A 3G-Theorem for Jordan Domains in ℝ²
We prove a new 3G-Theorem for the Laplace Green function G on an arbitrary Jordan domain D in ℝ². This theorem extends the recent one proved on a Dini-smooth Jordan domain.
A Bi-CG type iterative method for Drazin-inverse solution of singular inconsistent nonsymmetric linear systems of arbitrary index.
A bifurcation result for equations with anisotropic -Laplace-like operators.
A bifurcation theory for some nonlinear elliptic equations
We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ () ⎩ where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem () admits a non-zero, non-negative strong solution such that for all p ≥ 2. Moreover, the function is negative and decreasing in ]0,λ*[, where is the energy functional related to ().
A biharmonic elliptic problem with dependence on the gradient and the Laplacian.
A blowup analysis of the mean field equation for arbitrarily signed vortices
We study the noncompact solution sequences to the mean field equation for arbitrarily signed vortices and observe the quantization of the mass of concentration, using the rescaling argument.
A boundary blow-up for sub-linear elliptic problems with a nonlinear gradient term.
A boundary Harnack principle for infinity-Laplacian and some related results.
A boundary integral equation for Calderón's inverse conductivity problem.
Towards a constructive method to determine an L∞-conductivity from the corresponding Dirichlet to Neumann operator, we establish a Fredholm integral equation of the second kind at the boundary of a two dimensional body. We show that this equation depends directly on the measured data and has always a unique solution. This way the geometric optics solutions for the L∞-conductivity problem can be determined in a stable manner at the boundary and outside of the body.
A boundary value problem in the hyperbolic space.
A brief review of some application driven fast algorithms for elliptic partial differential equations
Some application driven fast algorithms developed by the author and his collaborators for elliptic partial differential equations are briefly reviewed here. Subsequent use of the ideas behind development of these algorithms for further development of other algorithms some of which are currently in progress is briefly mentioned. Serial and parallel implementation of these algorithms and their applications to some pure and applied problems are also briefly reviewed.
A -theory for the blow-up of second order elliptic equations of critical Sobolev growth.
A C1-P2 finite element without nodal basis
A new finite element, which is continuously differentiable, but only piecewise quadratic polynomials on a type of uniform triangulations, is introduced. We construct a local basis which does not involve nodal values nor derivatives. Different from the traditional finite elements, we have to construct a special, averaging operator which is stable and preserves quadratic polynomials. We show the optimal order of approximation of the finite element in interpolation, and in solving the biharmonic...
A calculus for a class of finitely degenerate pseudodifferential operators
For a class of degenerate pseudodifferential operators, local parametrices are constructed. This is done in the framework of a pseudodifferential calculus upon adding conditions of trace and potential type, respectively, along the boundary on which the operators degenerate.
A Capacitance Matrix Method for Dirichlet Problem on Polygon Region.
A Carleman function and the Cauchy problem for the Laplace equation.
A certified reduced basis method for parametrized elliptic optimal control problems
In this paper, we employ the reduced basis method as a surrogate model for the solution of linear-quadratic optimal control problems governed by parametrized elliptic partial differential equations. We present a posteriori error estimation and dual procedures that provide rigorous bounds for the error in several quantities of interest: the optimal control, the cost functional, and general linear output functionals of the control, state, and adjoint variables. We show that, based on the assumption...
A characterization of balls using the domain derivative.
A characterization of convex calibrable sets in with respect to anisotropic norms
A characterization of elliptic operators
We give a characterization of constant coefficients elliptic operators in terms of estimates of their iterations on smooth functions.