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73
Implicit Partial Differential Equations,” Birkhäuser
, 1999
"... Abstract. We study a Dirichlet problem associated to some nonlinear partial di¤erential equations under additional constraints that are relevant in non linear elasticity. We also give several examples related to the complex eikonal equation, optimal design, potential wells or nematic elastomers. ..."
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Cited by 40 (9 self)
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Abstract. We study a Dirichlet problem associated to some nonlinear partial di¤erential equations under additional constraints that are relevant in non linear elasticity. We also give several examples related to the complex eikonal equation, optimal design, potential wells or nematic elastomers.
The Dirichlet Problem for the Total Variation Flow
, 2001
"... We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we u ..."
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Cited by 27 (9 self)
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We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L¹ for entropy solutions. To prove the existence we use the nonlinear semigroup theory and we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions.
The boundedness of classical operators on variable Lp spaces
"... Abstract. We show that many classical operators in harmonic analysis —such as maximal operators, singular integrals, commutators and fractional integrals— are bounded on the variable Lebesgue space Lp(·) whenever the HardyLittlewood maximal operator is bounded on Lp(·). Further, we show that such o ..."
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Cited by 16 (0 self)
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Abstract. We show that many classical operators in harmonic analysis —such as maximal operators, singular integrals, commutators and fractional integrals— are bounded on the variable Lebesgue space Lp(·) whenever the HardyLittlewood maximal operator is bounded on Lp(·). Further, we show that such operators satisfy vectorvalued inequalities. We do so by applying the theory of weighted norm inequalities and extrapolation. As applications we prove the CalderónZygmund inequality for solutions of4u = f in variable Lebesgue spaces, and prove the Calderón extension theorem for variable Sobolev spaces.
Inequalities In Rearrangement Invariant Function Spaces
, 1995
"... Introduction. Set up by Hardy & Littlewood, a theory of rearrangements was popularized by the wellknown book [HLP]. Rearrangements of functions are frequently used in real and harmonic analysis, in investigations about singular integrals and function spaces  see, e.g., [He], [ON], [ONW], [SW ..."
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Cited by 14 (1 self)
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Introduction. Set up by Hardy & Littlewood, a theory of rearrangements was popularized by the wellknown book [HLP]. Rearrangements of functions are frequently used in real and harmonic analysis, in investigations about singular integrals and function spaces  see, e.g., [He], [ON], [ONW], [SW]. P'olya & Szego and their followers demonstrated a good many isoperimetric theorems and inequalities by means of rearrangements  see [PS], a source book on this matter. More recent investigations have shown 178 G. TALENTI that rearrangements of functions fit well also into the theory of elliptic secondorder partial differential equations  see, e.g., [Bae], [Ta3] and the bibliography therein. Several types of rearrangements are known  presentations are in [Ka] and [Bae]. Here we limit ourselves to rearrangements `a la Hardy & Littlewood. 1.2. Definitions and basic properties. Let G be a measurable subset of R<F
On Ito's formula for multidimensional Brownian motion
 and Related Fields
, 2000
"... . Consider a ddimensional Brownian motion X = (X 1 ; : : : ; X d ) and a function F which belongs locally to the Sobolev space W 1;2 . We prove an extension of Ito's formula where the usual second order terms are replaced by the quadratic covariations [f k (X); X k ] involving the wea ..."
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Cited by 9 (0 self)
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. Consider a ddimensional Brownian motion X = (X 1 ; : : : ; X d ) and a function F which belongs locally to the Sobolev space W 1;2 . We prove an extension of Ito's formula where the usual second order terms are replaced by the quadratic covariations [f k (X); X k ] involving the weak first partial derivatives f k of F . In particular we show that for any locally squareintegrable function f the quadratic covariations [f(X); X k ] exist as limits in probability for any starting point, except for some polar set. The proof is based on new approximation results for forward and backward stochastic integrals. Key words: Ito's formula, Brownian motion, stochastic integrals, quadratic covariation, Dirichlet spaces, polar sets. Supported in part by ONR grant # N000149610262 and NSF grant # 9401109INT 1. Introduction The behavior of a smooth function F on R d along the paths of ddimensional Brownian motion is described as follows by Ito's formula. Let P x be the distribu...
Repulsion and Quantization in AlmostHarmonic Maps, and Asymptotics of the Harmonic Map Flow
"... We present an analysis of boundedenergy lowtension maps between 2spheres. By deriving sharp estimates for the ratio of length scales on which bubbles of opposite orientation develop, we show that we can establish a `quantization estimate' which constrains the energy of the map to lie near to ..."
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Cited by 9 (4 self)
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We present an analysis of boundedenergy lowtension maps between 2spheres. By deriving sharp estimates for the ratio of length scales on which bubbles of opposite orientation develop, we show that we can establish a `quantization estimate' which constrains the energy of the map to lie near to a discrete energy spectrum. One application is to the asymptotics of the harmonic map flow; we find uniform exponential convergence in time, in the case under consideration. Contents 1 Introduction 2 1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Statement of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Almostharmonic map results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Heat flow results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Heuristics of the proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . ....
Rigidity In The Harmonic Map Heat Flow
, 1997
"... We establish various uniformity properties of the harmonic map heat flow, including uniform convergence in L 2 exponentially as t ! 1, and uniqueness of the positions of bubbles at infinite time. Our hypotheses are that the flow is between 2spheres, and that the limit map and any bubbles share th ..."
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Cited by 8 (4 self)
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We establish various uniformity properties of the harmonic map heat flow, including uniform convergence in L 2 exponentially as t ! 1, and uniqueness of the positions of bubbles at infinite time. Our hypotheses are that the flow is between 2spheres, and that the limit map and any bubbles share the same orientation. 1. Introduction Let us consider smooth maps OE : S 2 ! S 2 . We use z = x + iy as a complex coordinate on the domain, obtained by stereographic projection, and write the metric as oe 2 dzdz, where oe(z) = 2 1 + jzj 2 : Similarly we have a coordinate u on the target, and a metric ae 2 dudu. We are using the notation dz = dx + idy; dz = dx \Gamma idy; with analogues for du and du, and we will write u z = 1 2 (u x \Gamma iu y ); u z = 1 2 (u x + iu y ): To the map OE we associate the energy densities e @ (OE) = ae 2 (u) oe 2 ju z j 2 ; e @ (OE) = ae 2 (u) oe 2 ju z j 2 ; Received October 30, 1995. 593 594 peter miles topping and e(OE) =...
Coupled Map Lattices Via Transfer Operators On Functions Of Bounded Variation
 S.J. van Strien, S.M. Verduyn Lunel) Kon. Nederl. Akad. Wetensch. Verhandelingen, Afd. Natuurkunde, Eerste Reeks
, 1996
"... We describe the transfer operator approach to coupled map lattices (CML) in cases where the local map is expanding but has no Markov partition (e.g. a general tent map). The coupling is allowed to be nonlocal, but the total influence of all sites j<F NaN> 6= i on site i must be small. The mai ..."
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Cited by 7 (4 self)
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We describe the transfer operator approach to coupled map lattices (CML) in cases where the local map is expanding but has no Markov partition (e.g. a general tent map). The coupling is allowed to be nonlocal, but the total influence of all sites j<F NaN> 6= i on site i must be small. The main technical tool are latticesize independent estimates of LasotaYorke type which show that the transfer (PerronFrobenius) operator of the coupled system is quasicompact as an operator on the space of functions of bounded variation. 1 Introduction The purpose of this note is to summarize results from [11] and from the unpublished thesis [12]. Let L be a finite or countable index set, e.g. L = Zor L = Zn dZ. We investigate timediscrete dynamics on the state space X = [0; 1] L that are composed of independent chaotic actions on each component [0; 1] of X followed by some weak interaction that does not destroy the chaotic character of the whole system. More specifically, let ø : [0; 1] ! [0; 1]...