Multilevel Preconditioning for Variational Problems

The purpose of this paper is to construct a family of fundamental solutions for elliptic partial differential operators with quaternion constant coefficients. The elements of such family are expressed by means of functions, which depend jointly real analytically on the coefficients of the operators and on the spatial variable. We show some regularity properties in the frame of Schauder spaces for the corresponding single layer potentials. Ultimately, we exploit our construction by showing a real analyticity result for perturbations of the layer potentials corresponding to complex elliptic partial differential operators of order two. Copyright © 2012 John Wiley & Sons, Ltd.

We consider semilinear elliptic second-order partial differential inequalities of the form (*) in the whole space , where , q>0 and L is a linear elliptic second-order partial differential operator in divergence form. We assume that the coefficients of the operator L are measurable and locally bounded such that the quadratic form associated with the operator L is symmetric and non-negative definite. We obtain a Liouville comparison principle in terms of capacities associated with the operator L for solutions of (*) which are measurable and belong locally in to a Sobolev-type function space also associated with the operator L.

Boundary value problems for elliptic partial differential operators on bounded domains

Dense Subsets of [formula omitted]1-Solutions to Linear Elliptic Partial Differential Equations

For a linear second order elliptic partial differential operator A : V → V′, we consider the boundary value problems Au = f with stationary Gaussian random data f over the dual V′ of the separable Hilbert space V in which the solution u is sought. The operator A is assumed to be deterministic and bijective. The unique solution u = A−1f is a Gaussian random field over V. It is characterized by its mean field Eu and its covariance Cu ∈ V ⊗ V. For a class of Gaussian data f with piecewise analytic covariance kernels Cf ∈ V′ ⊗ V′, we characterize analytic regularity of the covariance Cu of the Gaussian solution u in families of countably normed, weighted Sobolev spaces. To this end, we investigate shift theorems for the (nonhypoelliptic) deterministic tensor PDEs (A ⊗ A)Cu = Cf proposed by Schwab and Todor, (Numer Math 95 (2003), 707–734) for the computation of covariance Cu of the random solution u. The nonhypoelliptic nature of A ⊗ A implies that sing supp(Cu) is in general strictly larger than sing supp(Cf). For a model problem, analyticity and singular support of the solution is characterized completely. Based on our regularity results, we outline an hp‐finite element strategy (Pentenrieder, Ph.D thesis, ETHZurich, Dissertation No. 18729, 2009; Pentenrieder and Schwab, Research Report 2010‐08, Seminar for Applied Mathematics, ETH Zürich, Submitted) to approximate Cu stemming from covariances of stationary Gaussian data f. In the second part (Pentenrieder and Schwab, Research Report 2010–08, Seminar for Applied Mathematics, ETH Zürich, Submitted) of this work, we prove that this discretization gives exponential rates of convergence of the FE approximations, in terms of the number of degrees of freedom. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012

Convergence of iterations of special Green integrals for overdetermined elliptic linear partial differential operatorsPof order p ≥ 1 is proved. Using this result we obtain necessary and sufficient conditions for the solvability of the equationPu=fin the Sobolev space Wp,2(D) and, as a corollary, necessary and sufficient conditions for the vanishing of the first cohomology group of elliptic differential complexes. Also a criterion for the solvability of aP‐Neumann problem for elliptic differential operators is proved.

On preconditioned eigensolvers and Invert–Lanczos processes

This paper deals with a new mixed finite element solution of the Dirichlet problem of fourth order elliptic linear partial differential operators with variable coefficients. As examples, bending problems of elastic plates having variable thickness have been considered, for which the algorithm of this method allows a simultaneous determination of the deflection, the curvature components and the „actual”︁ bending and twisting moments of the plate, the „actual”︁ bending and twisting moments being the components of the Lagrange multiplier of this method. Suitable error estimates are obtained under necessary regularity assumptions on the solution.

This paper considers the system of nonlinear Dirichlet boundary value problems \[ { L u ( x ) = λ f ( u ( x ) , v ( x ) ) L v ( x ) = μ g ( u ( x ) , v ( x ) ) } , x ∈ Ω , \left \{ \begin {gathered} Lu(x) = \lambda f(u(x),v(x)) \hfill \\ Lv(x) = \mu g(u(x),v(x)) \hfill \\ \end {gathered} \right \},\qquad x \in \Omega , \] a bounded domain in R n {{\mathbf {R}}^n} . Here L L is a strongly, uniformly elliptic linear partial differential operator, λ \lambda , μ \mu are real parameters, and f f , g : R 2 → R g:{{\mathbf {R}}^2} \to R are smooth with \[ f ( 0 , 0 ) = 0 = g ( 0 , 0 ) . f(0,0) = 0 = g(0,0). \] A detailed analysis of the solution set to the system is given from the point of view of several parameter bifurcation theory.

We show existence of an infinitesimally invariant measure $m$ for a large class of divergence and non-divergence form elliptic second order partial differential operators with locally Sobolev regular diffusion coefficient and drift of some local integrability order. Subsequently, we derive regularity properties of the corresponding semigroup which is defined in $L^s(\mathbb{R}^d,m)$, $s\in [1,\infty]$, including the classical strong Feller property and classical irreducibility. This leads to a transition function of a Hunt process that is explicitly identified as a solution to an SDE. Further properties of this Hunt process, like non-explosion, moment inequalities, recurrence and transience, as well as ergodicity, including invariance and uniqueness of $m$, and uniqueness in law, can then be studied using the derived analytical tools and tools from generalized Dirichlet form theory.

The Martin compactification of a bounded Lipschitz domain D⊂R n is shown to be D ¯ for a large class of uniformly elliptic second order partial differential operators on D.

‎‎‎The Morrey boundedness is proved for the Riesz transform and the inverse operator of the nondegenerate elliptic differential operator of divergence form generated by a vector-function in $(L^\infty)^{n^2}$‎, ‎and for the inverse operator of the Schrodinger operators whose nonnegative potentials satisfy a certain integrability condition‎. ‎In this note‎, ‎our result is not obtained directly from the estimates of integral formula‎, ‎which reflects the fact that the solution of the Kato conjecture did not use any integral expression of the operators‎. ‎One of the important tools in the proof is the decomposition of the functions in Morrey spaces based on the elliptic differential operators in question‎. ‎In some special cases where the integral kernel comes into play‎, ‎the boundedness property of the Littlewood-Paley operator was already obtained by Gong‎. ‎So‎, ‎the main novelties of this paper are the decomposition results associated with elliptic differential operators and the result in the case where the explicit formula of the integral kernel of the heat semigroup is unavailable‎.

Analytic K-theory on manifolds with corners

Moser-type estimates for functions whose gradient is in the Lorentz space L(n, q), 1≤q≤∞, are given. Similar results are obtained for solutions u∈H inf0 sup1 of Au=(f i ) x i , where A is a linear elliptic second order differential operator and |f|∈L(n, q), 2≤q≤∞.

We construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables x 1,x 2,? and of two parameters q,t are their eigenfunctions. These operators are defined as limits at N?? of renormalized Macdonald operators acting on symmetric polynomials in the variables x 1,?,x N . They are differential operators in terms of the power sum variables $p_{n}=x_{1}^{n}+x_{2}^{n}+\cdots$ and we compute their symbols by using the Macdonald reproducing kernel. We express these symbols in terms of the Hall---Littlewood symmetric functions of the variables x 1,x 2,?. Our result also yields elementary step operators for the Macdonald symmetric functions.