Singular integrals, harmonic functions, and differentiability properties of functions of several variables

Stein, Elias M.

doi:10.1090/pspum/010/0482394

Cited by 1,360 publications

(2,064 citation statements)

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“…Our approach is essentially a boundary integral method [71,73,79,31] which eliminates grid effects by eliminating the spatial grid; indeed, we eliminate the temperature field altogether. We must then solve a singular integral equation [17,96] for the velocity of the moving boundary, but it turns out that this can be done quite accurately. Given the velocity, it is still not altogether trivial to move the curve, because the velocity is curvature-dependent.…”

Section: Methodsmentioning

confidence: 99%

A boundary integral approach to unstable solidification

Strain¹

1989

Journal of Computational Physics

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We consider the supercooled Stefan problem with a general anisotropic curvature-and velocity-dependent boundary c<;mdition on the moving interface. This is a well-known model for pattern formation in unstable solidification.We reformulate the problem in terms of a quasilinear history-dependent singular integral equation for the velocity of the boundary. Using this equation, we carry out a new linear stability analysis of a planar solidification front with a general boundary condition. This analysis disagrees with the classical linear stability theory, because our approach includes transient effects due to initial conditions and linearizes the boundary conditions more accurately. Our analysis exhibits the smoothing role of velocity-dependence and the destabilizing effect of anisotropy.We then present numerical methods, also based on the integral equation formulation. These methods are able to follow the evolution of a periodic solidification front far into the nonlinear regime, with 0 (M) accuracy, where M is the time step. Previous work has been limited to short times and achieved slightly less than 0 (M 112 ) accuracy. Our methods also include a new algorithm for moving curves with curvature-dependent velocity.We present numerical results obtained with these new methods. After demonstrating first-order convergence for short time spans, we compare accurate numerical results with the predictions of the new and classical linear stability theories. Our results agree very well with the new theory, and disagree with the classical theory by as much as 25%. Then we study the long-time evolution of a dendritic front. We confirm the smoothing effect of velocity-dependence numerically. Our computations exhibit the beginnings of a sidebranching instability, followed by formation of a periodic cellular front, with an anisotropic boundary condition. Tip-splitting occurs instead, in the isotropic case .

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Section: Methodsmentioning

confidence: 99%

A boundary integral approach to unstable solidification

Strain¹

1989

Journal of Computational Physics

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“…Nevertheless, this restriction can be removed. In fact, at the point O we obtain the result L ∞ × · · · × L ∞ → BM O, which is the multilinear version of the theorem of Peetre [50], Spanne [52], and Stein [53] on the L ∞ → BM O boundedness of singular integrals. Full details about the theorem are given in [38], nevertheless, we want to illustrate here some of the arguments in the bilinear case in a somehow simplified form that clearly indicates the main ideas.…”

Section: Calderón-zygmund Kernels Interpolation and Endpoint Estimatesmentioning

confidence: 76%

On multilinear singular integrals of Calderón-Zygmund type

Grafakos

Torres²

2002

Publicacions Matemàtiques

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Abstract. A variety of results regarding multilinear Calderón-Zygmund singular integral operators is systematically presented. Several tools and techniques for the study of such operators are discussed. These include new multilinear endpoint weak type estimates, multilinear interpolation, appropriate discrete decompositions, a multilinear version of Schur's test, and a multilinear version of the T 1 Theorem suitable for the study of multilinear pseudodifferential and translation invariant operators. A maximal operator associated with multilinear singular integrals is also introduced and employed to obtain weighted norm inequalities.

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“…Here we present a direct multilinear proof adapted from the classical linear version due to Spanne [25], Peetre [23] and Stein [26], but prior to this proof we briefly discuss why we don't conclude here that S is bounded from…”

Section: Extending Square Function Boundsmentioning

confidence: 99%