| 1. |
- Lee, K. A., et al.
(författare)
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Regularity of a free boundary for viscosity solutions of nonlinear elliptic equations
- 2001
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Ingår i: Communications on Pure and Applied Mathematics. - 0010-3640 .- 1097-0312. ; 54:1, s. 43-56
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Tidskriftsartikel (refereegranskat)abstract
- Our objective in this paper is to analyze the regularity of the boundary of a set Omega2 that admits a solution u to the following overdetermined problem: F(D(2)u) = chi (Omega) in B, u = \ delu \ = 0 in B \ Omega. Here F is a convex, uniformly elliptic operator of homogeneity one, B is the unit ball in R-n, and the equation is satisfied in the viscosity sense. We also consider the case of concave F in R-2.
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| 2. |
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| 3. |
- Schatz, A. H., et al.
(författare)
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Stability, analyticity, and almost best approximation in maximum norm for parabolic finite element equations
- 1998
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Ingår i: Comm. Pure Appl. Math.. - 0010-3640 .- 1097-0312. ; 51:11-12, s. 1349-1385
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Tidskriftsartikel (refereegranskat)abstract
- We consider semidiscrete solutions in quasi-uniform finite element spaces of order O(hr) of the initial boundary value problem with Neumann boundary conditions for a second-order parabolic differential equation with time-independent coefficients in a bounded domain in R^N. We show that the semigroup on L∞, defined by the semidiscrete solution of the homogeneous equation, is bounded and analytic uniformly in h. We also show that the semidiscrete solution of the inhomogeneous equation is bounded in the space-time L∞-norm, modulo a logarithmic factor for r = 2, and we give a corresponding almost best approximation property.
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| 4. |
- Ameur, Yacin, et al.
(författare)
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Berezin Transform in Polynomial Bergman Spaces
- 2010
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Ingår i: Communications on Pure and Applied Mathematics. - : Wiley. - 0010-3640 .- 1097-0312. ; 63:12, s. 1533-1584
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Tidskriftsartikel (refereegranskat)abstract
- Fix a smooth weight function Q in the plane, subject to a growth condition from below Let K-m,K-n denote the reproducing kernel for the Hilbert space of analytic polynomials of degree at most n - 1 of finite L-2-norm with respect to the measure e-(mQ) dA Here dA is normalized area measure, and m is a positive real scaling parameter The (polynomial) Berezin measure dB(m,n)(< z0 >) (z) = K-m,K-n(z(0).z(0))(-1) vertical bar K-m,K-n(z.z(0))vertical bar(2)e(-mQ(z)) dA(z) for the point z(0) is a probability measure that defines the (polynomial) Berezin transform B-m,B-n f(z(0)) = integral(C) f dB(m,n)(< z0 >) for continuous f is an element of L-infinity (C). We analyze the semiclassical limit of the Berezin measure (and transform) as m -> +infinity while n = m tau + o(1), where tau is fixed, positive, and real We find that the Berezin measure for z(0) converges weak-star to the unit point mass at the point z(0) provided that Delta Q(z(0)) > 0 and that z(0) is contained in the interior of a compact set f(tau). defined as the coincidence set for an obstacle problem. As a refinement, we show that the appropriate local blowup of the Berezin measure converges to the standardized Gaussian measure in the plane For points z(0) is an element of C\f(tau), the Berezin measure cannot converge to the point mass at z(0) In the model case Q(z) = vertical bar z vertical bar(2), when f(tau) is a closed disk, we find that the Berezin measure instead converges to harmonic measure at z(0) relative to C\f(tau) Our results have applications to the study of the cigenvalues of random normal matrices The auxiliary results include weighted L-2-estimates for the equation partial derivative u = f when f is a suitable test function and the solution u is restricted by a polynomial growth bound at infinity.
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| 5. |
- Andersson, John, et al.
(författare)
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Optimal Regularity for the No-Sign Obstacle Problem
- 2013
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Ingår i: Communications on Pure and Applied Mathematics. - : Wiley. - 0010-3640 .- 1097-0312. ; 66:2, s. 245-262
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we prove the optimal C-1,C-1(B-1/2)-regularity for a general obstacle-type problem Delta u = f chi({u not equal 0}) in B-1, under the assumption that f * N is C-1,C-1(B-1), where N is the Newtonian potential. This is the weakest assumption for which one can hope to get C-1,C-1-regularity. As a by-product of the C-1,C-1-regularity we are able to prove that, under a standard thickness assumption on the zero set close to a free boundary point x(0), the free boundary is locally a C-1-graph close to x(0) provided f is Dini. This completely settles the question of the optimal regularity of this problem, which has been the focus of much attention during the last two decades.
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| 6. |
- Blackstone, Elliot, et al.
(författare)
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The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem
- 2023
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Ingår i: Communications on Pure and Applied Mathematics. - : Wiley. - 0010-3640 .- 1097-0312. ; 76:11, s. 3300-3345
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Tidskriftsartikel (refereegranskat)abstract
- The Bessel process models the local eigenvalue statistics near 0 of certain large positive definite matrices. In this work, we consider the probability (Figure presented.) where (Figure presented.) and (Figure presented.) is any non-negative integer. We obtain asymptotics for this probability as the size of the intervals becomes large, up to and including the oscillations of order 1. In these asymptotics, the most intricate term is a one-dimensional integral along a linear flow on a g-dimensional torus, whose integrand involves ratios of Riemann θ-functions associated to a genus g Riemann surface. We simplify this integral in two generic cases: (a) If the flow is ergodic, we compute the leading term in the asymptotics of this integral explicitly using Birkhoff's ergodic theorem. (b) If the linear flow has certain “good Diophantine properties”, we obtain improved estimates on the error term in the asymptotics of this integral. In the case when the flow is both ergodic and has “good Diophantine properties” (which is always the case for (Figure presented.), and “almost always” the case for (Figure presented.)), these results can be combined, yielding particularly precise and explicit large gap asymptotics.
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| 7. |
- Borcea, Julius, et al.
(författare)
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The Lee-Yang and Polya-Schur Programs. II. Theory of Stable Polynomials and Applications
- 2009
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Ingår i: Communications on Pure and Applied Mathematics. - : Wiley. - 0010-3640 .- 1097-0312. ; 62:12, s. 1595-1631
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Tidskriftsartikel (refereegranskat)abstract
- In the first part of this series we characterized all linear operators on spaces of multivariate polynomials preserving the property of being nonvanishing in products of open circular domains. For such sets this completes the multivariate generalization of the classification program initiated by Polya and Schur for univariate real polynomials. We build on these classification theorems to develop here a theory of multivariate stable polynomials. Applications and examples show that this theory provides a natural framework for dealing in a uniform way with Lee-Yang type problems in statistical mechanics, combinatorics, and geometric function theory in one or several variables. In particular, we answer a question of Hinkkanen on multivariate apolarity.
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| 8. |
- Constantin, Adrian, et al.
(författare)
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Exact steady periodic water waves with vorticity
- 2004
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Ingår i: Communications on Pure and Applied Mathematics. - : Wiley. - 0010-3640 .- 1097-0312. ; 57:4, s. 481-527
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Tidskriftsartikel (refereegranskat)abstract
- We consider the classical water wave problem described by the Euler equations with a free surface under the influence of gravity over a flat bottom. We construct two-dimensional inviscid periodic traveling waves with vorticity. They are symmetric waves whose profiles are monotone between each crest and trough. We use bifurcation and degree theory to construct a global connected set of such solutions. (C) 2004 Wiley Periodicals, Inc.
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| 9. |
- Dindos, Martin, et al.
(författare)
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Boundary value problems for second order elliptic operators satisfying a Carleson condition
- 2017
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Ingår i: Communications on Pure and Applied Mathematics. - : Wiley. - 0010-3640 .- 1097-0312. ; 70:7, s. 1316-1365
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Tidskriftsartikel (refereegranskat)abstract
- Let be a Lipschitz domain in Rn n ≥ 2, and L = divA∇· be a second order elliptic operator in divergence form. We establish solvability of the Dirichlet regularity problem with boundary data in H1,p(@) and of the Neumann problem with Lp(@) data for the operator L on Lipschitz domains with small Lipschitz con- stant. We allow the coefficients of the operator L to be rough obeying a certain Carleson condition with small norm. These results complete the results of [7] where the Lp(@) Dirichlet problem was considered under the same assumptions and [8] where the regularity and Neumann problems were considered on two dimensional domains.
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| 10. |
- Duits, Maurice
(författare)
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Gaussian Free Field in an Interlacing Particle System with Two Jump Rates
- 2013
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Ingår i: Communications on Pure and Applied Mathematics. - : Wiley. - 0010-3640 .- 1097-0312. ; 66:4, s. 600-643
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Tidskriftsartikel (refereegranskat)abstract
- We study the fluctuations of a random surface in a stochastic growth model on a system of interlacing particles placed on a two-dimensional lattice. There are two different types of particles, one with a low jump rate and the other with a high jump rate. In the large time limit, the random surface has a deterministic shape. Due to the different jump rates, the limit shape and the domain on which it is defined are not smooth. The main result is that the fluctuations of the random surface are governed by the Gaussian free field.
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