Sup norms of Cauchy data of eigenfunctions on manifolds with concave boundary

We prove that the Cauchy data of Dirichlet or Neumann Δ- eigenfunctions of Riemannian manifolds with concave (diffractive) boundary can only achieve maximal sup norm bounds if there exists a self-focal point on the boundary, i.e., a point at which a positive measure of geodesics leaving the point return to the point. In the case of real analytic Riemannian manifolds with real analytic boundary, maximal sup norm bounds on boundary traces of eigenfunctions can only be achieved if there exists a point on the boundary at which all geodesics loop back. As an application, the Dirichlet or Neumann eigenfunctions of Riemannian manifolds with concave boundary and non-positive curvature never have eigenfunctions whose boundary traces achieve maximal sup norm bounds. The key new ingredient is the Melrose–Taylor diffractive parametrix and Melrose’s analysis of the Weyl law.     

The Cauchy problem for nonlinear elliptic equations

This paper is devoted to investigation of the Cauchy problem for nonlinear equations with a small parameter. They are actually small perturbations of linear elliptic equations in which case the Cauchy problem is ill-posed. To study the Cauchy problem we invoke purely nonlinear methods, such as successive iterations and L^q$L^q$ Sobolev spaces with large q$q$. We also discuss linearisable problems.     

Recovering boundary data: The Cauchy Stokes system

This paper is concerned with the severely ill-posed Cauchy–Stokes problem. We are interested in a data completion problem which is exploited to detect small leaks to control water loss Kim et al. (2008) [1]. This inverse problem is rephrased into an optimization one: An energy-like error functional is introduced. We prove that the optimality condition of the first order is equivalent to solving an interfacial equation which turns out to be a Cauchy-Steklov-Poincaré operator. Numerical trials highlight the efficiency of the method.     

The exterior Dirichlet problem for quasi-linear elliptic equations with small boundary data

It is shown that if the order of non-uniformity of a quasi-linear elliptic equation is h,10,2(h–1)/h norm. For 0h1,existence of a bounded solution is guaranteed without any smallness assumption on the given boundary data.More precise information is given for the special case of the minimal surface equation.     

Stability estimates for the inverse boundary value problem by partial Cauchy data

We study the inverse conductivity problem with partial data in dimension n ≥ 3. We derive stability estimates for this inverse problem if the conductivity has regularity for 0 < σ < 1. Copyright © 2014 John Wiley & Sons, Ltd.     

Nonlinear random elliptic boundary value problem

In this paper we obtain a solvability result for random elliptic boundary value problem. Our result extends Browder's [2] main theorem to the random case. We use author's [4] fixed point theorem for this purpose.     

The fully non-linear Cauchy problem with small data

This paper is devoted to the Cauchy problem for a fully non-linear perturbation

On the Cauchy problem for elliptic equations in a disk

Letφ, ψ be smooth functions on the boundary of the unit diskB ₁. A second order uniformly elliptic operatorL and a functionu with second order derivatives inL ^p (1<p<2) are constructed with the following properties:u solvesLu=0 inB ₁ and satisfies the Cauchy dataφ, ψ on∂B ₁.     

Control of Cauchy system for an elliptic operator

The control of a Cauchy system for an elliptic operator seems to be globally an open problem. In this paper, we analyze this problem using a regularization method which consists in viewing a singular problem as a limit of a family of well-posed problems. Following this analysis and assuming that the interior of considered convex is non-empty, we obtain a singular optimality system （S.O.S.） for the considered control problem.     

Cauchy problem for heat-transfer equation with irregular elliptic operator

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 5, pp. 584–590, May, 1989.     

A second order,non-linear elliptic boundary value problem with generalized Goursat data

Summary In this paper the case of generalized Goursat data is considered for the non-linear partial differential equation Δu = f(x, y, u, u_x, u_y). The existence and uniqueness of a solution is demonstrated, under certain conditions, by employing the contraction mapping method in a suitable Banach space. This research was supported in part at the Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, by the National Science Foundation under Grants GP-2067, GP-3937, and in part by the Air Force Office of Scientific Research under Grant AFOSR 400-64, and at Georgetown University, Washington, D.C., by the National Science Foundation under Grant GP-1650 and GP-5023.     

An elliptic boundary value problem occurring in magnetohydrodynamics

We derive results concerning the spectral properties of an elliptic boundary value problem arising in the mathematical theory of magnetohydrodynamics which are of basic importance for the further development of this theory. The work of this author was supported in part by the FRD of South Africa