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An inequality on torsional rigidity is established. For tangential polygons this inequality is stronger than an inequality of Polya and Szego for convex domains. (A survey of related work, not in the journal submission, is presented in the arXiv version.), Comment: 109 pages, 16 figures. (First 2 Parts have been submitted, and accepted subject to minor revision. The supplementary material was motivated by one of the two referees wanting more on the "context" etc.)

This paper gives applications of the enclosure method introduced by the author to typical inverse obstacle and crack scattering problems in two dimensions. Explicit extraction formulae of the convex hull of unknown polygonal sound-hard obstacles and piecewise linear cracks from the far field pattern of the scattered field at a fixed wave number and at most two incident directions are given. The main new points of this paper are: a combination of the enclosure method and the Herglotz wave function; explicit construction of the density in the Herglotz wave function by using the idea of the Vekua transform. By virtue of the construction, one can avoid any restriction on the wave number in the extraction formulae. An attempt for the case when the far field pattern is given on limited angles is also given., 19 pages, final version

We study the resolvent for nontrapping obstacles on manifolds with Euclidean ends. It is well known that for such manifolds, the outgoing resolvent satisfies $\|\chi R(k) \chi\|_{L^2\to L^2}\leq C{k}^{-1}$ for ${k}>1$, but the constant $C$ has been little studied. We show that, for high frequencies, the constant is bounded above by $2/\pi$ times the length of the longest generalized bicharacteristic of $|\xi|_g^2-1$ remaining in the support of $\chi.$ We show that this estimate is optimal in the case of manifolds without boundary. We then explore the implications of this result for the numerical analysis of the Helmholtz equation., Comment: 40 pages

Using perfectly matched layers for the computation of resonances in open systems typically produces artificial or spurious resonances. We analyze the dependency of these artificial resonances with respect to the discretization parameters and the complex scaling function. In particular, we study the differences between a standard frequency independent complex scaling and a frequency dependent one. While the standard scaling leads to a linear eigenvalue problem, the frequency dependent scaling leads to a polynomial one. Our studies show that the location of artificial resonances is more convenient for the frequency dependent scaling than for a standard scaling. Moreover, the artificial resonances of a frequency dependent scaling are less sensitive to the discretization parameters. Hence, the use of a frequency dependent scaling simplifies the choice of the corresponding discretization parameters.

Solutions to the Helmholtz equation within an equilateral triangle which solve either the Dirichlet or Neumann problem are investigated. This is done by introducing a pair of differential operators, derived from symmetry considerations, which demonstrate interesting relationships among these solutions. One of these operators preserves the boundary condition while generating an orthogonal solution and the other leads to a bijection between solutions of the Dirichlet and Neumann problems.

Motivated by the state-based peridynamic framework, we introduce a new nonlocal Laplacian that exhibits double nonlocality through the use of iterated integral operators. The operator introduces additional degrees of flexibility that can allow for better representation of physical phenomena at different scales and in materials with different properties. We study mathematical properties of this state-based Laplacian, including connections with other nonlocal and local counterparts. Finally, we obtain explicit rates of convergence for this doubly nonlocal operator to the classical Laplacian as the radii for the horizons of interaction kernels shrink to zero.

This work extends monotonicity-based methods in inverse problems to the case of the Helmholtz (or stationary Schr\"odinger) equation $(\Delta + k^2 q) u = 0$ in a bounded domain for fixed non-resonance frequency $k>0$ and real-valued scattering coefficient function $q$. We show a monotonicity relation between the scattering coefficient $q$ and the local Neumann-Dirichlet operator that holds up to finitely many eigenvalues. Combining this with the method of localized potentials, or Runge approximation, adapted to the case where finitely many constraints are present, we derive a constructive monotonicity-based characterization of scatterers from partial boundary data. We also obtain the local uniqueness result that two coefficient functions $q_1$ and $q_2$ can be distinguished by partial boundary data if there is a neighborhood of the boundary where $q_1\geq q_2$ and $q_1\not\equiv q_2$.

The inverse scattering problem under consideration is to reconstruct both the shape and the impedance function of an impenetrable two-dimensional obstacle from the far field pattern for scattering of time-harmonic acoustic or E-polarized electromagnetic plane waves. We propose an inverse algorithm that is based on a system of nonlinear boundary integral equations associated with a single-layer potential approach to solve the forward scattering problem. This extends the approach we suggested for an inverse boundary value problem for harmonic functions in Kress and Rundell(2005) and is a counterpart of our earlier work on inverse scattering for shape and impedance in Kress and Rundell(2001). We present the mathematical foundation of the method and exhibit its feasibility by numerical examples.

In multiphase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise due to changes in material properties at an immersed interface or embedded boundary, which may have an irregular shape. Consequently, the solution and its gradient can be discontinuous, and numerical methods can be difficult to design. Here a new method is presented and analyzed, using a simple formulation of one-dimensional finite differences on a Cartesian grid, allowing for a relatively easy setup for one-, two-, or three-dimensional problems. The derivation is relatively simple and mainly involves centered finite difference formulas, with less reliance on the Taylor series expansions of typical immersed interface method derivations. The method preserves a sharp interface with discontinuous solutions, obtained from a small number of iterations (approximately five) of solving a symmetric linear system with updates to the right-hand side. Second-order accuracy is rigorously proven in one spatial dimension and demonstrated through numerical examples in two and three spatial dimensions. The method is tested here on the variable-coefficient Poisson equation, and it could be extended for use on time-dependent problems of heat transfer, fluid dynamics, or other applications.

International audience; Abstract The topological sensitivity analysis consists to provide an asymptotic expansion of a shape functional with respect to emerging of small holes in the interior of the domain occupied by the body. In this paper, such an expansion is obtained for the modified Helmholtz equation with an impedance condition prescribed on the boundary of a hole. The topological derivative is then used for numerical simulations for an inverse problem.

We present a regularization strategy that leads to well-conditioned boundary integral equation formulations of Helmholtz equations with impedance boundary conditions in two-dimensional Lipschitz domains. We consider both the case of classical impedance boundary conditions, as well as the case of transmission impedance conditions wherein the impedances are certain coercive operators. The latter type of problems is instrumental in the speed up of the convergence of Domain Decomposition Methods for Helmholtz problems. Our regularized formulations use as unknowns the Dirichlet traces of the solution on the boundary of the domain. Taking advantage of the increased regularity of the unknowns in our formulations, we show through a variety of numerical results that a graded-mesh based Nystr\"om discretization of these regularized formulations leads to efficient and accurate solutions of interior and exterior Helmholtz problems with impedance boundary conditions., Comment: arXiv admin note: text overlap with arXiv:1509.04415

We study Carleson inequalities in a framework of parabolic Hardy spaces. Similar results for parabolic Bergman spaces are discussed in [NSY1] (see also [NSY2]), where $\tau$-Carleson measures play an important roll. In the present case, $T_{\tau}$-Carleson measures are useful. We give an relation between these measures.

The aim of this paper is to consider the asymptotic behavior of boundary value problems in n-dimensional domains with periodically placed particles, with a general microscopic boundary condition on the particles and a p-Laplace diffusion operator on the interior, in the case in which the particles are of critical size. We consider the cases in which {1 , {n\geq 3} . In fact, in contrast to previous results in the literature, we formulate the microscopic boundary condition in terms of a Robin type condition, involving a general maximal monotone graph, which also includes the case of microscopic Dirichlet boundary conditions. In this way we unify the treatment of apparently different formulations, which before were considered separately. We characterize the so called “strange term” in the homogenized problem for the case in which the particles are balls of critical size. Moreover, by studying an application in Chemical Engineering, we show that the critically sized particles lead to a more effective homogeneous reaction than noncritically sized particles.

In this paper, the simulation of the time-fractional-forced-damped-wave equation (the diffusion advection forced wave) is given for different parameters. The common finite difference rules beside the backward Grunwald–Letnikov scheme are used to find the approximation solution of this model. The paper discusses also the effects of the memory, the internal force (resistance) and the external force on the travelling wave. We follow the waves till they reach their stationary waves. The Von-Neumann stability condition is also considered and discussed. Besides the simulation of the time evolution of the approximation solution of the classical and time-fractional model, the stationary solutions are also simulated. All the numerical results are compared for different values of time.

For a class of asymptotically periodic Schrödinger-Poisson systems with critical growth, the existence of ground states is established. The proof is based on the method of Nehari manifold and concentration compactness principle.

We are concerned with the acoustic scattering by an extended obstacle surrounded by point-like obstacles. The extended obstacle is supposed to be rigid while the point-like obstacles are modeled by point perturbations of the exterior Laplacian. In the first part, we consider the forward problem. Following two equivalent approaches (the Foldy formal method and the Krein resolvent method), we show that the scattered field is a sum of two contributions: one is due to the diffusion by the extended obstacle and the other arises from the linear combination of the interactions between the point-like obstacles and the interaction between the point-like obstacles with the extended one. In the second part, we deal with the inverse problem. It consists in reconstructing both the extended and point-like scatterers from the corresponding far-field pattern. To solve this problem, we describe and justify the factorization method of Kirsch. Using this method, we provide several numerical results and discuss the multiple scattering effect concerning both the interactions between the point-like obstacles and between these obstacles and the extended one.

International audience; This paper concerns the asymptotic expansion of the solution of the Dirichlet-Laplace problem in a domain with small inclusions. This problem is well understood for the Neumann condition in dimension greater than two or Dirichlet condition in dimension greater than three. The case of two circular inclusions in a bidimensional domain was considered in [1]. In this paper, we generalize the previous result to any shape and relax the assumptions of regularity and support of the data. Our approach uses conformal mapping and suitable lifting of Dirichlet conditions. We also analyze configurations with several scales for the distance between the inclusions (when the number is larger than 2).

In this note, we exhibit a three dimensional structure that permits to guide waves. This structure is obtained by a geometrical perturbation of a 3D periodic domain that consists of a three dimensional grating of equi-spaced thin pipes oriented along three orthogonal directions. Homogeneous Neumann boundary conditions are imposed on the boundary of the domain. The diameter of the section of the pipes, of order $\epsilon$ \textgreater{} 0, is supposed to be small. We prove that, for $\epsilon$ small enough, shrinking the section of one line of the grating by a factor of $\sqrt$ $\mu$ (0 \textless{} $\mu$ \textless{} 1) creates guided modes that propagate along the perturbed line. Our result relies on the asymptotic analysis (with respect to $\epsilon$) of the spectrum of the Laplace-Neumann operator in this structure. Indeed, as $\epsilon$ tends to 0, the domain tends to a periodic graph, and the spectrum of the associated limit operator can be computed explicitly., Comment: in Applied Mathematics Letters, Elsevier, 2016

Convenient for immediate computer implementation equivalents of Green’s functions are obtained for boundary-contact value problems posed for two-dimensional Laplace and Klein-Gordon equations on some regions filled in with piecewise homogeneous isotropic conductive materials. Dirichlet, Neumann and Robin conditions are allowed on the outer boundary of a simply-connected region, while conditions of ideal contact are assumed on interface lines. The objective in this study is to widen the range of effective applicability for the Green’s function version of the boundary integral equation method making the latter usable for equations with piecewise-constant coefficients.

Let $e_\l(x)$ be an eigenfunction with respect to the Laplace-Beltrami operator $\Delta_M$ on a compact Riemannian manifold $M$ without boundary: $\Delta_M e_\l=\l^2 e_\l$. We show the following gradient estimate of $e_\l$: for every $\l\geq 1$, there holds $\l\|e_\l\|_\infty/C\leq \|\nabla e_\l\|_\infty\leq C{\l}\|e_\l\|_\infty$, where $C$ is a positive constant depending only on $M$., Comment: 8 pages. The abstract is shortened to two sentences. The reference of the book by Yu Safarov and D. Vassiliev was added. An alternative proof of the gradient estimate for the unit band spectral projection operator is added in Section 4. The layout is changed

We describe an algorithm for solving steady one-dimensional convex-like Hamilton- Jacobi equations using a L 1 -minimization technique on piecewise linear approximations. For a large class of convex Hamiltonians, the algorithm is proven to be convergent and of optimal complexity whenever the viscosity solution is q-semiconcave. Numerical results are presented to illustrate the performance of the method.

A spectral method is described for solving coupled elliptic problems on an interior and an exterior domain. The method is formulated and tested on the two-dimensional interior Poisson and exterior Laplace problems, whose solutions and their normal derivatives are required to be continuous across the interface. A complete basis of homogeneous solutions for the interior and exterior regions, corresponding to all possible Dirichlet boundary values at the interface, are calculated in a preprocessing step. This basis is used to construct the influence matrix which serves to transform the coupled boundary conditions into conditions on the interior problem. Chebyshev approximations are used to represent both the interior solutions and the boundary values. A standard Chebyshev spectral method is used to calculate the interior solutions. The exterior harmonic solutions are calculated as the convolution of the free-space Green's function with a surface density; this surface density is itself the solution to an integral equation which has an analytic solution when the boundary values are given as a Chebyshev expansion. Properties of Chebyshev approximations insure that the basis of exterior harmonic functions represents the external near-boundary solutions uniformly. The method is tested by calculating the electrostatic potential resulting from charge distributions in a rectangle. The resulting influence matrix is well-conditioned and solutions converge exponentially as the resolution is increased. The generalization of this approach to three-dimensional problems is discussed, in particular the magnetohydrodynamic equations in a finite cylindrical domain surrounded by a vacuum.

For smooth bounded domains in $\mathbb{R}^{n}$ , we prove upper and lower $L^{2}$ bounds on the boundary data of Neumann eigenfunctions, and we prove quasiorthogonality of this boundary data in a spectral window. The bounds are tight in the sense that both are independent of the eigenvalues; this is achieved by working with an appropriate norm for boundary functions, which includes a spectral weight, that is, a function of the boundary Laplacian. This spectral weight is chosen to cancel concentration at the boundary that can happen for whispering gallery-type eigenfunctions. These bounds are closely related to wave equation estimates due to Tataru. Using this, we bound the distance from an arbitrary Helmholtz parameter $E\gt 0$ to the nearest Neumann eigenvalue in terms of boundary normal derivative data of a trial function $u$ solving the Helmholtz equation $(\Delta-E)u=0$ . This inclusion bound improves over previously known bounds by a factor of $E^{5/6}$ , analogously to a recently improved inclusion bound in the Dirichlet case due to the first two authors. Finally, we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of Neumann eigenpairs on smooth planar domains. We show that the new inclusion bound improves the relative accuracy in a computed Neumann eigenvalue (around the $42000$ th) from nine to fourteen digits, with negligible extra numerical effort.

This paper deals with an interior electromagnetic casting (free boundary) problem. We begin by showing that the associated shape optimization problem has a solution which is of class C2. Then, using the shape derivative and the maximum principle, we give a sufficient condition that the minimum obtained solves our problem.

In this paper, we present a local-structure-preserving local discontinuous Galerkin (LDG) method for the Laplace equation. The method is based on the standard LDG formulation and uses piecewise harmonic polynomials, which satisfy the partial differential equation (PDE) exactly inside each element, as the approximating solutions for the primitive variable u, leading to a significant reduction of the degrees of freedom for the final system and hence the computational cost, without sacrificing the convergence quality of the solutions. An a priori error estimate in the energy norm is established. Numerical experiments are performed to verify optimal convergence rates of the local-structure-preserving LDG method in the energy norm and in the L-norm, as well as to compare it with the standard LDG method to demonstrate comparable performance of the two methods even though the new local-structure-preserving LDG method is computational less expensive.

We consider an inverse scattering problem of determining a periodic structure by near-field observations of the total field. We prove the global uniqueness results in both cases of the transverse electric polarization and the transverse magnetic polarization within the class of rectangular periodic structures by a single choice of any wave number. The proof is based on the analyticity of solutions to the Helmholtz equation.

We consider the problem of recovering a two-dimensional periodic structure from scattered waves measured above the structure. Following an approach by Kirsch and Kress, this inverse problem is reformulated as a nonlinear optimization problem. We develop a theoretical basis for the reconstruction method in the case of an arbitrary Lipschitz grating profile. The convergence analysis is based on new perturbation and stability results for the forward problem.

We consider an indirect boundary integral equation formulation for the mixed Dirichlet Neumann boundary value problem for the Laplace equation on a plane domain with a polygonal boundary. The resulting system of integral equations is solved by a collocation method which uses a mesh grading transformation and a cosine approximating space. The mesh grading transformation method yields fast convergence of the collocation solution by smoothing the singularities of the exact solution. A complete stability and solvability analysis of the transformed integral equations is given by use of a Mellin transform technique, in a setting in which each arc of the polygon has associated with it a periodic Sobolev space.

In this work we establish a connection between two classical notions, unrelated so far: Harmonic functions on the one hand and absolutely monotonic functions on the other hand. We use this to prove convexity type and propagation of smallness results for harmonic functions on the lattice., 20 pages; revised according to referees' comments; new references; two theorems added; simplified proof; conjecture restated; title changed; more friendly introduction

Poisson equation is frequently encountered in mathematical modeling for scientific and engineering applications. Fast Poisson numerical solvers for 2D and 3D problems are, thus, highly requested. In this paper, we consider solving the Poisson equation $\nabla^{2}u = f(x,y)$ in the Cartesian domain $\Omega = [-1,1] \times [-1,1]$, subject to all types of boundary conditions, discretized with the Chebyshev pseudospectral method. The main purpose of this paper is to propose a reflexive decomposition scheme for orthogonally decoupling the linear system obtained from the discretization into independent subsystems via the exploration of a special reflexive property inherent in the second-order Chebyshev collocation derivative matrix. The decomposition will introduce coarse-grain parallelism suitable for parallel computations. This approach can be applied to more general linear elliptic problems discretized with the Chebyshev pseudospectral method, so long as the discretized problems possess reflexive property. Numerical examples with error analysis are presented to demonstrate the validity and advantage of the proposed approach.

In this paper, we study the geometry associated with Schr\"{o}dinger operator via Hamiltonian and Lagrangian formalism. Making use of a multiplier technique, we construct the heat kernel with the coefficient matrices of the operator both diagonal and non-diagonal. For applications, we compute the heat kernel of a Schr\"{o}dinger operator with terms of lower order, and obtain a globally closed solution to a matrix Riccati equations as a by-product. Besides, we finally recover and generalise several classical results on some celebrated operators.

This paper is concerned with the variational approach in weighted Sobolev spaces to time-harmonic elastic wave scattering by one-dimensional unbounded rough surfaces. The rough surface is supposed to be the graph of a bounded and uniformly Lipschitz continuous function, on which the total elastic displacement satisfies either the Dirichlet or impedance boundary condition. We establish uniqueness and existence results at arbitrary frequency for both elastic plane wave and point source (spherical) wave incidence in the two-dimensional case. In particular, our approach covers the elastic scattering from periodic structures (diffraction gratings), and we prove quasiperiodicity of the scattered field whenever the incident field is quasiperiodic. Moreover, the diffraction grating problem is also uniquely solvable in the presented weighted Sobolev spaces for a broad class of non-quasiperiodic incident waves.

The topic of the present paper is the reflection of electromagnetic plane waves by rough surfaces, that is, by smooth and bounded perturbations of planar faces. Moreover, the contrast between the cover material and the substrate beneath the rough surface is supposed to be low. In this case, a modification of Stearns’ formula based on Born approximation and Fourier techniques is derived for a special class of surfaces. This class contains the graphs of functions where the interface function is a radially modulated almost periodic function. For the Born formula to converge, a sufficient and almost necessary condition is given. A further technical condition is defined, which guarantees the existence of the corresponding far field of the Born approximation. This far field contains plane waves, far-field terms such as those for bounded scatterers, and, additionally, a new type of terms. The derived formulas can be used for the fast numerical computations of far fields and for the statistics of random rough surfaces. Copyright © 2013 John Wiley & Sons, Ltd.

We study the asymptotic behaviour of the eigenvalues of the Laplace-Beltrami operator on a compact hypersurface in \mathds{R}^{n+1} as it is flattened into a singular double-sided flat hypersurface. We show that the limit spectral problem corresponds to the Dirichlet and Neumann problems on one side of this flat (Euclidean) limit, and derive an explicit three-term asymptotic expansion for the eigenvalues where the remaining two terms are of orders \varepsilon^2\log\varepsilon and \varepsilon^2.

We propose an asymptotic numerical method called the Gaussian beam approach for the boundary value problem of high frequency Helmholtz equation. The basic idea is to ap- proximate the traveling waves with a summation of Gaussian beams by the least squares algorithm. Gaussian beams are asymptotic solutions of linear wave equations in the high frequency regime. We deduce the ODE systems satisfied by the Gaussian beams up to third order. The key ingredient of the proposed method is the construction of a finite-dimensional beam space which has a good approximating property. If the exact solutions of boundary value problems contain some strongly evanescent wave modes, the Gaussian beam approach might fail. To remedy this problem, we re- sort to the domain decomposition technique to separate the domain of definition into a boundary layer region and its complementary interior region. The former is handled by a domain-based dis- cretization method, and the latter by the Gaussian beam approach. Schwarz iterations should then be performed based on suitable transmission boundary conditions at the interface of two regions. Numerical tests demonstrate that the proposed method is very promising.

Let $\Omega\subset\mathbb R^n$ be a bounded domain and for $x\in\Omega$ let $\tau(x)$ be the expected exit time from $\Omega$ of a diffusing particle starting at $x$ and advected by an incompressible flow $u$. We are interested in the question which flows maximize $\|\tau\|_{L^\infty(\Omega)}$, that is, they are most efficient in the creation of hotspots inside $\Omega$. Surprisingly, among all simply connected domains in two dimensions, the discs are the only ones for which the zero flow $u\equiv 0$ maximises $\|\tau\|_{L^\infty(\Omega)}$. We also show that in any dimension, among all domains with a fixed volume and all incompressible flows on them, $\|\tau\|_{L^\infty(\Omega)}$ is maximized by the zero flow on the ball., Comment: 15 pages, 1 figure

Periodic arrays are structures consisting of geometrically identical subdomains, usually named periodic cells. In this paper, by taking the Helmholtz equation as a model, we consider the definition and evaluation of scattering operators for general semi-infinite periodic arrays. The well-posedness of the Helmholtz equation is established via the limiting absorption principle. A method based on the doubling procedure and extrapolation technique is first proposed to compute the scattering operators of Sommerfeld-to-Sommerfeld type. The advantages of this method are the robustness and simplicity of implementation. However, it suffers from the heavy computational cost and the resonance wavenumbers. To overcome these shortcomings, we propose another more efficient method based on a conjecture about the asymptotic behavior of limiting absorption principle solutions. Numerical evidences suggest that this method presents the same results as the first one.

We study the Schrodinger equation which comes from the paraxial approximationof the Helmholtz equation in the case where the direction of propagation is tilted with respect tothe boundary of the domain. This model has been proposed in [12]. Our primary interest here isin the boundary conditions successively in a half-plane, then in a quadrant of R 2 . The half-planeproblem has been used in [11] to build a numerical method, which has been introduced in the HERAplateform of CEA.Key words. Laser plasma interaction, paraxial approximation of Helmholtz equation, W.K.B.approximation, transparent and absorbing boundary condition, Schrodinger equation.AMS subject classifications. 35E05, 35L05, 35J05, 78A40 1. Introduction. The simulation of a laser wave according to the paraxial ap-proximation of the Maxwell equation has been intensively studied for a long time whenthe simulation domain is rectangular and the direction of propagation is parallel toone of the principal axis of simulation domain(see for instance [16] and referencestherein).We are concerned here with the case where the direction of propagation is notparallel to a principal axis of the simulation domain, and cannot even be consideredas having a small incidence angle with it. It may be crucial for example if we wantto simulate several beams with different directions, possibly crossing each other inthe same domain. This tilted frame model has been considered some years ago byphysicists for dealing with beam crossing problems (see [24]), and is of particularinterest in the framework of CEA’s Laser Megajoule experiment (see [23, 25]).In [12, 11], a new model was proposed to deal with this case, and a numericalscheme was introduced and coupled with a time-dependent interaction model (thisscheme was then used in the HERA plateform of CEA). We focus here on the theo-retical study of this new model, of “advection-Schrodinger” type.This model is derived from the paraxial approximation, intensively used in opticmodels and in a lot of models related to laser-plasma interaction in Inertial Confine-ment Fusion (cf [9, 15, 17, 20, 3, 21] and their references). According to laws of optics,the laser electromagnetic field may be modeled by the following frequency Maxwellequation (the Helmholtz problem)

Harmonic functions u : R n ! R m are equivalent to integral manifolds of an exterior differential system with independence condition (M,I,!). To this system one asso- ciates the space of conservation laws C. They provide necessary conditions for g : S n 1 ! M to be the boundary of an integral submanifold. We show that in a local sense these condi- tions are also sufficient to guarantee the existence of an integral manifold with boundary g(S n 1 ). The proof uses standard linear elliptic theory to produce an integral manifold G : D n ! M and the completeness of the space of conservation laws to show that this candidate has g(S n 1 ) as its boundary. As a corollary we obtain a new elementary proof of the characterization of boundaries of holomorphic disks in C m in the local case. a @¯ 1 = 0. The 1-forms ' satisfying d' 2 (2,0) (2,0) provide moment conditions for the boundaries of holomorphic curves. That is, for any holomorphic curve G : X ! C m with boundary g : @X ! C m we find