The Cauchy problem of the vacuum Einstein’s equations aims to find a semimetric gαβ of a spacetim... more The Cauchy problem of the vacuum Einstein’s equations aims to find a semimetric gαβ of a spacetime with vanishing Ricci curvature Rα,β and prescribed initial data. Under the harmonic gauge condition, the equations Rα,β = 0 are transferred into a system of quasi-linear wave equations which are called the reduced Einstein equations. The initial data for Einstein’s equations are a proper Riemannian metric hab and a second fundamental form Kab. A necessary condition for the reduced Einstein equation to satisfy the vacuum equations is that the initial data satisfy Einstein constraint equations. Hence the data (hab,Kab) cannot serve as initial data for the reduced Einstein equations. Previous results in the case of asymptotically flat spacetimes provide a solution to the constraint equations in one type of Sobolev spaces, while initial data for the evolution equations belong to a different type of Sobolev spaces. The goal of our work is to resolve this incompatibility and to show that und...
Null quadrature domains are unbounded domains in R n (n ≥ 2) with external gravitational force ze... more Null quadrature domains are unbounded domains in R n (n ≥ 2) with external gravitational force zero in some generalized sense. In this paper we prove that the complement of null quadrature domain is a convex set with real analytic boundary. We establish the quadratic growth estimate for the Schwarz potential of a null quadrature domain which reduces our main result to Theorem II of [8] on the regularity of solution to the classical global free boundary problem for Laplacian. We also show that any null quadrature domain with non-zero upper Lebesgue density at infinity is half-space.
Modern Developments in Multivariate Approximation, 2003
The main result of this paper gives a sufficient condition for removability of an isolated singul... more The main result of this paper gives a sufficient condition for removability of an isolated singularity of a harmonic function. The condition is given in terms of Newtonian capacity. In addition, an application to an approximation problem is presented.
We cast the non–isentropic Einstein–Euler system into a symmetric hyperbolic form. Such systems a... more We cast the non–isentropic Einstein–Euler system into a symmetric hyperbolic form. Such systems are very suited to treat initial value problems of hyperbolic type. We obtain this form by using the pressure p and not the density ρ as a variable. However, the system becomes degenerate when the pressure p approaches zero, and in these cases we regularise the system by replacing the pressure with an appropriate new matter variable, the Makino variable.
The characterization of null quadrature domains in R n (n ≥ 3) has been an open problem throughou... more The characterization of null quadrature domains in R n (n ≥ 3) has been an open problem throughout the past two and a half decades. A substantial contribution was done by Friedman and Sakai [10]; they showed that if the complement is bounded, then null quadrature domains are exactly the complement of ellipsoids. The first result with unbounded complements appeared in [15], there it is assumed the complement is contained in an infinitely cylinder. The aim of this paper is to show the relation between null quadrature domains and Newton's theorem on the gravitational force induced by homogeneous homoeoidal ellipsoids. We also succeed to make progress in the classification problem and we show that if the boundary of null quadrature domain is contained in a strip and the complement satisfies a certain capacity condition at infinity, then it must be a half-space or a complement of a strip. In addition, we present a Phragmén-Lindelöf type theorem which seems to be forgotten in the literature. 1 |x − y| dy = const. for all x ∈ K.
The characterization of null quadrature domains in R (n ≥ 3) has been an open problem throughout ... more The characterization of null quadrature domains in R (n ≥ 3) has been an open problem throughout the past two and a half decades. A substantial contribution was done by Friedman and Sakai [10]; they showed that if the complement is bounded, then null quadrature domains are exactly the complement of ellipsoids. The first result with unbounded complements appeared in [15], there it is assumed the complement is contained in an infinitely cylinder. The aim of this paper is to show the relation between null quadrature domains and Newton’s theorem on the gravitational force induced by homogeneous homoeoidal ellipsoids. We also succeed to make progress in the classification problem and we show that if the boundary of null quadrature domain is contained in a strip and the complement satisfies a certain capacity condition at infinity, then it must be a half-space or a complement of a strip. In addition, we present a Phragmén-Lindelöf type theorem which seems to be forgotten in the literature.
We consider the problem of reconstruction of planar domains from their moments. Specifically, we ... more We consider the problem of reconstruction of planar domains from their moments. Specifically, we consider domains with boundary which can be represented by a union of a finite number of pieces whose graphs are solutions of a linear differential equation with polynomial coefficients. This includes domains with piecewisealgebraic and, in particular, piecewise-polynomial boundaries. Our approach is based on one-dimensional reconstruction method of [5] and a kind of "separation of variables" which reduces the planar problem to two one-dimensional problems, one of them parametric. Several explicit examples of reconstruction are given. Another main topic of the paper concerns "invisible sets" for various types of incomplete moment measurements. We suggest a certain point of view which stresses remarkable similarity between several apparently unrelated problems. In particular, we discuss zero quadrature domains (invisible for harmonic polynomials), invisibility for powers of a given polynomial, and invisibility for complex moments (Wermer's theorem and further developments). The common property we would like to stress is a "rigidity" and symmetry of the invisible objects.
We show global existence of classical solutions for the nonlinear Nordström theory with a source ... more We show global existence of classical solutions for the nonlinear Nordström theory with a source term and a cosmological constant under the assumption that the source term is small in an appropriate norm, while in some cases no smallness assumption on the initial data is required. In this theory, the gravitational field is described by a single scalar function that satisfies a certain semi-linear wave equation. We consider spatial periodic deviation from the background metric, that is why we study the semi-linear wave equation on the three-dimensional torus $${\mathord {\mathbb T}}^3$$ T 3 in the Sobolev spaces $$H^m({\mathord {\mathbb T}}^3)$$ H m ( T 3 ) . We apply two methods to achieve the existence of global solutions, the first one is by Fourier series, and in the second one, we write the semi-linear wave equation in a non-conventional way as a symmetric hyperbolic system. We also provide results concerning the asymptotic behavior of these solutions and, finally, a blow-up res...
Using a compactness argument, weintroduce a PhragmenLindelof type theoremforfunctionswithboundedL... more Using a compactness argument, weintroduce a PhragmenLindelof type theoremforfunctionswithboundedLaplacian. Thetechniqueisveryusefulinstudy- ing unbounded free boundary problems near the innity point and also in approxi- mating integrable harmonic functions by those that decrease rapidly at innity. The method isflexible in the sense that it can be applied to any operator which admits the standard elliptic estimate.
In the unit ball B(0,1), let u and Ω (a domain in R N) solve the following overdetermined problem... more In the unit ball B(0,1), let u and Ω (a domain in R N) solve the following overdetermined problem: ∆u = χΩ in B(0,1), 0 ∈ ∂Ω, u = |∇u | = 0 in B(0,1) \ Ω, where χΩ denotes the characteristic function, and the equation is satisfied in the sense of distributions. If the complement of Ω does not develop cusp singularities at the origin then we prove ∂Ω is analytic in some small neighborhood of the origin. The result can be modified to yield for more general divergence form operators. As an application of this, then, we obtain the regularity of the boundary of a domain without the Pompeiu property, provided its complement has no cusp singularities. 1.
We show the continuity of the flow map for quasilinear symmetric hyperbolic systems with general ... more We show the continuity of the flow map for quasilinear symmetric hyperbolic systems with general right-hand sides in different functional setting, including weighted Sobolev spaces H s,δ. An essential tool to achieve the continuity of the flow map is a new type of energy estimate, which we all it a low regularity energy estimate. We then apply these results to the Euler-Poisson system which describes various systems of physical interests.
This note deals with two topics of linear algebra. We give a simple and short proof of the multip... more This note deals with two topics of linear algebra. We give a simple and short proof of the multiplicative property of the determinant and provide a constructive formula for rotations. The derivation of the rotation matrix relies on simple matrix calculations and thus can be presented in an elementary linear algebra course. We also classify all invariant subspaces of equiangular rotations in 4D.
We prove that if Ω is a simply connected quadrature domain (QD) of a distribution with compact su... more We prove that if Ω is a simply connected quadrature domain (QD) of a distribution with compact support and the point of infinity belongs to the boundary, then the boundary has an asymptotic curve that is a straight line, parabola or infinite ray. In other words, such QDs in the plane are perturbations of null QDs.
This paper deals with the applications of weighted Besov spaces to elliptic equations on asymptot... more This paper deals with the applications of weighted Besov spaces to elliptic equations on asymptotically flat Riemannian manifolds, and in particular to the solutions of Einstein's constraints equations. We establish existence theorems for the Hamiltonian and the momentum constraints with constant mean curvature and with a background metric that satisfies very low regularity assumptions. These results extend the regularity results of Holst, Nagy and Tsogtgerel about the constraint equations on compact manifolds in the Besov space B s p,p [23], to asymptotically flat manifolds. We also consider the Brill-Cantor criterion in the weighted Besov spaces. Our results improve the regularity assumptions on asymptotically flat manifolds [15, 29], as well as they enable us to construct the initial data for the Einstein-Euler system.
Null quadrature domains are unbounded domains in R n (n ≥ 2) with external gravitational force ze... more Null quadrature domains are unbounded domains in R n (n ≥ 2) with external gravitational force zero in some generalized sense. In this paper we prove a quadratic growth estimate of the Schwarz potential of a null quadrature domain and conclude by a theorem of Caffarelli, Karp and Shahgolian that any null quadrature domain is the complement of a convex set with analytic boundary. Using this result we prove that a null quadrature domain with a non-zero upper Lebesgue density at infinity is half-space.
The Cauchy problem of the vacuum Einstein's equations aims to find a semimetric g αβ of a spaceti... more The Cauchy problem of the vacuum Einstein's equations aims to find a semimetric g αβ of a spacetime with vanishing Ricci curvature R α,β and prescribed initial data. Under the harmonic gauge condition, the equations R α,β = 0 are transferred into a system of quasi-linear wave equations which are called the reduced Einstein equations. The initial data for Einstein's equations are a proper Riemannian metric h ab and a second fundamental form K ab. A necessary condition for the reduced Einstein equation to satisfy the vacuum equations is that the initial data satisfy Einstein constraint equations. Hence the data (h ab , K ab) cannot serve as initial data for the reduced Einstein equations. Previous results in the case of asymptotically flat spacetimes provide a solution to the constraint equations in one type of Sobolev spaces, while initial data for the evolution equations belong to a different type of Sobolev spaces. The goal of our work is to resolve this incompatibility and to show that under the harmonic gauge the vacuum Einstein equations are well-posed in one type of Sobolev spaces.
This is the second part of our work concerning the well posedness of the coupled Einstein-Euler s... more This is the second part of our work concerning the well posedness of the coupled Einstein-Euler system in an asymptotically flat spacetime. Here we prove a local in time existence and uniqueness theorem of classical solutions of the evolution equations. We use the condition that the energy density might vanish or tends to zero at infinity and that the pressure is a certain function of the energy density, conditions which are used to describe simplified stellar models. In order to achieve our goals we are enforced, by the complexity of the problem, to deal with these equations in a new type of weighted Sobolev spaces of fractional order. Beside their construction, we develop tools for PDEs and techniques for hyperbolic equations in these spaces. The well posedness is obtained in these spaces. The results obtained are related to and generalize earlier works of Rendall [23] for the Euler-Einstein system under the restriction of time symmetry and of Gamblin [10] for the simpler Euler-Poisson system.
The Cauchy problem of the vacuum Einstein’s equations aims to find a semimetric gαβ of a spacetim... more The Cauchy problem of the vacuum Einstein’s equations aims to find a semimetric gαβ of a spacetime with vanishing Ricci curvature Rα,β and prescribed initial data. Under the harmonic gauge condition, the equations Rα,β = 0 are transferred into a system of quasi-linear wave equations which are called the reduced Einstein equations. The initial data for Einstein’s equations are a proper Riemannian metric hab and a second fundamental form Kab. A necessary condition for the reduced Einstein equation to satisfy the vacuum equations is that the initial data satisfy Einstein constraint equations. Hence the data (hab,Kab) cannot serve as initial data for the reduced Einstein equations. Previous results in the case of asymptotically flat spacetimes provide a solution to the constraint equations in one type of Sobolev spaces, while initial data for the evolution equations belong to a different type of Sobolev spaces. The goal of our work is to resolve this incompatibility and to show that und...
Null quadrature domains are unbounded domains in R n (n ≥ 2) with external gravitational force ze... more Null quadrature domains are unbounded domains in R n (n ≥ 2) with external gravitational force zero in some generalized sense. In this paper we prove that the complement of null quadrature domain is a convex set with real analytic boundary. We establish the quadratic growth estimate for the Schwarz potential of a null quadrature domain which reduces our main result to Theorem II of [8] on the regularity of solution to the classical global free boundary problem for Laplacian. We also show that any null quadrature domain with non-zero upper Lebesgue density at infinity is half-space.
Modern Developments in Multivariate Approximation, 2003
The main result of this paper gives a sufficient condition for removability of an isolated singul... more The main result of this paper gives a sufficient condition for removability of an isolated singularity of a harmonic function. The condition is given in terms of Newtonian capacity. In addition, an application to an approximation problem is presented.
We cast the non–isentropic Einstein–Euler system into a symmetric hyperbolic form. Such systems a... more We cast the non–isentropic Einstein–Euler system into a symmetric hyperbolic form. Such systems are very suited to treat initial value problems of hyperbolic type. We obtain this form by using the pressure p and not the density ρ as a variable. However, the system becomes degenerate when the pressure p approaches zero, and in these cases we regularise the system by replacing the pressure with an appropriate new matter variable, the Makino variable.
The characterization of null quadrature domains in R n (n ≥ 3) has been an open problem throughou... more The characterization of null quadrature domains in R n (n ≥ 3) has been an open problem throughout the past two and a half decades. A substantial contribution was done by Friedman and Sakai [10]; they showed that if the complement is bounded, then null quadrature domains are exactly the complement of ellipsoids. The first result with unbounded complements appeared in [15], there it is assumed the complement is contained in an infinitely cylinder. The aim of this paper is to show the relation between null quadrature domains and Newton's theorem on the gravitational force induced by homogeneous homoeoidal ellipsoids. We also succeed to make progress in the classification problem and we show that if the boundary of null quadrature domain is contained in a strip and the complement satisfies a certain capacity condition at infinity, then it must be a half-space or a complement of a strip. In addition, we present a Phragmén-Lindelöf type theorem which seems to be forgotten in the literature. 1 |x − y| dy = const. for all x ∈ K.
The characterization of null quadrature domains in R (n ≥ 3) has been an open problem throughout ... more The characterization of null quadrature domains in R (n ≥ 3) has been an open problem throughout the past two and a half decades. A substantial contribution was done by Friedman and Sakai [10]; they showed that if the complement is bounded, then null quadrature domains are exactly the complement of ellipsoids. The first result with unbounded complements appeared in [15], there it is assumed the complement is contained in an infinitely cylinder. The aim of this paper is to show the relation between null quadrature domains and Newton’s theorem on the gravitational force induced by homogeneous homoeoidal ellipsoids. We also succeed to make progress in the classification problem and we show that if the boundary of null quadrature domain is contained in a strip and the complement satisfies a certain capacity condition at infinity, then it must be a half-space or a complement of a strip. In addition, we present a Phragmén-Lindelöf type theorem which seems to be forgotten in the literature.
We consider the problem of reconstruction of planar domains from their moments. Specifically, we ... more We consider the problem of reconstruction of planar domains from their moments. Specifically, we consider domains with boundary which can be represented by a union of a finite number of pieces whose graphs are solutions of a linear differential equation with polynomial coefficients. This includes domains with piecewisealgebraic and, in particular, piecewise-polynomial boundaries. Our approach is based on one-dimensional reconstruction method of [5] and a kind of "separation of variables" which reduces the planar problem to two one-dimensional problems, one of them parametric. Several explicit examples of reconstruction are given. Another main topic of the paper concerns "invisible sets" for various types of incomplete moment measurements. We suggest a certain point of view which stresses remarkable similarity between several apparently unrelated problems. In particular, we discuss zero quadrature domains (invisible for harmonic polynomials), invisibility for powers of a given polynomial, and invisibility for complex moments (Wermer's theorem and further developments). The common property we would like to stress is a "rigidity" and symmetry of the invisible objects.
We show global existence of classical solutions for the nonlinear Nordström theory with a source ... more We show global existence of classical solutions for the nonlinear Nordström theory with a source term and a cosmological constant under the assumption that the source term is small in an appropriate norm, while in some cases no smallness assumption on the initial data is required. In this theory, the gravitational field is described by a single scalar function that satisfies a certain semi-linear wave equation. We consider spatial periodic deviation from the background metric, that is why we study the semi-linear wave equation on the three-dimensional torus $${\mathord {\mathbb T}}^3$$ T 3 in the Sobolev spaces $$H^m({\mathord {\mathbb T}}^3)$$ H m ( T 3 ) . We apply two methods to achieve the existence of global solutions, the first one is by Fourier series, and in the second one, we write the semi-linear wave equation in a non-conventional way as a symmetric hyperbolic system. We also provide results concerning the asymptotic behavior of these solutions and, finally, a blow-up res...
Using a compactness argument, weintroduce a PhragmenLindelof type theoremforfunctionswithboundedL... more Using a compactness argument, weintroduce a PhragmenLindelof type theoremforfunctionswithboundedLaplacian. Thetechniqueisveryusefulinstudy- ing unbounded free boundary problems near the innity point and also in approxi- mating integrable harmonic functions by those that decrease rapidly at innity. The method isflexible in the sense that it can be applied to any operator which admits the standard elliptic estimate.
In the unit ball B(0,1), let u and Ω (a domain in R N) solve the following overdetermined problem... more In the unit ball B(0,1), let u and Ω (a domain in R N) solve the following overdetermined problem: ∆u = χΩ in B(0,1), 0 ∈ ∂Ω, u = |∇u | = 0 in B(0,1) \ Ω, where χΩ denotes the characteristic function, and the equation is satisfied in the sense of distributions. If the complement of Ω does not develop cusp singularities at the origin then we prove ∂Ω is analytic in some small neighborhood of the origin. The result can be modified to yield for more general divergence form operators. As an application of this, then, we obtain the regularity of the boundary of a domain without the Pompeiu property, provided its complement has no cusp singularities. 1.
We show the continuity of the flow map for quasilinear symmetric hyperbolic systems with general ... more We show the continuity of the flow map for quasilinear symmetric hyperbolic systems with general right-hand sides in different functional setting, including weighted Sobolev spaces H s,δ. An essential tool to achieve the continuity of the flow map is a new type of energy estimate, which we all it a low regularity energy estimate. We then apply these results to the Euler-Poisson system which describes various systems of physical interests.
This note deals with two topics of linear algebra. We give a simple and short proof of the multip... more This note deals with two topics of linear algebra. We give a simple and short proof of the multiplicative property of the determinant and provide a constructive formula for rotations. The derivation of the rotation matrix relies on simple matrix calculations and thus can be presented in an elementary linear algebra course. We also classify all invariant subspaces of equiangular rotations in 4D.
We prove that if Ω is a simply connected quadrature domain (QD) of a distribution with compact su... more We prove that if Ω is a simply connected quadrature domain (QD) of a distribution with compact support and the point of infinity belongs to the boundary, then the boundary has an asymptotic curve that is a straight line, parabola or infinite ray. In other words, such QDs in the plane are perturbations of null QDs.
This paper deals with the applications of weighted Besov spaces to elliptic equations on asymptot... more This paper deals with the applications of weighted Besov spaces to elliptic equations on asymptotically flat Riemannian manifolds, and in particular to the solutions of Einstein's constraints equations. We establish existence theorems for the Hamiltonian and the momentum constraints with constant mean curvature and with a background metric that satisfies very low regularity assumptions. These results extend the regularity results of Holst, Nagy and Tsogtgerel about the constraint equations on compact manifolds in the Besov space B s p,p [23], to asymptotically flat manifolds. We also consider the Brill-Cantor criterion in the weighted Besov spaces. Our results improve the regularity assumptions on asymptotically flat manifolds [15, 29], as well as they enable us to construct the initial data for the Einstein-Euler system.
Null quadrature domains are unbounded domains in R n (n ≥ 2) with external gravitational force ze... more Null quadrature domains are unbounded domains in R n (n ≥ 2) with external gravitational force zero in some generalized sense. In this paper we prove a quadratic growth estimate of the Schwarz potential of a null quadrature domain and conclude by a theorem of Caffarelli, Karp and Shahgolian that any null quadrature domain is the complement of a convex set with analytic boundary. Using this result we prove that a null quadrature domain with a non-zero upper Lebesgue density at infinity is half-space.
The Cauchy problem of the vacuum Einstein's equations aims to find a semimetric g αβ of a spaceti... more The Cauchy problem of the vacuum Einstein's equations aims to find a semimetric g αβ of a spacetime with vanishing Ricci curvature R α,β and prescribed initial data. Under the harmonic gauge condition, the equations R α,β = 0 are transferred into a system of quasi-linear wave equations which are called the reduced Einstein equations. The initial data for Einstein's equations are a proper Riemannian metric h ab and a second fundamental form K ab. A necessary condition for the reduced Einstein equation to satisfy the vacuum equations is that the initial data satisfy Einstein constraint equations. Hence the data (h ab , K ab) cannot serve as initial data for the reduced Einstein equations. Previous results in the case of asymptotically flat spacetimes provide a solution to the constraint equations in one type of Sobolev spaces, while initial data for the evolution equations belong to a different type of Sobolev spaces. The goal of our work is to resolve this incompatibility and to show that under the harmonic gauge the vacuum Einstein equations are well-posed in one type of Sobolev spaces.
This is the second part of our work concerning the well posedness of the coupled Einstein-Euler s... more This is the second part of our work concerning the well posedness of the coupled Einstein-Euler system in an asymptotically flat spacetime. Here we prove a local in time existence and uniqueness theorem of classical solutions of the evolution equations. We use the condition that the energy density might vanish or tends to zero at infinity and that the pressure is a certain function of the energy density, conditions which are used to describe simplified stellar models. In order to achieve our goals we are enforced, by the complexity of the problem, to deal with these equations in a new type of weighted Sobolev spaces of fractional order. Beside their construction, we develop tools for PDEs and techniques for hyperbolic equations in these spaces. The well posedness is obtained in these spaces. The results obtained are related to and generalize earlier works of Rendall [23] for the Euler-Einstein system under the restriction of time symmetry and of Gamblin [10] for the simpler Euler-Poisson system.
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Papers by Lavi Karp