On the discrepancy principle for some Newton type methods for solving nonlinear inverse problems

We consider the computation of stable approximations to the exact solution of nonlinear ill-posed inverse problems F(x) = y with nonlinear operators F : X → Y between two Hilbert spaces X and Y by the Newton type methods

Principal Eigenvalues for Isaacs Operators with Neumann Boundary Conditions

In this paper we show the existence of two principal eigenvalues associated to general non-convex fully nonlinear elliptic operators with Neumann boundary conditions in a bounded C ² domain. We study these objects and we establish some of their basic properties. Finally, Lipschitz regularity, uniqueness and existence results for the solution of the Neumann problem are given.      

Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions

Consider the equation −Δu = 0 in a bounded smooth domain , complemented by the nonlinear Neumann boundary condition ∂_ν u = f(x, u) − u on ∂Ω. We show that any very weak solution of this problem belongs to L ^∞(Ω) provided f satisfies the growth condition |f(x, s)| ≤ C(1 + |s| ^p ) for some p ∈ (1, p*), where . If, in addition, f(x, s) ≥ −C + λs for some λ > 1, then all positive very weak solutions are uniformly a priori bounded. We also show by means of examples that p* is a sharp critical exponent. In particular, using variational methods we prove the following multiplicity result: if N ∈ {3, 4} and f(x, s) =  s ^p then there exists a domain Ω and such that our problem possesses at least two positive, unbounded, very weak solutions blowing up at a prescribed point of ∂Ω provided . Our regularity results and a priori bounds for positive very weak solutions remain true if the right-hand side in the differential equation is of the form h(x, u) with h satisfying suitable growth conditions.     

Solutions of elliptic problems with nonlinearities of linear growth

In this paper, we study existence of nontrivial solutions to the elliptic equation

Uniqueness of solutions for some nonlinear Dirichlet problems

We consider here a class of nonlinear Dirichlet problems, in a bounded domain , of the form

Multiplicity of Positive Solutions for a Class of Inhomogeneous Neumann Problems Involving the p(x)-Laplacian

We study the existence and multiplicity of positive solutions for the inhomogeneous Neumann boundary value problems involving the p(x)-Laplacian of the form

Sign changing solutions of semilinear elliptic problems in exterior domains

We prove the existence of a sign changing solution to the semilinear elliptic problem , in an exterior domain Ω having finite symmetries.     

Existence of type (II) regions and convexity and concavity of potential functionals corresponding to jumping nonlinear problems

In this paper we study the jumping nonlinear problem

Bifurcation problems for superlinear elliptic indefinite equations with exponential growth

This paper deals with the existence and the behaviour of global connected branches of positive solutions of the problem

Positive solutions for second-order superlinear repulsive singular Neumann boundary value problems

In this paper we establish the multiplicity of positive solutions to second-order superlinear repulsive singular Neumann boundary value problems. It is proved that such a problem has at least two positive solutions under reasonable conditions. Our nonlinearity may be repulsive singular in its dependent variable and superlinear at infinity. The proof relies on a nonlinear alternative of Leray-Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones.      

Stabilized boundary element methods for exterior Helmholtz problems

In this paper we describe and analyze some modified boundary element methods to solve the exterior Dirichlet boundary value problem for the Helmholtz equation. As in classical combined field integral equations also the proposed approach avoids spurious modes. Moreover, the stability of related modified boundary element methods can be shown even in the case of Lipschitz boundaries. The proposed regularization is done based on boundary integral operators which are already included in standard boundary element formulations. Numerical examples are given to compare the proposed approach with other already existing regularized formulations.     

On approximate solutions to the first initial boundary value problem for the m-Hessian evolution equations

We adapt to degenerate m-Hessian evolution equations the notion of m-approximate solutions introduced by N. Trudinger for m-Hessian elliptic equations, and we present close to necessary and sufficient conditions guaranteeing the existence and uniqueness of such solutions for the first initial boundary value problem. Dedicated to Professor Felix Browder     

Inverse positivity for general Robin problems on Lipschitz domains

It is proved that elliptic boundary value problems in divergence form can be written in many equivalent forms. This is used to prove regularity properties and maximum principles for problems with Robin boundary conditions with negative or indefinite boundary coefficient on Lipschitz domains by rewriting them as a problem with positive coefficient. It is also shown that such methods cannot be applied to domains with an outward pointing cusp. Applications to the regularity of the harmonic Steklov eigenfunctions on Lipschitz domains are given. Received: 26 June 2008; Revised: 12 September 2008     

A nondegeneracy result for a nonlinear elliptic equation

Let Ω be a smooth bounded domain of with N ≥ 5. In this paper we prove, for ɛ > 0 small, the nondegeneracy of the solution of the problem

On the Multiplicity of Nodal Solutions to a Singularly Perturbed Neumann Problem

We consider the problem

Singular solutions of some nonlinear parabolic equations with spatially inhomogeneous absorption

We study the limit behaviour of solutions of with initial data k δ ₀ when k → ∞, where h is a positive nondecreasing function and p > 1. If h(r) = r ^β , β > N(p − 1) − 2, we prove that the limit function u _∞ is an explicit very singular solution, while such a solution does not exist if β ≤  N(p − 1) − 2. If lim inf _{r→ 0} r ² ln (1/h(r))  >  0, u _∞ has a persistent singularity at (0, t) (t ≥  0). If , u _∞ has a pointwise singularity localized at (0, 0).     

A unifying theory of a posteriori error control for discontinuous Galerkin FEM

A unified a posteriori error analysis is derived in extension of Carstensen (Numer Math 100:617–637, 2005) and Carstensen and Hu (J Numer Math 107(3):473–502, 2007) for a wide range of discontinuous Galerkin (dG) finite element methods (FEM), applied to the Laplace, Stokes, and Lamé equations. Two abstract assumptions (A1) and (A2) guarantee the reliability of explicit residual-based computable error estimators. The edge jumps are recast via lifting operators to make arguments already established for nonconforming finite element methods available. The resulting reliable error estimate is applied to 16 representative dG FEMs from the literature. The estimate recovers known results as well as provides new bounds to a number of schemes. C. Carstensen and M. Jensen supported by the DFG Research Center MATHEON “Mathematics for key technologies” in Berlin and the Hausdorff Institute of Mathematics in Bonn, Germany. C. Carstensen, T. Gudi, and M. Jensen supported by DST-DAAD (PPP-05) project no. 32307481.     

Existence and Stability Results for Renormalized Solutions to Noncoercive Nonlinear Elliptic Equations with Measure Data

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is

Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves

In this paper we address several theoretical questions related to the numerical approximation of the scattering of acoustic waves in two or three dimensions by penetrable non-homogeneous obstacles using convolution quadrature (CQ) techniques for the time variable and coupled boundary element method/finite element method for the space variable. The applicability of CQ to waves requires polynomial type bounds for operators related to the operator Δ − s ² in the right half complex plane. We propose a new systematic way of dealing with this problem, both at the continuous and semidiscrete-in-space cases. We apply the technique to three different situations: scattering by a group of sound-soft and -hard obstacles, by homogeneous and non-homogeneous obstacles.     

Discontinuous Galerkin least-squares finite element methods for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities

In this paper, a discontinuous Galerkin least-squares finite element method is developed for singularly perturbed reaction-diffusion problems with discontinuous coefficients and boundary singularities by recasting the second-order elliptic equations as a system of first-order equations. In a companion paper (Lin in SIAM J Numer Anal 47:89–108, 2008) a similar method has been developed for problems with continuous data and shown to be well-posed, uniformly convergent, and optimal in convergence rate. In this paper the method is modified to take care of conditions that arise at interfaces and boundary singularities. Coercivity and uniform error estimates for the finite element approximation are established in an appropriately scaled norm. Numerical examples confirm the theoretical results.