The Geometry of Filtering

In the study of elliptic differential operators of second order, we found that the automorphism pseudogroups are of finite type ([4]). However this fact takes a complete change in elliptic differential operators of first order.So as to make the objects which can be dealt with clear, we introduce the concept of pseudoellipticity of first order differential operators (Definition 1.1), which is naturally satisfied by first order elliptic differential operators.

Let {λ2} and {ϕ λ } be the eigenvalues and an orthonormal system of eigenvectors of a second order elliptic differential operator Δ on a compact manifoldM of dimensionN. We prove that the Riesz means of order δ, defined by\(R_\Lambda ^\delta f = \sum\limits_{\lambda< \Lambda } {\left( {1 - \frac{{\lambda ^2 }}{{\Lambda ^2 }}} \right)^\delta \hat f(} \lambda ) \varphi _\lambda \), are uniformly bounded from the Hardy spaceH p (M) into Weak-L p (M), if 0<p<1 and δ=N/p−(N+1)/2.

Analysis and Algebra on Differentiable Manifolds

Let $\\Omega\\subset{\\mathbb R}^n$ be bounded with a smooth boundary $\\Gamma$\nand let $S$ be the symmetric operator in $L^2(\\Omega)$ given by the minimal\nrealization of a second order elliptic differential operator. We give a\ncomplete classification of the Markovian self-adjoint extensions of $S$ by\nproviding an explicit one-to-one correspondence between such extensions and the\nclass of Dirichlet forms in $L^2(\\Gamma)$ which are additively decomposable by\nthe bilinear form of the Dirichlet-to-Neumann operator plus a Markovian form.\nBy such a result two further equivalent classifications are provided: the first\none is expressed in terms of an additive decomposition of the bilinear forms\nassociated to the extensions, the second one uses the additive decomposition of\nthe resolvents provided by Krein's formula. The Markovian part of the\ndecomposition allows to characterize the operator domain of the corresponding\nextension in terms of Wentzell-type boundary conditions. Some properties of the\nextensions, and of the corresponding Dirichlet forms, semigroups and heat\nkernels, like locality, regularity, irreducibility, recurrence, transience,\nultracontractivity and Gaussian bounds are also discussed.\n

Boundary value problems for elliptic partial differential operators on bounded domains

When L is a second order ordinary or elliptic differential operator, the principal eigenvalue for the Dirichlet problem and the corresponding principal (positive) eigenfunction u are known to exist and u is unique up to normalization. If further L has the form ε Σ a i j ∂ 2 / ∂ x i ∂ x i + Σ b i ∂ / ∂ x i \varepsilon \Sigma {a_{ij}}{\partial ^2}/\partial {x_i}\partial {x_i} + \Sigma {b_i}\partial /\partial {x_i} then results are known regarding the behavior of the principal eigenvalue λ = λ ε \lambda = {\lambda _\varepsilon } as ε ↓ 0 \varepsilon \downarrow 0 . These results are very sharp in case the vector ( b i ) ({b_i}) has a unique asymptotically stable point in the domain ω \omega where the eigenvalue problem is considered. In this paper the case where L is an ordinary differential operator degenerating on the boundary of ω \omega is considered. Existence and uniqueness of a principal eigenvalue and eigenfunction are proved and results on the behavior of λ ε {\lambda _\varepsilon } as ε ↓ 0 \varepsilon \downarrow 0 are established.

Special generic maps are smooth maps between smooth manifolds with only definite fold points as their singularities. The problem of whether a closed $n$-manifold admits a special generic map into Euclidean $p$-space for $1 \leq p \leq n$ was studied by several authors including Burlet, de Rham, Porto, Furuya, Eliasberg, Saeki, and Sakuma. In this paper, we study rational homology $n$-spheres that admit special generic maps into $\mathbb{R}^{p}$ for $p<n$. We use the technique of Stein factorization to derive a necessary homological condition for the existence of such maps for odd $n$. We examine our condition for concrete rational homology spheres including lens spaces and total spaces of linear $S^{3}$-bundles over $S^{4}$, and obtain new results on the (non-)existence of special generic maps.

Maximum principles for subharmonic functions in the framework of quasi-regular local semi-Dirichlet forms admitting lower bounds are presented. As applications, we give weak and strong maximum principles for (local) subsolutions of a second order elliptic differential operator on the domain of Euclidean space under conditions on coefficients, which partially generalize the results by Stampacchia.

Stochastic mechanics of electron transfer in graphene

We estimate the constant in the strengthened Cauchy-Bunyakowski-Schwarz inequality for hierarchical bilinear finite element spaces and elliptic partial differential equations with coefficients corresponding to anisotropy (orthotropy). It is shown that there is a nontrivial universal estimate, which does not depend on anisotropy. Moreover, this estimate is sharp and the same as for hierarchical linear finite element spaces.

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.

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

Let L be a second order elliptic differential operator on a Riemannian manifold E with no zero order terms. We say that a function h is L-harmonic if Lh 0. Every positive L-harmonic function has a unique representation

Let L be a second order elliptic differential operator in ${\bf R}^d$ and let E be a bounded domain in ${\bf R}^d$ with smooth boundary $\partial E$. A pair $(\Gamma,\nu)$ is associated with every positive solution of a semilinear differential equation $Lu=\psi(u)$ in E, where $\Gamma$ is a closed subset of~$\partial E$ and~$\nu$ is a Radon measure on $O=\partial E\setminus \Gamma$. We call this pair the {\it rough trace of u on~$\partial E$}. (In [E. B. Dynkin and S. E. Kuznetsov, Comm. Pure Appl. Math., 51 (1998), pp. 897--936], we introduced a fine trace allowing us to distinguish solutions with identical rough traces.) The case of $\psi(u)=u^\alpha$ with $\alpha>1$ was investigated using various methods by Le Gall, Dynkin, and Kuznetsov and by Marcus and V\'eron. In this paper we cover a wide class of functions~$\psi$ and simplify substantially the proofs contained in our earlier papers.

We investigate a second order elliptic differential operator $A_{β, μ}$ on a bounded, open set $Ω\subset\mathbb{R}^{d}$ with Lipschitz boundary subject to a nonlocal boundary condition of Robin type. More precisely we have $0\leq β\in L^{\infty}(\partialΩ)$ and $μ\colon \partial Ω\to \mathscr{M}(\overlineΩ)$, and boundary conditions of the form \[ \partial_ν^{\mathscr{A}}u(z)+β(z)u(z)=\int_{\overlineΩ}u(x)μ(z)(dx),\ z\in\partialΩ, \] where $\partial_ν^{\mathscr{A}}$ denotes the weak conormal derivative with respect to our differential operator. Under suitable conditions on the coefficients of the differential operator and the function $μ$ we show that $A_{β, μ}$ generates a holomorphic semigroup $T_{β,μ}$ on $L^{\infty}(Ω)$ which enjoys the strong Feller property. In particular, it takes values in $C(\overlineΩ)$. Its restriction to $C(\overlineΩ)$ is strongly continuous and holomorphic. We also establish positivity and contractivity of the semigroup under additional assumptions and study the asymptotic behavior of the semigroup.