Showing papers in "Communications in Partial Differential Equations in 1997"
TL;DR: In this paper, the uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions was studied and the authors proposed a method to solve the problem using partial differential equations.
Abstract: (1997). Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions. Communications in Partial Differential Equations: Vol. 22, No. 5-6, pp. 1009-1027.
307 citations
TL;DR: In this paper, the authors proved maximal Lp-Lp a-priori estimates for the solution of the parabolic evolution equation provided T may be represented by a heat kernel satisfying certain bounds.
Abstract: Let A be the generator of an analytic semigroup Ton L2(Ω), where Ω is a homogeneous space with doubling property. We prove maximal Lp-Lp a—priori estimates for the solution of the parabolic evolution equation u'(t)=Au(t)+f(t), u(0)=0 provided Tmay be represented by a heat—kernel satisfying certain bounds (and in particular a Gaussian bound). 1991 Mathematics Subject Classification:35K22, 58D25, 47D06
283 citations
TL;DR: In this article, the optimal regularity of wave maps was shown to be optimal for equations of wave map type, where wave maps are assumed to be wave maps of the wave type.
Abstract: (1997). Remark on the optimal regularity for equations of wave maps type. Communications in Partial Differential Equations: Vol. 22, No. 5-6, pp. 99-133.
159 citations
TL;DR: In this paper, the authors show that weak solutions of Navier-Stokes equations in T2 turn out to be smooth as long as the density remains bounded in L∞(T2).
Abstract: Regularity of weak solutions of the compressible isentropic Navier-Stokes equations is proven for small time in dimension N = 2 or 3 under periodic boundary conditions. In this paper, the initial density is not required to have a positive lower bound and the pressure law is assumed to satisfy a condition that reduces to τ > 1 when N = 2 and p(φ) = aφτ. Moreover,weak solutions in T2turn out to be smooth as long as the density remains bounded in L∞( T2).
151 citations
TL;DR: In this paper, the existence, uniqueness and Holder regularity of weak solutions of Hessian equations, determined by the elementary symmetric functions, with Lp inhomogeneous terms, were investigated.
Abstract: We consider the existence, uniqueness and Holder regularity of weak solutions of Hessian equations, determined by the elementary symmetric functions, with Lp inhomogeneous terms. The notion of weak solution is defined by approximation and our treatment draws on the classical theory, together with recent Lp estimates resulting from our isoperimetric inequalities for quermassintegrals on non-convex domains.
149 citations
TL;DR: In this article, the authors considered the initial value problem in the context of conservation laws, where the characteristics X (t, x) given by have to be defined in a generalized sense.
Abstract: We are interested in the initial value problem This is the familiar (multi-dimensional) linear transport equation, in conservation form. We are going to assume a very weak regularity on the coefficients Obviously the characteristics X (t, x), given by have to be defined in a generalized sense. We choose the construction of Filippov, which is classically used in the context of conservation laws. Our definition requires the uniqueness of these generalized characteristics. To illustrate this point, two examples are considered. 19 refs.
146 citations
TL;DR: In this article, the authors studied the regularity of solutions to boundary value problems for the Laplace operator on Lipschitz domains {Omega} in R{sup d} and its relationship with adaptive and other nonlinear methods for approximating these solutions.
Abstract: This paper studies the regularity of solutions to boundary value problems for the Laplace operator on Lipschitz domains {Omega} in R{sup d} and its relationship with adaptive and other nonlinear methods for approximating these solutions The smoothness spaces which determine the efficiency of such nonlinear approximation in L{sub p}({Omega}) are the Besov spaces B{sub {tau}}{sup {alpha}}(L{sub {tau}}({Omega})), {tau} := ({alpha}/d + 1/p){sup -1} Thus, the regularity of the solution in T this scale of Besov spaces is investigated with the aim of determining the largest a for which the solution is in B{sub {tau}}{sup {alpha}}(L{sub {tau}}({Omega})) The regularity theorems given in this paper build upon the recent results of Jerison and Kenig The proof of the regularity theorem uses characterizations of Besov spaces by wavelet expansions 14 refs, 1 fig
139 citations
TL;DR: In this paper, the authors consider the Harniltonian system consisting of a scalar wave field and a single particle coupled in a translation invariant manner and prove that solutions of finite energy converge, in suitable local energy seminorms, to the set of stationary solutions.
Abstract: We consider the Harniltonian system consisting of scalar wave field and a single particle coupled in a translation invariant manner. The point particle is subject to a confining external potential. The stationary solutions of the system are a Coulomb type wave field centered at those particle positions for which the external force vanishes. We prove that solutions of finite energy converge, in suitable local energy seminorms, to the set of stationary solutions in the long time limit t f oo. The rate of relaxation to a stable stationary solution is determined by spatial decay of initial data. 'Supported partly by French-Russian A.M.Liapunov Center of Moscow State University, by research grants of RFBR (9601-00527) and of Volkswagen-Stiftung.
125 citations
TL;DR: In this paper, a mixed boundary value problem for elliptic second order equations was studied under weak assumptions on the data and the dependence of the solution with respect to perturbations of the boundary sets carrying the Dirichlet and the Neumann conditions.
Abstract: We study a mixed boundary value problem for elliptic second order equations obtaining optimal regularity results under weak assumptions on the data. We also consider the dependence of the solution with respect to perturbations of the boundary sets carrying the Dirichlet and the Neumann conditions.
117 citations
TL;DR: In this article, the convergence of weak solutions of Navier Stokes equations with a large Coriolis term as the Rossby and Ekman numbers go to zero is studied.
Abstract: In this paper we study the convergence of weak solutions of the Navier Stokes equations with a large Coriolis term as the Rossby and Ekman numbers go to zero, and in particular the so called Ekman boundary layers, and justify some classical expansions in geophysical fluid dynamics (see [14], chapter 4).
99 citations
TL;DR: It is a classical result, Kruzhkov as mentioned in this paper, that the Cauchy problem for the scalar multidimensional conservation law, with u{sub 0} {element_of} L{sup 1} (R{sup n}) {intersection} L {sup {infinity}}(R{Sup n}) has a unique global weak solution u(x, t) satisfying the Kruzkov entropy conditions, and weak solutions of are constructed via finite difference approximations, Conway and Smoller, or as zero-viscosity limits of par
Abstract: It is a classical result, Kruzhkov, that the Cauchy problem for the scalar multidimensional conservation law, with u{sub 0} {element_of} L{sup 1}(R{sup n}) {intersection} L{sup {infinity}}(R{sup n}) has a unique global weak solution u(x, t) satisfying the Kruzhkov entropy conditions Weak solutions of are constructed via finite difference approximations, Conway and Smoller, or as zero-viscosity limits of parabolic regularizations, Volpert and Kruzhkov, and the solution operator defines a contraction semigroup in L{sup 1}(R{sup n}), Crandall. 28 refs.
TL;DR: In this article, the authors propose a global solution of a linearized inverse problem for the wave equation, which is the same as the one we consider in this paper, but with a different solution.
Abstract: (1997). Global solution of a linearized inverse problem for the wave equation. Communications in Partial Differential Equations: Vol. 22, No. 5-6, pp. 127-149.
TL;DR: In this paper, the boundedness of solutions to variational problems under general growth conditions is investigated under the assumption that the solution of a variational problem is a solution to a general growth condition.
Abstract: (1997). Boundedness of solutions to variational problems under general growth conditions. Communications in Partial Differential Equations: Vol. 22, No. 9-10, pp. 1629-1646.
TL;DR: In this paper, the existence of positive solutions to semilinear elliptic equations on r or r is not proven through the method of moving planes, but it is shown that positive solutions can be found on r and r.
Abstract: (1997). Non—existence of positive solutions to semilinear elliptic equations on r” or r” through the method of moving planes. Communications in Partial Differential Equations: Vol. 22, No. 9-10, pp. 1671-1690.
TL;DR: For a class of Dirichlet problems in two dimensions, generalizing the model case, this paper showed existence of a critical so that there are exactly 0, 1 or 2 nontrivial solutions.
Abstract: For a class of Dirichlet problems in two dimensions, generalizing the model case we show existence of a critical so that there are exactly 0, 1 or 2 nontrivial solutions (in fact, positive), depend...
TL;DR: In this paper, the existence of a smooth solution in an elliptic region in the self-similar plane to the pressure gradient system arising from the wave-particle splitting of the two-dimensional compressible Euler system of equations is established.
Abstract: We establish the existence of a smooth solution in tis elliptic region in the self—similar plane to the pressure—gradient system arisen from the wave—particle splitting of the two—dimensional compressible Euler system of equations. The pressure—gradient system takes the form Here (u,v) is the velocity, ρ is the density which is independent of time resulted from the splitting procedurep is the pressure, and is the energy. The natural (parabolically degenerate) boundary value is used.
TL;DR: Given a sequence of uniformly bounded open sets of the plane, whose boundaries are connected and uniformly bounded in length, converging to some open set (in the sense that the complements converge...
Abstract: Given a sequence of uniformly bounded open sets of the plane, whose boundaries are connected and uniformly bounded in length, converging to some open set (in the sense that the complements converge...
TL;DR: In this article, a method to uniquely reconstruct a potential V(x) from the scattering operator associated to the nonlinear Schrodinger equation and the corresponding unperturbed equation where.
Abstract: In this paper I present a method to uniquely reconstruct a potential V(x) from thescattering operator associated to the nonlinear Schrodinger equation and thecorresponding unperturbed equation where . I uniquely reconstruct the potential V by considering scattering states that have small amplitude and high velocity. In the small amplitude limit the main contribution to scattering comes from the potential V andsince moreover, the scattering state has high velocity the classical translation dominated the solution and the quantum spreading is a lower order term. These two effects lead to asimplification of the scattering process that allows me to uniquely reconstruct the potentialV.
TL;DR: In this paper, it was shown that a-priori weak Lp dissipativity implies strong L ∞ dissipativity for a class of weakly coupled quasilinear parabolic systems satisfying general structure conditions.
Abstract: We show that a-priori weak Lp dissipativity implies strong L ∞ dissipativity for a class of weakly coupled quasilinear parabolic systems satisfies general structure conditions. The existence of glo...
TL;DR: In this article, Harnack's inequality for cooperative weakly coupled elliptic systems is studied in the context of Partial Differential Equations (PDE), and the authors propose a solution to the problem.
Abstract: (1999). Harnack's inequality for cooperative weakly coupled elliptic systems. Communications in Partial Differential Equations: Vol. 24, No. 9-10, pp. 1555-1571.
TL;DR: In this paper, the Baouendi-Goulaouic operator was shown to be a non-isotropic Gevrey hypoelliptic operator for a class of partial differential operators that are sums of squares of real vector fields satisfying the Hormander bracket condition.
Abstract: We prove sharp non-isotropic Gevrey hypoellipticiy for a class ofo partial differential operators that are sums of squares of real vector fields (and their generalizations) satisfying the Hormander bracket condition. These include the Baouendi-Goulaouic operator. Our results, which refine those of Chirst [8], are proved entirely by L 2 methods and a careful study of brackets of vector fields. Applications to a recent conjecture of Traves are given.
TL;DR: In this article, the authors studied the problem of finding a small energy solution to (P_μ) with critical nonlinearity and Neumann boundary conditions, where Ω is a smooth bounded domain in ℝ^n, n≥3, and μ is a strictly positive parameter.
Abstract: We are interested in elliptic problems with critical nonlinearity and Neumann boundary conditions, namely (P_μ) : -Δu + μu = u^(n+2)/(n-2), u>0 in Ω, ∂u/∂ν = 0 on ∂Ω — where Ω is a smooth bounded domain in ℝ^n, n≥3, and μ is a strictly positive parameter. We show, for n≥7, and u a small energy solution to (P_μ), that u concentrates as μ goes to infinity at a point of the boundary such that the mean curvature H is positive, and critical if it is strictly positive. Conversely we show, for n≥5, and α>0 a critical value of H inducing a difference of topology between the level sets of H, that there exists for μ large enough a solution to (P_μ) which concentrates at a point y of the boundary such that H(y) = α and H'(y) = 0. Lastly, if n≥6 and y_1, … , y_k are k distinct critical points of H, there exists for μ large enough a solution to (P_μ) which concentrates at each of the points y_i, 1≤i≤k.
TL;DR: In this paper, the inverse scattering of Schrodinger operators with short-range (resp. long-range) electric and magnetic potentials is studied. And the authors show that the electric potential and the magnetic field are uniquely determined by the first two terms of this asymptotic expansion.
Abstract: In this paper, we study the inverse scattering of Schrodinger operators with short-range (resp. long-range) electric and magnetic potentials. We develop a stationary approach to determine the high energy asymptotics of the scattering operator (resp. modified scattering operator). As a corollary, we show that the electric potential and the magnetic field are uniquely determined by the first two terms of this asymptotic expansion.
TL;DR: In this paper, the existence of large positive solutions of some nonlinear elliptic $si:equations on singularly perturbed doamins was investigated and shown to be true.
Abstract: (1997). Existence of large positive solutions of some nonlinear elliptic $si:equations on singularly perturbed doamins. Communications in Partial Differential Equations: Vol. 22, No. 9-10, pp. 1731-1769.
TL;DR: In this article, a comparison principle was established for singular degenerate parabolic equations including the p-Laplace diffusion equation, which is a natural extension of the paper by Lshii and Souganidis.
Abstract: We consider singular degenerateparabolic equations including the p-Laplace diffusion equation. We establish a comparison principle which is a natural extension of the paper[13] by Lshii and Souganidis. Once we get a comparison priciple we can constructt the unique global-in-time viscosity solution to the Cauchy problem for the p-Laplace diffusion equation. The solution is bounded, uniformly continous in (0,T)×R N if the initial data is bounded, uniformly continuous on R N
TL;DR: In this article, the existence and stability of positive solutions for a general calss of semilinear elliptic boundary value problems of superlinear type with indedefinite weight functions were shown.
Abstract: In this work we show the existence and stability of positive solutions for a general calss of semilinear elliptic boundary value problems of superlinear type with indedefinite weight functions. Optimal necessary and sufficient conditions are found.
TL;DR: In this paper, the Ginzburg-Landau equation is considered for the case where the unit ball is a unit ball in R{sup 2} with center at the origin.
Abstract: In this paper, we consider the the Ginzburg-Landau equation -{del}u = 1/2 u (1 - {vert_bar}u{vert_bar}{sup 2}) in B u = g on {partial_derivative}B where B is the unit ball in R{sup 2} with center at the origin, g(e{sup i0} = e{sup id)}, d = 1,2,3,..., u:B {yields} C is smooth, and {epsilon} > 0 is a small parameter.
TL;DR: In this paper, the authors study the conormal regularity of solutions of two dirnensional incompressible Euler systems whose initial data are vortez patches with singular boundary and obtain the following result for vories patches: if the initial boundary is regular apart from a closed subset, it remains regular for all time apart from the closed subset transported by the pow.
Abstract: We study here regularity of solutions of the two dirnensional incompressible Euler system whose initial data are vortez patches with singular boundary. We show the persislance of conormal regularity apart from a closed subset. As a consequence, we obtain the following result for vories patches : if the initial boundary is regular apart from a closed subset, it remairw regular for all time apart from the closed subset transported by the pow.