spherical Coxeter orbifolds.) A version of Singer’s conjecture in dimension 3 follows: That the reduced ℓ ²-homology of Σ _L vanishes.      

On Transference of Multipliers on Matrix Weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Spaces

We consider a periodic matrix weight W defined on ℝ ^d and taking values in the N×N positive-definite matrices. For such weights, we prove transference results between multiplier operators on L _p (ℝ ^d ;W) and L_p(\mathbb T^d;W)L_{p}(\mathbb {T}^{d};W), 1<p<∞, respectively. As a specific application, we study transference results for homogeneous multipliers of degree zero.     

K-functional, weighted moduli of smoothness, and best weighted polynomial approximation on a simplex

An inverse theorem for the best weighted polynomial approximation of a function in (S) is established. We also investigate Besov spaces generated by Freud weight and their connection with algebraic polynomial approximation in , wherew _α is a Jacobi-type weight onS, 0<p ≤ ∞,S is a simplex andW _λ is a Freud weight. For Ditzian-TotikK-functionals onL _p(S), 1 ≤p ≤ ∞, we obtain a new equivalence expression.     

Wavelet decompositions of L 2-functionals

Based on distribution-theoretical definitions of L ₂ and Sobolev spaces given by Werner in [P. Werner (1970). A distribution-theoretical approach to certain Lebesgue and Sobolev spaces. J. Math. Anal. Appl., 29, 19–78.] real interpolation, Besov type spaces and approximation spaces with respect to multiresolution approximations are considered. The key for the investigation are generalized moduli of smoothness introduced by Haf in [H. Haf (1992). On the approximation of functionals in Sobolev spaces by singular integrals. Applicable Analysis, 45, 295–308.]. Those moduli of smoothness allow to connect the concept of L ₂-functionals with more recent developments in multiscale analysis, see e.g. [W. Dahmen (1995). Multiscale analysis, approximation, and interpolation spaces. In: C.K. Chui and L.L. Schumaker (Eds.), Approximation Theory VIII, Vol. 2: Wavelets and Multilevel Approximation, pp. 47–88.]. In particular, we derive wavelet characterizations for the Sobolev spaces introduced by Werner and establish stable wavelet decompositions of L ₂-functionals. Generalizations to more general spaces of functionals and applications are also mentioned.     

Fast decoding of quasi-perfect Lee distance codes

A code D over Z ² _n is called a quasi-perfect Lee distance-(2t + 1) code if d _L(V,W) ≥ 2t + 1 for every two code words V,W in D, and every word in Z ² _n is at distance ≤ t + 1 from at least one code word, where D _L(V,W) is the Lee distance of V and W. In this paper we present a fast decoding algorithm for quasi-perfect Lee codes. The basic idea of the algorithm comes from a geometric representation of D in the 2-dimensional plane. It turns out that to decode a word it suffices to calculate its distance to at most four code words.