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164
Wellposedness in sobolev spaces of the full water wave problem in 3d
 J. Amer. Math. Soc
, 1997
"... We consider the motion of the interface separating an inviscid, incompressible, irrotational fluid from a region of zero density in threedimensional space; we assume that the fluid region is below the vacuum, the fluid is under the influence of gravity and the surface tension is zero. Assume that t ..."
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Cited by 154 (3 self)
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We consider the motion of the interface separating an inviscid, incompressible, irrotational fluid from a region of zero density in threedimensional space; we assume that the fluid region is below the vacuum, the fluid is under the influence of gravity and the surface tension is zero. Assume that the density of mass of the fluid is one,
Steady transonic shocks and free boundary problems in infinite cylinders for the Euler equations
"... We are concerned with the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as the following secondorde ..."
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Cited by 68 (23 self)
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We are concerned with the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as the following secondorder nonlinear equation of mixed
Implementation of a boundary element method for high frequency scattering by convex polygons
 ADVANCES IN BOUNDARY INTEGRAL METHODS (PROCEEDINGS OF THE 5TH UK CONFERENCE ON BOUNDARY INTEGRAL METHODS
"... In this paper we consider the problem of timeharmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. H ..."
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Cited by 46 (20 self)
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In this paper we consider the problem of timeharmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretisation of a wellknown second kind combinedlayerpotential integral equation. We provide a proof that this equation and its adjoint are wellposed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.
Operator Theory and Harmonic Analysis
, 1996
"... Contents 1. Spectral Theory of Bounded Operators (A) Spectra and resolvents of bounded operators on Banach spaces (B) Holomorphic functional calculi of bounded operators 2. Spectral Theory of Unbounded Operators (C) Spectra and resolvents of closed operators in Banach spaces (D) Holomorphic function ..."
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Cited by 41 (11 self)
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Contents 1. Spectral Theory of Bounded Operators (A) Spectra and resolvents of bounded operators on Banach spaces (B) Holomorphic functional calculi of bounded operators 2. Spectral Theory of Unbounded Operators (C) Spectra and resolvents of closed operators in Banach spaces (D) Holomorphic functional calculi of operators of type S!+ 3. Quadratic Estimates (E) Quadratic norms of operators of type S!+ in Hilbert spaces (F) Boundedness of holomorphic functional calculi 4. Operators with Bounded Holomorphic Functional Calculi (G) Accretive operators (H) Operators of type S! and spectral projections 5. Singular Integrals (I) Convolutions and the functional calculus of \Gammai d dx (J) The Hilbert transform and Hardy spaces 6. Calder'onZygmund Theory (K) Maximal functions and the Calder'onZygmund decomposition (L) Singular integral operators 7. Functional Ca
A new approach to absolute continuity of elliptic measure, with applications to nonsymmetric equations
 Adv. in Math
"... In the late 50’s and early 60’s, the work of De Giorgi [De Gi] and Nash [N], and then Moser [Mo] initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients. Weak solutions in a domain Ω, a priori only in a Sobolev space W 2 1,loc ..."
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Cited by 41 (8 self)
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In the late 50’s and early 60’s, the work of De Giorgi [De Gi] and Nash [N], and then Moser [Mo] initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients. Weak solutions in a domain Ω, a priori only in a Sobolev space W 2 1,loc
Solvability of elliptic systems with square integrable boundary data, preprint Preprint arXiv:0809.4968v1 [math.AP
"... Abstract. We consider second order elliptic divergence form systems with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A for which the boundary value problem with L2 Dirichlet or Neumann data is well posed, is an open set. Furthermore ..."
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Cited by 32 (17 self)
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Abstract. We consider second order elliptic divergence form systems with complex measurable coefficients A that are independent of the transversal coordinate, and prove that the set of A for which the boundary value problem with L2 Dirichlet or Neumann data is well posed, is an open set. Furthermore we prove that these boundary value problems are well posed when A is either Hermitean, block or constant. Our methods apply to more general systems of PDEs and as an example we prove perturbation results for boundary value problems for differential forms. MSC classes: 35J25, 35J55, 47N20 1.
Weighted maximal regularity estimates and solvability of elliptic systems
 I. Invent. Math
"... Abstract. We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with L2 boundary data. Our methods yield full characterization of weak solutions, whose gradients have L2 estimates of a nontangential maximal function or of the ..."
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Cited by 30 (11 self)
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Abstract. We develop new solvability methods for divergence form second order, real and complex, elliptic systems above Lipschitz graphs, with L2 boundary data. Our methods yield full characterization of weak solutions, whose gradients have L2 estimates of a nontangential maximal function or of the square function, via an integral representation acting on the conormal gradient, with a singular operatorvalued kernel. The coefficients A may depend on all variables, but are assumed to be close to coefficients A0 that are independent of the coordinate transversal to the boundary, in the Carleson sense ‖A − A0‖C defined by Dahlberg. We obtain a number of a priori estimates and boundary behaviour under finiteness of ‖A − A0‖C. For example, the nontangential maximal function of a weak solution is controlled in L2 by the square function of its gradient. This estimate is new for systems in such generality, even for real nonsymmetric equations in dimension 3 or higher. The existence of a proof a priori to wellposedness, is also a new fact. As corollaries, we obtain wellposedness of the Dirichlet, Neumann and Dirichlet regularity problems under smallness of ‖A−A0‖C and wellposedness for A0, improving earlier results for real symmetric equations. Our methods build on an algebraic reduction to a first order system first made for coefficients A0 by the two authors and A. McIntosh in order to use functional calculus related to the Kato conjecture solution, and the main analytic tool for coefficients A is an operational calculus to prove weighted maximal regularity estimates.
The Dirichlet problem in Lipschitz domains with boundary data in Besov spaces for higher order elliptic systems with rough coefficients
, 2005
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Optimal size estimates for the inverse conductivity problem with one measurement
 Proc. Amer. Math. Soc
"... Abstract. We prove upper and lower estimates on the measure of an inclusion D in a conductor Ω in terms of one pair of current and potential boundary measurements. The growth rates of such estimates are essentially best possible. 1. ..."
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Cited by 24 (5 self)
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Abstract. We prove upper and lower estimates on the measure of an inclusion D in a conductor Ω in terms of one pair of current and potential boundary measurements. The growth rates of such estimates are essentially best possible. 1.