A Family of Fundamental Solutions of Elliptic Partial Differential Operators with Real Constant Coefficients

We present a construction of a family of fundamental solutions for elliptic partial differential operators with real constant coefficients. The elements of such a family are expressed by means of jointly real analytic functions of the coefficients of the operators and of the spatial variable. The aim is to write detailed expressions for such functions. Such expressions are then exploited to prove regularity properties in the frame of Schauder spaces and jump properties of the corresponding single layer potentials.     

Arithmetic Laplacians

We develop an arithmetic analogue of elliptic partial differential equations. The role of the space coordinates is played by a family of primes, and that of the space derivatives along the various primes are played by corresponding Fermat quotient operators subjected to certain commutation relations. This leads to arithmetic linear partial differential equations on algebraic groups that are analogues of certain operators in analysis constructed from Laplacians. We classify all such equations on one-dimensional groups, and analyze their spaces of solutions.     

Gaussian upper bounds for heat kernels of second order complex elliptic operators with unbounded diffusion coefficients on arbitrary domains

In this paper we obtain Gaussian upper bounds for the integral kernel of the semigroup associated with second order elliptic differential operators with complex unbounded measurable coefficients defined in a domain Ω of ? ^N and subject to various boundary conditions. In contrast to the previous literature the diffusions coefficients are not required to be bounded or regular. A new approach based on Davies-Gaffney estimates is used. It is applied to a number of examples, including degenerate elliptic operators arising in Financial Mathematics and generalized Ornstein-Uhlenbeck operators with potentials.     

Bilinear embedding for real elliptic differential operators in divergence form with potentials

We present a simple Bellman function proof of a bilinear estimate for elliptic operators in divergence form with real coefficients and with nonnegative potentials. The constants are dimension-free. The p-range of applicability of this estimate is (1,∞) for any real accretive (nonsymmetric) matrix A of coefficients.     

The family index theorem and bifurcation of solutions of nonlinear elliptic BVP

We obtain some new bifurcation criteria for solutions of general boundary value problems for nonlinear elliptic systems of partial differential equations. The results are of different nature from the ones that can be obtained via the traditional Lyapunov–Schmidt reduction. Our sufficient conditions for bifurcation are derived from the Atiyah–Singer family index theorem and therefore they depend only on the coefficients of derivatives of leading order of the linearized differential operators. They are computed explicitly from the coefficients without the need of solving the linearized equations. Moreover, they are stable under lower order perturbations.     

Localized Boundary-Domain Singular Integral Equations Based on Harmonic Parametrix for Divergence-Form Elliptic PDEs with Variable Matrix Coefficients

Employing the localized integral potentials associated with the Laplace operator, the Dirichlet, Neumann and Robin boundary value problems (BVPs) for general variable-coefficient divergence-form second-order elliptic partial differential equations are reduced to some systems of localized boundary-domain singular integral equations. Equivalence of the integral equations systems to the original BVPs is proved. It is established that the corresponding localized boundary-domain integral operators belong to the Boutet de Monvel algebra of pseudo-differential operators. Applying the Vishik–Eskin theory based on the factorization method, the Fredholm properties and invertibility of the operators are proved in appropriate Sobolev spaces.     

Square function estimates on layer potentials for higher‐order elliptic equations

In this paper we establish square‐function estimates on the double and single layer potentials for divergence form elliptic operators, of arbitrary even order 2m , with variable t‐independent coefficients in the upper half‐space. This generalizes known results for variable‐coefficient second‐order operators, and also for constant‐coefficient higher‐order operators.     

Localized boundary‐domain singular integral equations of Dirichlet problem for self‐adjoint second‐order strongly elliptic PDE systems

The paper deals with the three‐dimensional Dirichlet boundary value problem (BVP) for a second‐order strongly elliptic self‐adjoint system of partial differential equations in the divergence form with variable coefficients and develops the integral potential method based on a localized parametrix. Using Green's representation formula and properties of the localized layer and volume potentials, we reduce the Dirichlet BVP to a system of localized boundary‐domain integral equations. The equivalence between the Dirichlet BVP and the corresponding localized boundary‐domain integral equation system is studied. We establish that the obtained localized boundary‐domain integral operator belongs to the Boutet de Monvel algebra. With the help of the Wiener–Hopf factorization method, we investigate corresponding Fredholm properties and prove invertibility of the localized operator in appropriate Sobolev (Bessel potential) spaces. Copyright © 2016 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons, Ltd.     

Analyticity of solutions of analytic non-linear general elliptic boundary value problems, and some results about linear problems

The main aim of this paper is to discuss the problem concerning the analyticity of the solutions of analytic non-linear elliptic boundary value problems. It is proved that if the corresponding first variation is regular in Lopatinskiĭ sense, then the solution is analytic up to the boundary. The method of proof really covers the case that the corresponding first variation is regularly elliptic in the sense of Douglis-Nirenberg-Volevich, and hence completely generalize the previous result of C. B. Morrey. The author also discusses linear elliptic boundary value problems for systems of elliptic partial differential equations where the boundary operators are allowed to have singular integral operators as their coefficients. Combining the standard Fourier transform technique with analytic continuation argument, the author constructs the Poisson and Green’s kernel matrices related to the problems discussed and hence obtain some representation formulae to the solutions. Some a priori estimates of Schauder type and L _p type are obtained. __________ Translated from Acta Sci. Natur. Univ. Jilin, 1963, (2): 403–447 by GAO Wenjie.     

Fredholm property of a family of operators on the five-dimensional Heisenberg group

An expansion in Euclidean spherical harmonics on the ball in the Heisenberg group of dimension five decomposes the Dirichlet problem for the Laplacian in an infinite number of two-dimensional problems. Fundamental solutions are obtained for each of the partial differential operators in these problems, thus reducing them further (via layer potentials) to one-dimensional integral equations. The main result in this article states that the corresponding integral operators are Fredholm in appropriate weighted L² spaces.     

Bounds on the non-real spectrum of differential operators with indefinite weights

Ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non-negative operators in Krein spaces. Under the assumption that 0 and $\infty $ are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non-real spectrum is contained in a compact set and with definite type real spectrum outside this set. The main results are quantitative estimates for this set, which are applied to Sturm–Liouville and second order elliptic partial differential operators with indefinite weights on unbounded domains.     

Hypoelliptic systems connected with Newton's polyhedron

We introduce a general class of hypoelliptic systems of partial differential operators with variable coefficients generalizing the so‐called multi‐quasi‐elliptic differential operators. We study the regularity of solutions of these systems. More complete results are obtained in the case of two independent variables. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gap in the essential spectrum of an elliptic formally self-adjoint system of differential equations

For a formally self-adjoint elliptic system of partial differential equations with periodic coefficients in the space ℝ ⁿ , we show that, by conferring contrast properties to the coefficients of differential operators, one can open a gap in the essential spectrum of the system. We suggest a method based on the derivation of an asymptotically sharp generalized Korn inequality and the use of the maximin principle; this method applies to perforated media as well as to periodic layered and quasi-cylindrical waveguides.     

Maximum Principles for Boundary-Degenerate Second Order Linear Elliptic Differential Operators

We prove weak and strong maximum principles, including a Hopf lemma, for C ² subsolutions to equations defined by linear, second-order, linear, elliptic partial differential operators whose principal symbols vanish along a portion of the domain boundary. The boundary regularity property of the C ² subsolutions along this boundary vanishing locus ensures that these maximum principles hold irrespective of the sign of the Fichera function. Boundary conditions need only be prescribed on the complement in the domain boundary of the principal symbol's vanishing locus. We obtain uniqueness and a priori maximum principle estimates for C ² solutions to boundary value and obstacle problems defined by these boundary-degenerate elliptic operators with partial Dirichlet or Neumann boundary conditions. We also prove weak maximum principles and uniqueness for W ^{1, 2} solutions to the corresponding variational equations and inequalities defined with the aide of weighted Sobolev spaces. The domain is allowed to be unbounded when the operator coefficients and solutions obey certain growth conditions.     

A direct boundary integral equation method for transmission problems

A system of integral equations for the field and its normal derivative on the boundary in acoustic or potential scattering by a penetrable homogeneous object in arbitrary dimensions is presented. The system contains the operators of the single and double layer potentials, of the normal derivative of the single layer, and of the normal derivative of the double layer potential. It defines a strongly elliptic system of pseudodifferential operators. It is shown by the method of Mellin transformation that a corresponding property, namely a Gårding's inequality in the energy norm, holds also in the case of a polygonal boundary of a plane domain. This yields asymptotic quasioptimal error estimates in Sobolev spaces for the corresponding Galerkin approximation using finite elements on the boundary only.     

A direct boundary integral equation method for transmission problems

A system of integral equations for the field and its normal derivative on the boundary in acoustic or potential scattering by a penetrable homogeneous object in arbitrary dimensions is presented. The system contains the operators of the single and double layer potentials, of the normal derivative of the single layer, and of the normal derivative of the double layer potential. It defines a strongly elliptic system of pseudodifferential operators. It is shown by the method of Mellin transformation that a corresponding property, namely a Gårding's inequality in the energy norm, holds also in the case of a polygonal boundary of a plane domain. This yields asymptotic quasioptimal error estimates in Sobolev spaces for the corresponding Galerkin approximation using finite elements on the boundary only.     

Elastic potentials at corners in Sobolev spaces with asymptotics

This paper is aimed at studying the single and double layer potentials related to the boundary value problems of elasticity theory for anisotropic case for the plane, corner domains. We start from the systems of second order elliptic differential equations with constant coefficients, write the fundamental solution and form the single and double layer (elastic) potentials. Applying the pseudo‐differential calculus we obtain the continuity results of the elastic potentials at corners in cone Sobolev spaces without and with asymptotics and characterize asymptotics of solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Reconstruction in the Calderón Problem with Partial Data

We consider the problem of recovering the coefficient σ(x) of the elliptic equation ?·(σ?u) = 0 in a body from measurements of the Cauchy data on possibly very small subsets of its surface. We give a constructive proof of a uniqueness result by Kenig, Sjöstrand, and Uhlmann. We construct a uniquely specified family of solutions such that their traces on the boundary can be calculated by solving an integral equation which involves only the given partial Cauchy data. The construction entails a new family of Green's functions for the Laplacian, and corresponding single layer potentials, which may be of independent interest.     

On the Lipschitz continuity of derivatives for some scalar nonlinearities

Vector nonlinearities determined by a scalar function arise in various mathematical models. The numerical solution of the corresponding partial differential equations often rely on the Lipschitz continuity of the derivative of the nonlinear operator. In this paper a simple sufficient condition is given for the required Lipschitz continuity, also providing an easily computable estimate of the Lipschitz constant. Some discussion is included for the corresponding elliptic operators.     

Optimal elliptic Sobolev regularity near three-dimensional multi-material Neumann vertices

We study the optimal elliptic regularity (within the scale of Sobolev spaces) of anisotropic div-grad operators in three dimensions at a multi-material vertex on the Neumann part of the boundary of a 3D polyhedral domain. The gradient of any solution of the corresponding elliptic partial differential equation (in a neighborhood of the vertex) is p-integrable with p > 3.