A remark on unique continuation along and across lower‐dimensional planes for the wave equation

We prove unique continuation of solutions of the wave equation along and across lower‐dimensional planes containing the t‐axis. This is a sharpening and a generalization of a result of Cheng, Ding and Yamamoto as well as a simplification of the proof. Copyright © 2008 John Wiley & Sons, Ltd.     

Energy‐minimizing coarse spaces for two‐level Schwarz methods for multiscale PDEs

Two‐level overlapping Schwarz methods for elliptic partial differential equations combine local solves on overlapping domains with a global solve of a coarse approximation of the original problem. To obtain robust methods for equations with highly varying coefficients, it is important to carefully choose the coarse approximation. Recent theoretical results by the authors have shown that bases for such robust coarse spaces should be constructed such that the energy of the basis functions is minimized. We give a simple derivation of a method that finds such a minimum energy basis using one local solve per coarse space basis function and one global solve to enforce a partition of unity constraint. Although this global solve may seem prohibitively expensive, we demonstrate that a one‐level overlapping Schwarz method is an effective and scalable preconditioner and we show that such a preconditioner can be implemented efficiently using the Sherman–Morrison–Woodbury formula. The result is an elegant, scalable, algebraic method for constructing a robust coarse space given only the supports of the coarse space basis functions. Numerical experiments on a simple two‐dimensional model problem with a variety of binary and multiscale coefficients confirm this. Numerical experiments also show that, when used in a two‐level preconditioner, the energy‐minimizing coarse space gives better results than other coarse space constructions, such as the multiscale finite element approach. Copyright © 2009 John Wiley & Sons, Ltd.     

S‐inequality for certain product measures

In this paper we prove the S‐inequality for certain product probability measures and ideals in . As a result, for the Weibull and Gamma product distributions we derive concentration of measure type estimates as well as optimal comparison of moments.     

A penalty‐FEM for navier‐stokes type variational inequality with nonlinear damping term

In this article, we consider a penalty finite element (FE) method for incompressible Navier‐Stokes type variational inequality with nonlinear damping term. First, we establish penalty variational formulation and prove the well‐posedness and convergence of this problem. Then we show the penalty FE scheme and derive some error estimates. Finally, we give some numerical results to verify the theoretical rate of convergence. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 918–940, 2017     

Area‐minimizing projective planes in 3‐manifolds

Let (M,g) be a compact Riemannian manifold of dimension 3, and let ? denote the collection of all embedded surfaces homeomorphic to \input amssym ${\Bbb R}{ \Bbb P}^2$ . We study the infimum of the areas of all surfaces in ?. This quantity is related to the systole of (M,g). It makes sense whenever ? is nonempty. In this paper, we give an upper bound for this quantity in terms of the minimum of the scalar curvature of (M,g). Moreover, we show that equality holds if and only if (M,g) is isometric to \input amssym ${\Bbb R}{ \Bbb P}^3$ up to scaling. The proof uses the formula for the second variation of area and Hamilton's Ricci flow. © 2010 Wiley Periodicals, Inc.     

The sharp Sobolev inequality and the Banchoff-Pohl inequality on surfaces

Let be a complete two dimensional simply connected Riemannian manifold with Gaussian curvature . If is a compactly supported function of bounded variation on , then satisfies the Sobolev inequality

Conversely, letting be the characteristic function of a domain recovers the sharp form of the isoperimetric inequality for simply connected surfaces with . Therefore this is the Sobolev inequality ``equivalent' to the isoperimetric inequality for this class of surfaces. This is a special case of a result that gives the equivalence of more general isoperimetric inequalities and Sobolev inequalities on surfaces.

Under the same assumptions on , if is a closed curve and is the winding number of about , then the Sobolev inequality implies

which is an extension of the Banchoff-Pohl inequality to simply connected surfaces with curvature .

     

Harnack inequality for two‐weight subelliptic p ‐Laplace operators

In this note, we prove a Harnack inequality for two‐weight subelliptic p ‐Laplace operators together with an upper bound of the Harnack constant associated with such inequality. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Relative isoperimetric inequality for minimal surfaces outside a convex set

Let C be a closed convex set in a complete simply connected Riemannian manifold M with sectional curvature bounded above by a nonpositive constant K. Assume that Σ is a compact minimal surface outside C such that Σ is orthogonal to ∂C along ∂Σ ∩ ∂C. If ∂Σ ∼ ∂C is radially connected from a point , then we prove a sharp relative isoperimetric inequality

Minimal surfaces with constraints of inequality type

In this paper we establish sufficient conditions for bifurcations of minimal surfaces with nonstrict inequality constraints.     

Comparison results for nonlinear elliptic equations with lower–order terms

We consider a solution u of the homogeneous Dirichlet problem for a class of nonlinear elliptic equations in the form A(u) + g(x, u) = f, where the principal term is a Leray–Lions operator defined on and g(x, u) is a term having the same sign as u and satisfying suitable growth assumptions. We prove that the rearrangement of u can be estimated by the solution of a problem whose data are radially symmetric.     

A Turán-type inequality for the gamma function

We prove that the following Turán-type inequality holds for Euler's gamma function. For all odd integers n?1 and real numbers x>0 we have

α?Γ(n−1)(x)Γ(n+1)(x)−Γ(n)²(x),     

A Brunn-Minkowski inequality for the Monge-Ampère eigenvalue

We prove a Brunn-Minkowski-type inequality for the eigenvalue Λ of the Monge-Ampère operator: Λ^-1/2n is concave in the class of domains in Rⁿ endowed with Minkowski addition. The equality case is explicitly described too. The main device of the proof is a notion of addition for convex functions, called infimal convolution, which corresponds to the Minkowski addition of the graphs of the involved functions.