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H 2 convergence of solutions of a biharmonic problem on a truncated convex sector near the angle π

Abdelkader Tami, Mounir Tlemcani (2021)

Applications of Mathematics

We consider a biharmonic problem Δ 2 u ω = f ω with Navier type boundary conditions u ω = Δ u ω = 0 , on a family of truncated sectors Ω ω in 2 of radius r , 0 < r < 1 and opening angle ω , ω ( 2 π / 3 , π ] when ω is close to π . The family of right-hand sides ( f ω ) ω ( 2 π / 3 , π ] is assumed to depend smoothly on ω in L 2 ( Ω ω ) . The main result is that u ω converges to u π when ω π with respect to the H 2 -norm. We can also show that the H 2 -topology is optimal for such a convergence result.

H functional calculus for an elliptic operator on a half-space with general boundary conditions

Giovanni Dore, Alberto Venni (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let A be the L p realization ( 1 &lt; p &lt; ) of a differential operator P ( D x , D t ) on n × + with general boundary conditions B k ( D x , D t ) u ( x , 0 ) = 0 ( 1 k m ). Here P is a homogeneous polynomial of order 2 m in n + 1 complex variables that satisfies a suitable ellipticity condition, and for 1 k m B k is a homogeneous polynomial of order...

H P -finite element approximations on non-matching grids for partial differential equations with non-negative characteristic form

Andrea Toselli (2003)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We propose and analyze a domain decomposition method on non-matching grids for partial differential equations with non-negative characteristic form. No weak or strong continuity of the finite element functions, their normal derivatives, or linear combinations of the two is imposed across the boundaries of the subdomains. Instead, we employ suitable bilinear forms defined on the common interfaces, typical of discontinuous Galerkin approximations. We prove an error bound which is optimal with respect...

H p spaces associated with Schrödinger operators with potentials from reverse Hölder classes

Jacek Dziubański, Jacek Zienkiewicz (2003)

Colloquium Mathematicae

Let A = -Δ + V be a Schrödinger operator on d , d ≥ 3, where V is a nonnegative potential satisfying the reverse Hölder inequality with an exponent q > d/2. We say that f is an element of H A p if the maximal function s u p t > 0 | T t f ( x ) | belongs to L p ( d ) , where T t t > 0 is the semigroup generated by -A. It is proved that for d/(d+1) < p ≤ 1 the space H A p admits a special atomic decomposition.

Hamilton-Jacobi flows and characterization of solutions of Aronsson equations

Petri Juutinen, Eero Saksman (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this note, we verify the conjecture of Barron, Evans and Jensen [3] regarding the characterization of viscosity solutions of general Aronsson equations in terms of the properties of associated forward and backwards Hamilton-Jacobi flows. A special case of this result is analogous to the characterization of infinity harmonic functions in terms of convexity and concavity of the functions r max y B r ( x ) u ( y ) and r min y B r ( x ) u ( y ) , respectively.

Hardy spaces H¹ for Schrödinger operators with certain potentials

Jacek Dziubański, Jacek Zienkiewicz (2004)

Studia Mathematica

Let K t t > 0 be the semigroup of linear operators generated by a Schrödinger operator -L = Δ - V with V ≥ 0. We say that f belongs to H ¹ L if | | s u p t > 0 | K t f ( x ) | | | L ¹ ( d x ) < . We state conditions on V and K t which allow us to give an atomic characterization of the space H ¹ L .

Hardy-Poincaré type inequalities derived from p-harmonic problems

Iwona Skrzypczak (2014)

Banach Center Publications

We apply general Hardy type inequalities, recently obtained by the author. As a consequence we obtain a family of Hardy-Poincaré inequalities with certain constants, contributing to the question about precise constants in such inequalities posed in [3]. We confirm optimality of some constants obtained in [3] and [8]. Furthermore, we give constants for generalized inequalities with the proof of their optimality.

Harmonic averages, exact difference schemes and local Green’s functions in variable coefficient PDE problems

Owe Axelsson, János Karátson (2013)

Open Mathematics

A brief survey is given to show that harmonic averages enter in a natural way in the numerical solution of various variable coefficient problems, such as in elliptic and transport equations, also of singular perturbation types. Local Green’s functions used as test functions in the Petrov-Galerkin finite element method combined with harmonic averages can be very efficient and are related to exact difference schemes.

Harmonic spaces associated with adjoints of linear elliptic operators

Peter Sjögren (1975)

Annales de l'institut Fourier

Let L be an elliptic linear operator in a domain in R n . We imposse only weak regularity conditions on the coefficients. Then the adjoint L * exists in the sense of distributions, and we start by deducing a regularity theorem for distribution solutions of equations of type L * u = given distribution. We then apply to L * R.M. Hervé’s theory of adjoint harmonic spaces. Some other properties of L * are also studied. The results generalize earlier work of the author.

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