Consider the degenerate elliptic operator Lδ := −∂2 x − δ 2 x2 ∂2 y on Ω := (0, 1) × (0, l), for δ > 0, l > 0. We prove well-posedness and regularity results for the degenerate elliptic equation Lδu = f in Ω, u|∂Ω = 0 using weighted Sobolev spaces Km a . In particular, by a proper choice of the parameters in the weighted Sobolev spaces Km a , we establish the existence and uniqueness of the sol...

Multivariate multiple linear regression (MMLR), which occurs in a number of practical applications, generalizes traditional least squares (multivariate regression) to right-hand sides. We extend recent MLR analyses sketched MMLR general Schatten p-norms by interpreting the problem as multiplicative perturbation. Our work represents an extension Maher's results on p-norms. derive expressions for...

Let u and uVn be the solution and, respectively, the finite element solution of the Poisson’s equation ∆u = f with zero boundary conditions. We construct for any m ∈ N and any polygon P a sequence of finite dimensional subspaces Vn such that ‖u−uVn‖H1 ≤ C dim(Vn)‖f‖Hm−1 , where f ∈ Hm−1(P) is arbitrary and C is a constant that depends only on P (we do not assume u ∈ Hm+1(P)). Although the final...

We present a simple example for the failure of Calder\'on-Zygmund estimate $\bar{\partial}$-operator when Sobolev $(k,p)$-norms are replaced by $C^k$-norms. This is discussed in context elliptic bootstrapping, Fredholm theory, and regularity $J$-holomorphic curves.

Let u and uV ∈ V be the solution and, respectively, the discrete solution of the non-homogeneous Dirichlet problem u = f onP, u|∂P = 0. For any m ∈ N and any bounded polygonal domain P, we provide a construction of a new sequence of finite dimensional subspaces Vn such that ‖u− uVn‖H 1 ≤ C dim(Vn)−m/2‖f ‖Hm−1 , where f ∈ Hm−1(P) is arbitrary and C is a constant that depends only on P and not on...

This paper provides a framework for developing computationally efficient multilevel preconditioners and representations for Sobolev norms. Specifically, given a Hilbert space V and a nested sequence of subspaces V1 ⊂ V2 ⊂ . . . ⊂ V , we construct operators which are spectrally equivalent to those of the form A = ∑ k μk(Qk −Qk−1). Here μk , k = 1, 2, . . . , are positive numbers and Qk is the or...

The unique solvability of parabolic equations in Sobolev spaces with mixed norms is presented. The second order coefficients (except a) are assumed to be only measurable in time and one spatial variable, and VMO in the other spatial variables. The coefficient a is measurable in one spatial variable and VMO in the other variables.

We consider the wave equation in an unbounded conical domain, with initial conditions and boundary conditions of Dirichlet or Neumann type. We give a uniform decay estimate of the solution in terms of weighted Sobolev norms of the initial data. The decay rate is the same as in the full space case.