Moser-type inequalities in Lorentz spaces

For the numerical solution of linear variational problems involving elliptic partial differential operators in n ≥ 2 space dimensions, iterative solution schemes are indispensable on account of their problem size. Our guiding principle is to devise iterative solvers which are optimal in the number of arithmetic operations, i.e., which are of linear complexity in the total number of unknowns. For these algorithms, asymptotically optimal preconditioners are required. The class of preconditioners for which this can be shown are of multilevel type, requiring nested approximation spaces to approximate the solution of the system on a fine user-specified grid. For smooth solutions of standard second and fourth order elliptic PDEs (partial differential equations) in variational form, approximations based on tensor products of higher-order B-splines yield high accuracy.For such problem classes, this survey collects the main ingredients for multilevel preconditioners in terms of higher order B-splines. There are three types of multilevel preconditioners for which asymptotic optimality can be shown. Two of them, the so-called additive preconditioners, are specified for isogeometric analysis involving linear elliptic partial differential operators in terms of variants of the BPX (Bramble-Pasciak-Xu) preconditioner and wavelet preconditioners. The third class are the so-called multiplicative preconditioners, specifically, multigrid methods.An essential ingredient for all these multilevel preconditioners are intergrid operators which transform vectors or matrices between grids of different grid spacing. For higher order B-splines, these intergrid operators can be derived from their refinement relations. In addition to a presentation of the theoretical ingredients, the performance of the different preconditioners will be demonstrated by some numerical examples.KeywordsVariational InequalityRiesz BasisBiorthogonal WaveletElliptic PDEsSpline SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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

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

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.

Let L be a uniformly elliptic linear second order differential operator in divergence form with bounded measurable real coefficients in a bounded domain G ⊂ ℝn (n ⩾ 2). We define classes of continuous functions in G that contain generalized solutions of the equation Lƒ = 0 and have the property that the compact sets removable for such solutions in these classes can be completely described in terms of Hausdorff measures.

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.

A note on estimates for elliptic systems with L1 data

Let Ω be a bounded domain in $ℝ^n$ with smooth boundary ∂Ω and let L denote a second order linear elliptic differential operator and a mapping from $L^2(Ω)$ into itself with compact inverse, with eigenvalues $-λ_{i}$, each repeated according to its multip

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.

The covariant Weyl (spin s = 1/2) and Maxwell (s = 1) equations in certain local charts (u, φ) of a space‐time (M, g) are considered. It is shown that the condition g00(x) > 0 for all x ε u is necessary and sufficient to rewrite them in a unified manner as evolution equations δtφ = L(s)φ. Here L(s) is a linear first order differential operator on the pre—Hilbert space (C (Ut, 2s+1). (…)), where Ut ⊂ IR3 is the image of the coordinate map of the spacelike hyper‐surface t = const, and (φ, C) = ƒUt ϕ *Q d(3)x with a suitable Hermitian n × n‐ matrix Q = Q(t,x). The total energy of the spinor field ϕ with respect to Ut is then simply given by E = 〈ϕ,ϕ〉. In this way inequalities for the energy change rate with respect to time, δt|ϕ|2 = 2Re (ϕ, L(s)ϕ) are obtained. As an application, the Kerr—Newman black hole is studied, yielding quantitative estimates for the energy change rate. These estimates especially confirm the energy conservation of the Weyl field and the well—known superradiance of electromagnetic waves.

A homogenization result for a family of oscillating integral energies u_{\epsilon} \mapsto \int_{\Omega} f(x,\frac{x}{\epsilon},u_{\epsilon}(x))\,dx,\quad \epsilon \to 0^+ is presented, where the fields u_{\epsilon} are subjected to first order linear differential constraints depending on the space variable x . The work is based on the theory of \mathscr A -quasiconvexity with variable coefficients and on two-scale convergence techniques, and generalizes the previously obtained results in the case in which the differential constraints are imposed by means of a linear first order differential operator with constant coefficients. The identification of the relaxed energy in the framework of \mathscr A -quasiconvexity with variable coefficients is also recovered as a corollary of the homogenization result.

This paper concerns homogenization for the elliptic operator in L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (Π), Π = R × (0,a), defined by the differential expression B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ε</sup> = Σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j=1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> g <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> (x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> /ε, x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ) D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> + Σ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j=1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (h <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> (x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> /ε, x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ) D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> + D <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> h <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">j</sub> *(x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> /ε, x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> )) + Q(x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> /ε, x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ) + λQ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sub> (x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> /ε, x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ) with periodic, Neumann or Dirichlet boundary conditions. All the coefficients are assumed to be periodic of period 1 with respect to the first variable. Sharp-order approximations for the inverse of B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">λ</sub> ε in the norms of B(L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (Π)) and B(L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (Π), H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> (Π)) are obtained, with error terms being O(ε).

In this article, we are concerned with the following eigenvalue problem of a second order linear elliptic operator: $-D\Delta \phi -2\alpha\nabla m(x)\cdot \nabla\phi+V(x)\phi=\lambda\phi\ \ { in }\Omega, $ complemented by a general boundary condition, including Dirichlet boundary condition and Robin boundary condition, $ \frac{\partial\phi}{\partial n}+\beta(x)\phi=0 \ \ { on }\partial\Omega, $ where $\beta\in C(\partial\Omega)$ is allowed to be positive, sign-changing, or negative, and $n(x)$ is the unit exterior normal to $\partial\Omega$ at $x$. The domain $\Omega\subset\mathbb{R}^N$ is bounded and smooth, the constants $D>0$ and $\alpha>0$ are, respectively, the diffusive and advection coefficients, and $m\in C^2(\bar\Omega),\,V\in C(\bar\Omega)$ are given functions. We aim to investigate the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as the diffusive coefficient $D\to0$ or $D\to\infty$. Our results, together with those of [X. F. Chen and Y. Lou, Indiana Univ. Math. J., 61 (2012), pp. 45--80; A. Devinatz, R. Ellis, and A. Friedman, Indiana Univ. Math. J., 23 (1973/74), pp. 991--1011; and A. Friedman, Indiana U. Math. J., 22 (1973), pp. 1005--1015] where the Neumann boundary case (i.e., $\beta=0$ on $\partial\Omega$) and Dirichlet boundary case were studied, reveal the important effect of advection and boundary conditions on the asymptotic behavior of the principal eigenvalue.

In this paper the notion of a gap between two linear operators is extended to the case of linear differential operators defined on different open sets. Estimates of the gap between second order uniformly elliptic partial differential operators subject to homogeneous Dirichlet boundary conditions defined on different open sets Ω1 and Ω2 are obtained in terms of the geometrical characteristics of vicinity of Ω1 and Ω2. These estimates can be used for obtaining spectral stability estimates for the eigenvalues and eigenfunctions of the aforementioned operators.

The estimate \|{D^{k-1}u}\|_{L^{n/(n-1)}} \le \|{A(D)u}\|_{L^1} is shown to hold if and only if A(D) is elliptic and canceling. Here A(D) is a homogeneous linear differential operator A(D) of order k on \mathbb R^n from a vector space V to a vector space E . The operator A(D) is defined to be canceling if \bigcap_{\xi \in \mathbb R^n \setminus \{0\}} A(\xi)[V]=\{0\}. This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous linear differential operator L(D) of order k on \mathbb R^n from a vector space E to a vector space F is introduced. It is proved that L(D) is cocanceling if and only if for every f \in L^1(\mathbb R^n; E) such that L(D)f=0 , one has f \in \dot{W}^{-1, n/(n-1)}(\mathbb R^n; E) . The results extend to fractional and Lorentz spaces and can be strengthened using some tools of J. Bourgain and H. Brezis.