By Luigi Ambrosio (auth.), Antonio Bove, Daniele Del Santo, M.K. Venkatesha Murthy (eds.)

ISBN-10: 0817648607

ISBN-13: 9780817648602

This number of unique articles and surveys addresses the hot advances in linear and nonlinear elements of the speculation of partial differential equations.

Key issues include:

* Operators as "sums of squares" of actual and intricate vector fields: either analytic hypoellipticity and regularity for terribly low regularity coefficients;

* Nonlinear evolution equations: Navier–Stokes method, Strichartz estimates for the wave equation, instability and the Zakharov equation and eikonals;

* neighborhood solvability: its reference to subellipticity, neighborhood solvability for structures of vector fields in Gevrey classes;

* Hyperbolic equations: the Cauchy challenge and a number of features, either confident and unfavourable results.

Graduate scholars at a variety of degrees in addition to researchers in PDEs and similar fields will locate this a good resource.

List of contributors:

L. Ambrosio N. Lerner

H. Bahouri X. Lu

S. Berhanu J. Metcalfe

J.-M. Bony T. Nishitani

N. Dencker V. Petkov

S. Ervedoza J. Rauch

I. Gallagher M. Reissig

J. Hounie L. Stoyanov

E. Jannelli D. S. Tartakoff

K. Kajitani D. Tataru

A. Kurganov F. Treves

G. Zampieri

E. Zuazua

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Extra resources for Advances in Phase Space Analysis of Partial Differential Equations: In Honor of Ferruccio Colombini's 60th Birthday

Example text

We may assume that Ω1 is bounded by C and a real analytic, simple closed Jordan curve Σ. Consider the Riemann surface (Ω, L). 1 in [FK], this Riemann surface is conformal to either the punctured disk, an annulus of the form {z : a < |z| < b} for some a, b > 0, or the punctured plane where each one is equipped with the standard structure. Since ∂Ω intersects ∂D on an arc where the orbit of L is two-dimensional, there is an open set Ω2 such that Ω2 \ Ω has nonempty interior and Ω2 is contained in an orbit of L in U of dimension two.

Since X and Y are assumed to have no common zeros, the orbits of L in Ω are immersed submanifolds of Ω of dimension one or two; moreover, the twodimensional orbits are open subsets of Ω. Let O ⊂ Ω be a two-dimensional 2 2 (T (Ω)) orbit of L in Ω and consider X ∧ Y ∈ C ∞ (Ω; (T (Ω))). Since has a global nonvanishing section e1 ∧ e2 , X ∧ Y is a real multiple of e1 ∧ e2 and this gives a meaning to the requirement that X ∧ Y does not change sign on any two-dimensional orbit O of {X, Y } in Ω. The vector field L satisfies A Generalization of the Rudin–Carleson Theorem 39 condition (P) at p ∈ Ω if there is a disk U ⊂ Ω centered at p such that X ∧ Y does not change sign on any two-dimensional orbit of L in U .

Consider the structure (Ω1 , L). This structure cannot be conformal to the punctured plane since it is prolongable. Assume F : (Ω1 , L) −→ {z : a < |z| < b} is conformal for some a, b > 0. The methods in [BH1] show that F extends as a homeomorphism up to the boundary piece Σ and we may assume that it maps Σ onto {w : |w| = b}. It then follows that as z → C, F (z) −→ {w : |w| = a}. Let h ∈ C(D) be a solution of L in D. ˜ holomorphic on {z : a < |z| < b} such that h = h ˜ ◦ F on Ω1 . There exists h ˜ extends continuously up Since h is continuous up to C and is constant on C, h to {w : |w| = a} and is constant on this circle.

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Advances in Phase Space Analysis of Partial Differential Equations: In Honor of Ferruccio Colombini's 60th Birthday by Luigi Ambrosio (auth.), Antonio Bove, Daniele Del Santo, M.K. Venkatesha Murthy (eds.)


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