A condition on the potential for the existence of doubly periodic solutions of a semi-linear fourth-order partial differential equation.
A Conjugate Gradient Method and a Multigrid Algorithm for Morley' s Finite Element Approximation of the Biharmonic Equation.
A conservative particle approximation for a boundary advection-diffusion problem
A construction of a parametrix for an elliptic boundary value problem.
A construction of parametrices of elliptic boundary value problems
A continuity result for a quasilinear elliptic free boundary problem
We investigate a two dimensional quasilinear free boundary problem, and show that the free boundary is a union of graphs of continuous functions.
A convergence result and numerical study for a nonlinear piezoelectric material in a frictional contact process with a conductive foundation
We consider two static problems which describe the contact between a piezoelectric body and an obstacle, the so-called foundation. The constitutive relation of the material is assumed to be electro-elastic and involves the nonlinear elastic constitutive Hencky's law. In the first problem, the contact is assumed to be frictionless, and the foundation is nonconductive, while in the second it is supposed to be frictional, and the foundation is electrically conductive. The contact is modeled with the...
A convergent adaptive finite element method with optimal complexity.
A Counterexample in Elliptic Regularity Theory.
A counterexample to an endpoint bilinear Strichartz inequality.
A counterexample to Schauder estimates for elliptic operators with unbounded coefficients
We consider a homogeneous elliptic Dirichlet problem involving an Ornstein-Uhlenbeck operator in a half space of . We show that for a particular initial datum, which is Lipschitz continuous and bounded on , the second derivative of the classical solution is not uniformly continuous on . In particular this implies that the well known maximal Hölder-regularity results fail in general for Dirichlet problems in unbounded domains involving unbounded coefficients.
A Counter-Example to the Boundary Regularity of Solutions to Elliptic Quasilinear Systems.
A counterexample to the “hot spots" conjecture.
A counterexample to the -Hodge decomposition
We construct a bounded domain with the cone property and a harmonic function on Ω which belongs to for all 1 ≤ p < 4/3. As a corollary we deduce that there is no -Hodge decomposition in for all p > 4 and that the Dirichlet problem for the Laplace equation cannot be in general solved with the boundary data in for all p > 4.
A counterexample to the regularity of the degenerate complex Monge-Ampère equation
We modify an example due to X.-J. Wang and obtain some counterexamples to the regularity of the degenerate complex Monge-Ampère equation on a ball in ℂⁿ and on the projective space ℙⁿ.
A deformation theorem in the noncompact nonsmooth setting and its applications.
A Dicontinuous Solution of a Mildly Nonlinear Elliptic System.
A difference method for a nonlinear system of elliptic equations with mixed derivatives
A difference scheme for an elliptic system of nonlinear differential-functional equations with Dirichlet type boundary conditions. The existence and uniqueness of solution
A diffused interface whose chemical potential lies in a Sobolev space
We study a singular perturbation problem arising in the scalar two-phase field model. Given a sequence of functions with a uniform bound on the surface energy, assume the Sobolev norms of the associated chemical potential fields are bounded uniformly, where and is the dimension of the domain. We show that the limit interface as tends to zero is an integral varifold with a sharp integrability condition on the mean curvature.