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62
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 ..."
Abstract

Cited by 74 (0 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,
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 28 (17 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 15 (6 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 13 (3 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
L p spectral theory of higherorder elliptic differential operators
 Bull. London Math. Soc
, 1997
"... 2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518 ..."
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Cited by 13 (1 self)
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2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518
Bounds of Riesz transforms on L p spaces for second order elliptic operators
 Ann. Inst. Fourier
"... Abstract. Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on R n or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p> 2, a necessary and sufficient condition is obtained for the boundedness of the Rie ..."
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Cited by 10 (1 self)
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Abstract. Let L = −div(A(x)∇) be a second order elliptic operator with real, symmetric, bounded measurable coefficients on R n or on a bounded Lipschitz domain subject to Dirichlet boundary condition. For any fixed p> 2, a necessary and sufficient condition is obtained for the boundedness of the Riesz transform ∇(L) −1/2 on the L p space. As an application, for 1 < p < 3 + ε, we establish the L p boundedness of Riesz transforms on Lipschitz domains for operators with V MO coefficients. The range of p is sharp. The closely related boundedness of ∇(L) −1/2 on weighted L 2 spaces is also studied. 1.
A Minmax Principle, Index of the Critical Point, and Existence of SignChanging Solutions to Elliptic Boundary Value Problems
, 1998
"... In this article we apply the minmax principle we developed in [6] to obtain signchanging solutions for superlinear and asymptotically linear Dirichlet problems. We prove that, when isolated, the local degree of any solution given by this minmax principle is +1. By combining the results of [6] with ..."
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Cited by 8 (2 self)
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In this article we apply the minmax principle we developed in [6] to obtain signchanging solutions for superlinear and asymptotically linear Dirichlet problems. We prove that, when isolated, the local degree of any solution given by this minmax principle is +1. By combining the results of [6] with the degreetheoretic results of Castro and Cossio in [5], in the case where the nonlinearity is asymptotically linear, we provide su#cient conditions for: i) the existence of at least four solutions (one of which changes sign exactly once), ii) the existence of at least five solutions (two of which change sign), and iii) the existence of precisely two signchanging solutions. For a superlinear problem in thin annuli we prove: i) the existence of a nonradial signchanging solution when the annulus is su#ciently thin, and ii) the existence of arbitrarily many signchanging nonradial solutions when, in addition, the annulus is two dimensional. The reader is referred to [7] where the existence...
THE REGULARITY AND NEUMANN PROBLEM FOR NONSYMMETRIC ELLIPTIC OPERATORS
, 2006
"... We will consider the Dirichlet problem Lu=0, in Ω ..."
ANALYTICITY OF LAYER POTENTIALS AND L 2 SOLVABILITY OF BOUNDARY VALUE PROBLEMS FOR DIVERGENCE FORM ELLIPTIC EQUATIONS WITH COMPLEX L ∞ COEFFICIENTS
, 705
"... Abstract. We consider divergence form elliptic operators of the form L= − div A(x)∇, defined inR n+1 ={(x, t)∈R n ×R}, n≥2, where the L ∞ coefficient matrix A is (n+ 1)×(n+1), uniformly elliptic, complex and tindependent. We show that for such operators, boundedness and invertibility of the corresp ..."
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Cited by 8 (6 self)
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Abstract. We consider divergence form elliptic operators of the form L= − div A(x)∇, defined inR n+1 ={(x, t)∈R n ×R}, n≥2, where the L ∞ coefficient matrix A is (n+ 1)×(n+1), uniformly elliptic, complex and tindependent. We show that for such operators, boundedness and invertibility of the corresponding layer potential operators on L2 (Rn)=L 2 (∂Rn+1 +), is stable under complex, L ∞ perturbations of the coefficient matrix. Using a variant of the Tb Theorem, we also prove that the layer potentials are bounded and invertible on L2 (Rn) whenever A(x) is real and symmetric (and thus, by our stability result, also when A is complex,‖A − A0‖ ∞ is small enough and A0 is real, symmetric, L ∞ and elliptic). In particular, we establish solvability of the Dirichlet and Neumann (and Regularity) problems, with L2 (resp. ˙L 2 1) data, for small complex perturbations of a real symmetric matrix. Previously, L2 solvability results for complex (or even real but nonsymmetric) coefficients were known to hold only for perturbations of constant matrices (and then only for the Dirichlet problem), or in the special case that the coefficients A j,n+1 = 0=An+1, j, 1 ≤ j≤n, which corresponds to the Kato square root problem.